From the Equations of Motion to the Canonical Commutation Relations
aa r X i v : . [ qu a n t - ph ] M a y FROM THE EQUATIONS OF MOTION TOTHE CANONICAL COMMUTATIONRELATIONS
November 2, 2018
E.ErcolessiDipartimento di Fisica and INFN. Universita’ di Bologna,46 v. Irnerio, I-40126 Bologna. Italy. e-mail: [email protected]
G.MarmoDipartimento di Scienze Fisiche and INFN. Universita’ di Napoli ”Federico II”,v.Cinthia, I-80125 Napoli. Italy. e-mail: [email protected]
G.MorandiDipartimento di Fisica and INFN. Universita’ di Bologna,6/2 v.le B. Pichat, I-40127 Bologna. Italy. e-mail: [email protected]
Abstract
The problem of whether or not the equations of motion of a quantumsystem determine the commutation relations was posed by E.P.Wigner in1950. A similar problem (known as ”The Inverse Problem in the Calcu-lus of Variations” ) was posed in a classical setting as back as in 1887 byH.Helmoltz and has received great attention also in recent times. The aimof this paper is to discuss how these two apparently unrelated problemscan actually be discussed in a somewhat unified framework. After review-ing briefly the Inverse Problem and the existence of alternative structuresfor classical systems, we discuss the geometric structures that are intrin-sically present in Quantum Mechanics, starting from finite-level systemsand then moving to a more general setting by using the Weyl-Wigner ap-proach, showing how this approach can accomodate in an almost naturalway the existence of alternative structures in Quantum Mechanics as well.
Keywords: Classical and Quantum Alternative Structures; Wigner problem;Quantization; Geometric Quantum MechanicsPACS: 03.65.-w; 03.65.Ta; 45.20.Jjj 1 ndex
GN S construction 894.3.1 The
GN S construction . . . . . . . . . . . . . . . . . . . 894.3.2 Geometric structures over a C ∗ -algebra . . . . . . . . . . 924.4 Recovering a Hilbert Space out of R n . . . . . . . . . . . . . . . 944.5 Compatible Hermitian structures and Bihamiltonian vector fields 984.6 The infinite-dimensional case . . . . . . . . . . . . . . . . . . . . 104 KMS
Condition inPhase Space . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Nijenhuis torsions and Nijenhuis Tensors 170
Nijenhuis Torsions and Tensors on Smooth Manifolds . . . . . . . . . . 170Nijenhuis Torsions and Tensors on Associative Algebras . . . . . . . . 171A Digression on: Hochschild Cohomologies . . . . . . . . . . . . . . . 172Making Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
B Recursion Operators 174
Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 H− weak and ω -weak Recursion Operators. Strong Recursion Operators176Factorizable Recursion Operators . . . . . . . . . . . . . . . . . . . . . 178 C Symplectic Fourier Transform 181
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Equivariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1834
Introduction and Motivations
Back in 1950, E.P.Wigner [229] (see also Refs.[25, 150, 196]) raised the problemof whether the equations of motion determine or not the quantum commutationrelations. A few papers [199, 235] followed immediately, and the same prob-lem was considered by S.Schweber [211] in the framework of Quantum FieldTheory. It also originated the interest for parastatistics [79, 91, 92]. Physicistswere apparently motivated in this research by the search of a way out of theapparently uncontrollable divergences that were plaguing Relativistic QuantumField Theory.As reported by F.Dyson [63], also Feynman addressed the same problem,looking for commutation relations not associated with Lagrangian descriptions.One would have also avoided in this way [35] the introduction of gauge poten-tials. In the classical setting the problem, known as the ”Inverse Problem inthe Calculus of Variations” [186], was stated and clearly formulated already byH.Helmoltz [99]. An example of a system admitting of two alternative Hamil-tonian descriptions had already been given by J.L.Lagrange [118] when dealingwith linear problems.With the advent of Relativity. T.Levi-Civita [127] considered a similar prob-lem when looking for a Lagrangian description of massless particles in GeneralRelativity. P.Bergmann also noticed, in his famous book on Relativity [21],that, when the Lagrangian function is itself a constant of the motion, as it hap-pens, e.g., for geodesic motions in General Relativity, then any function of theLagrangian can be shown to provide, under very mild assumptions, a possiblealternative Lagrangian description of the same dynamical system.Other motivations for interest in the same problem arose from the so-called”no-interaction theorem” [10, 46, 162] concerning the covariant canonical de-scription of relativistic interacting particles [9]. Here too alternative Lagrangiandescriptions were sought that could allow to evade the theorem [47]. The so-called ”quadratic Hamiltonian theorem” [48] was also considered in the samespirit.A complete mathematical investigation of the inverse problem was initiatedby J.Douglas [58] (who was also one of the first Field medalists) back in 1941.Many investigators considered in particular the problem with reference to theN¨other theorem [1] connecting symmetries and constants of the motion [186].A first differential-geometric formulation of the problem appeared in themid-Seventies [151]. A few years later, R.M.Santilli [208] initiated a systematicpresentation of the problem for both particles and fields.The Inverse Problem arises quite naturally if one starts from the ”experi-mentalist’s” point of view [167] that the trajectories (think of the observationsin a bubble-chamber experiment) are the first raw data that are provided bythe direct observation of a dynamical evolution. It is therefore natural to start5rom the trajectories to build up a vector field and, afterwards, to look for La-grangian and/or Hamiltonian descriptions. A first attempt in this direction hadbeen made by E.K.Kasner [111] already in 1913.As the ”raw data” are usually given on some configuration space, the firstproblem one is faced with are the ambiguities that are present when trying to gofrom a second-order differential equation on a configuration space to a first-orderone (i.e. a vector field) on a larger carrier space. This problem was analyzed indetail in Ref.[167].To clearly identify and formulate the problem, it is very useful to considerlinear dynamical systems first, and to investigate the existence of Hamiltoniandescriptions from the point of view of Poisson brackets.In this context, writing the equations of motion in Hamiltonian form, i.e.: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dq i dtdp i dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) n × n n × n − n × n n × n (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂H∂q i ∂H∂p i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1.1)or, in collective coordinates: . ξ i = Λ ij ∂H∂ξ j = A i j ξ j (1.2)amounts to looking for a decomposition [83] of the matrix representing the (lin-ear) dynamics, say A , into the product of a skew-symmetric matrix Λ, whichstays for the Poisson tensor and defines the Poisson brackets and, if it is non-singular, the symplectic structure, and of a symmetric matrix H which repre-sents the Hamiltonian, i.e.: A = Λ · H (1.3)Out of all possible such decompositions we obtain all the alternative quadraticHamiltonian descriptions for a given dynamical system. It is easy to realize (seebelow, Chapt.3) that all symmetries for A , once applied to the factorization, willtake from one factorization to another one unless they correspond to canonicaltransformations.When going from a linear vector space to a generic differentiable manifold,matrices should be replaced by tensor fields and, when ”moving from a point”to a neighboring one, we will have to take into account also differential relations(partial differential equations will arise in addition to algebraic relations).One may trace the existence of alternative Lagrangian and/or Hamiltoniandescriptions to the existence of a large group of symmetries for the dynamics,some of them being non-canonical symmetries.The most obvious transformation taking one Lagrangian into another one isa scale transformation. For instance, we might scale the mass in a Lagrangiancontaining only a kinetic term, or we could do that, thanks to the equivalenceprinciple [21], for a massive particle moving in a gravitational field.When moving to the quantum descriptions, it becomes already clear that thescaling of the Lagrangian will give rise to a selection of the ”allowed” periodicmotion within a Bohr-Sommerfeld quantization scheme which will depend on6he scale. This is not surprising, as the Lagrangian times the period is measuredin units of Planck’s constant.This observation shows that we should not expect that the quantum descrip-tion of a dynamical evolution would trivially exhibit properties similar to theclassical ones.On the other hand, there is a strong belief that Classical Mechanics shouldbe a suitable limit of Quantum Mechanics. To quote from Dirac’s book [56]: “Classical mechanics must be a limiting case of quantum mechanics. Weshould thus expect to find that important concepts in classical mechanics corre-spond to important concepts in quantum mechanics and, from an understandingof the general nature of the analogy between classical and quantum mechan-ics, we may hope to get laws and theorems in quantum mechanics appearing assimple generalizations of well known results in classical mechanics.” This, along with the existence of alternative Hamiltonian descriptions forsolitonic equations [180], strongly suggests that a proper formulation of bi-Hamiltonian descriptions should exist for quantum dynamical systems as well.Here one can be more or less demanding. For instance, one may require thatknown situations of bi-Hamiltonian descriptions of specific classical dynamicalsystems be fully recovered in the quantum framework. As we shall see, theserequirements may have far-reaching consequences in the acceptable formulationsof Quantum Mechanics.For instance, one of the fundamental principles of Quantum Mechanics asformulated by Dirac [56] is the existence of a superposition rule for wave func-tions in order to deal with interference phenomena. This is usually translatedinto the requirement [56] that the carrier space should be a vector space.On the other hand, the approach in terms of C ∗ -algebras shows clearly thatthe Hilbert space we arrive at with the GN S construction [95] depends onthe initial state we choose, which is obviously ”prepared”, so-to-speak, ”in thelaboratory”.A spin-off of this construction is also the need for a clear distinction betweenthe ”abstract” C ∗ -algebra and its specific realizations in terms of operatorsacting on the Hilbert space that results from the GN S construction.Considering next more closely the Dirac prescription of replacing Poissonbrackets with commutator brackets, one finds that, while in the classical caseall possible Poisson brackets generate derivations for the pointwise product offunctions on the carrier space (i.e. the classical observables), in the quantumsetting another result by Dirac (see Chapt. IV of Ref.[56]) shows that theassociative product of operators identifies completely (up to a scale factor) theassociated Lie algebra structure (the commutator brackets). In some sense,therefore, the associative product and the Lie product strongly determine eachother in the quantum case.Many of these issues will be closely scrutinized in the present Report, whichhas been organized in the following way.The remainder of this Chapter and Chapt.2 serve to, so-to-speak, ”set thestage” for the analysis of the following Chapters, discussing, to begin with,how the Schr¨odinger equation can be recast in the form of a Hamiltonian7ystem, both in the finite and the infinite-dimensional case, and how alter-native Hamiltonian descriptions of the same quantum system can be generated.As bi-Hamiltonian systems are usually associated with complete integrability[50, 54, 134], Chapt.2 reviews some general problems concerning complete (Li-ouville) integrability and related invariant structures. In Chapt.3 we discuss theexistence of alternative structures at the classical level starting, as anticipatedin these introductory notes, with a discussion of the case of linear vector fields.Chapt.4 moves to the quantum setting. Also in order to set the problem withina framework similar to that of the classical case, and to take into account thefact that pure states in Quantum Mechanics are a manifold rather than a vec-tor space, we begin with a discussion of how geometric (tensorial) structuresthat are somehow hidden by the linear vector space structure of the Hilbertspace emerge nonetheless as fundamental structures. We emphasize there howthe proper carrier space for quantum dynamical system is instead the (no morelinear) complex projective space associated with the Hilbert space. We con-clude by discussing here too possible bi-Hamiltonian descriptions of quantumsystems and with a brief account of the extensions of the concepts developedalong the Chapter to the infinite-dimensional case. In Chapt.5 we discuss theWigner-Weyl approach to Quantum Mechanics, beginning with a review of theWeyl map, illustrated also with a good number of examples, we continue withthe Wigner map, the Moyal product, Quantum Mechanics in phase space andwe discuss also the quantum-classical transition. In the following Chapt.6 wediscuss how one can induce either on the same space or on spaces that are dif-feomorphically related alternative linear structures, i.e. linear structures on thesame carrier space that are however not linearly related. We discuss how alter-native linear structures can offer a way of ”reformulating”, in a sense explainedin the text, the von Neumann uniqueness theorem [222], as well as their rˆole inStatistical Mechanics. Chapt.7 contains some further generalizations and ourconcluding remarks.In order to make the paper more readable, some technical matters have beendiscussed in details in the Appendices, that expert readers can of course skipreading. We begin by considering the Schr¨odinger equation: ddt ψ ( t ) = − i ~ Hψ ( t ) ; ψ (0) = ψ (1.4)on a finite-dimensional (complex) Hilbert space H , deferring the discussion ofsome infinite-dimensional examples to the end of this Chapter. Hence, for thetime being: H ≈ C n for some n , As H is a vector space, there is a natural8dentification of the tangent space at any point ψ ∈ H with H itself: T ψ H ≈ H . In other words, vectors in a Hilbert space play a double rˆole , as ”points” in thespace and as tangent vectors at a given point. Which rˆole they play should be(hopefully) clear from the context. More generally, we have the identification: T H ≈ H × H , with T H the tangent bundle of H .As in the case of differentiable manifolds, ψ = ψ ( t ) , ψ (0) = ψ will definea curve in H , and hence the quantity ( dψ ( t ) /dt ) | t =0 will define the tangentvector at the curve at ψ ∈ H . A smooth assignment of tangent vectors at everypoint ψ ∈ H will define then a vector field , i.e. a smooth (and global) sectionof T H : Γ : H → T H ; ψ ( ψ, φ ) , ψ ∈ H , φ ∈ T ψ H ≈ H (1.5)where the second argument may depend in a smooth way on ψ and with thetangent bundle projection: π : ( ψ, φ ) ψ (1.6)such that: π ◦ Γ = Id H . We will employ the notation: Γ ( ψ ) for the vectorfield evaluated at the point ψ with tangent vector at ψ given by Eqn.(1.5). Thelatter defines a flow on H determined by the differential equation: ddt ψ ( t ) = φ ( ψ ( t )) , ψ (0) = ψ (1.7)Every vector field will define a derivation on the algebra of functions just asin the case of real manifolds. Specifically, if: φ = ( dψ ( t ) /dt ) | t =0 , ψ (0) = ψ and: f : H 7→ R is a function, then, in intrinsic terms:( L Γ ( f )) ( ψ ) = ddt f ( ψ ( t )) | t =0 (1.8)will define the Lie derivative along Γ on the algebra of functions.In local coordinates, choosing, e.g., an orthonormal ( O.N. from now on)basis { e i } ni ( n = dim H ), vectors (and tangent vectors) will be represented by n -tuples of complex numbers ( ψ = (cid:0) ψ , ..., ψ n (cid:1) , ψ j =: h e j | ψ i and so on), and :( L Γ ( f )) ( ψ ) = φ i ( ψ ) ∂f∂ψ i ( ψ ) (1.9)Notice that, in the infinite-dimensional case (for a separable and infinite-dimensional Hilbert space), ”functions” will become functionals , and ordinaryderivatives will have to be replaced by properly defined functional derivatives. Constant as well as linear (with respect to the linear structure identified bythe vector space) vector fields will play a role in what follows. The former arecharacterized by: φ = const. in the second argument of Eqn.(1.5), and give riseto the one-parameter group: R ∋ t ψ ( t ) = ψ + tφ (1.10) As in any linear vector space. As ψ j is complex: ψ j = q j + ip j , q j , p j ∈ R , the derivative here has to be understoodsimply as: ∂/∂ψ j = ∂/∂q j − i∂/∂p j . φ ( ψ ) being a linear and homogeneousfunction of ψ , i.e.: φ = Aψ for some linear operator A . Eqn.(1.7) integrates inthis case to : ψ ( t ) = exp { tA } ψ (1.11)Of particular interest is the dilation vector field ∆:∆ : ψ ( ψ, ψ ) (1.12)which corresponds to: A = Id H . In this case Eqns.(1.7) and (1.11) become: ddt ψ ( t ) = ψ ⇒ ψ ( t ) = e t ψ (1.13)Eqn.(1.12) exhibits clearly the fact that the dilation field leads to an identifi-cation of H with the fiber T ψ H . The latter carrying a natural linear structure,Eqn.(1.12) provides a tensorial characterization of the linear structure of thebase space H by means of the vector field ∆. For more details, see, e.g., Ref.[52].With every linear operator A there is therefore associated the linear vectorfield: X A : H → T H ; ψ → ( ψ, A ψ ) (1.14)In local coordinates, this vector field can be written as: X A =: A i j ψ j ∂∂ψ i (1.15)and is of course entirely defined by the representative matrix: A = (cid:13)(cid:13) A i j (cid:13)(cid:13) of thelinear operator. In particular, then:∆ = ψ i ∂∂ψ i (1.16)Notice however that, while linear operators form an associative algebra, vec-tor fields do not : they form instead only a Lie algebra. An associative algebracan be recovered by using the same matrix A to define instead the (1 ,
1) tensor : T A =: A i j dψ j ⊗ ∂∂ψ i (1.17)Then it is easy to check that the vector field X A is recovered from T A and thedilation field as: X A = T A (∆) (1.18) in the finite-dimensional case there are of course no problems in exponentiating a linearoperator. Not considering questions of domain, which are of no relevance in the finite-dimensionalcase. Notice that, while X A depends on the choice of the origin of the coordinates, T A does not,i.e. it has an affine character. H will definea linear vector field that we will denote for short as Γ H :Γ H : H → T H ; Γ H : ψ ( ψ, − ( i/ ~ ) Hψ ) (1.19)and then: L Γ H ψ ≡ ddt ψ = − i ~ Hψ (1.20)In this sense, the Schr¨odinger equation (1.4) can be viewed as a classical evolu-tion equation on a complex vector space.At variance with the infinite-dimensional case, every linear vector field iscomplete in finite dimensions. Then, if in addition we require conservation ofprobability, Wigner’s theorem [227] states that the associated one-parametergroup has to be unitary and, by Stone-von Neumann’s theorem [201], H hasto be essentially self-adjoint, i.e. it will be symmetric with a unique self-adjointextension. In the sequel we will refer always to the latter, and will simplysay that H is self-adjoint. In the finite-dimensional case no distinctions be-tween Hermitian, symmetric and self-adjoint operators [201] need to be made,of course.Let now: h : H × H → C (1.21)be a Hermitian structure on H , i.e. let: h ( φ, ψ ) =: h φ | ψ i (1.22)define an Hermitian scalar product on H with the usual properties, namely; • h ( φ, ψ ) = h ( ψ, φ ) • h ( φ, φ ) ≥ , h ( φ, φ ) = 0 ↔ φ = 0 • h ( λφ, ψ ) = λh ( φ, ψ ) , h ( φ, λψ ) = λh ( φ, ψ ) Remark 1 If h is viewed more properly as a (0 , tensor field, then φ and ψ in Eqn.(1.22) have to be viewed as tangent vectors at a point in H , and a morecomplete (albeit a bit more cumbersome) notation should be: h ( ϕ ) (Γ φ ( ϕ ) , Γ ψ ( ϕ )) = h φ | ψ i (1.23) We use here the notation Γ H instead of X H as a reminder of the fact that we had toinclude the ”extra” factor ( − i/ ~ ) in its definition. To be a bit more precise, pure states in Quantum Mechanics are described by elementsof the projective Hilbert space P H (for instance, one-dimensional projectors of the form: P ψ = | ψ ih ψ | / h ψ | ψ i , | ψ i ∈ H . The Hermitian structure on H induces a binary product: h ., . i on P H via: (cid:10) P ψ , P φ (cid:11) =: T r (cid:8) P ψ P φ (cid:9) = |h φ | ψ i| / ( h φ | φ i h ψ | ψ i ) and yields a transitionprobability. Wigner’s theorem states then that any bijective map on P H preserving transitionprobabilities can be realized as a unitary or anti-unitary transformation on the original Hilbertspace. here h ( ϕ ) stands for h evaluated at point ϕ ∈ H . As the r.h.s. of this equationdoes not depend on ϕ , this implies : L Γ H h φ | ψ i ≡ L Γ H ( h ( φ, ψ )) = 0 and, usingEqn.(1.4): L Γ H ( h ( φ, ψ )) = ( L Γ H h ) ( φ, ψ ) + h ( L Γ H φ, ψ ) + h ( φ, L Γ H ψ ) == ( L Γ H h ) ( φ, ψ ) + i ~ {h Hφ | ψ i − h φ | Hψ i} (1.24) which implies in turn, as H is self-adjoint, that: L Γ H h = 0 (1.25) i.e. that the Hermitian structure be invariant under the (unitary) flow of Γ H (and viceversa), or, stated equivalently, that Γ H be a Killing vector field for theHermitian structure. If instead the Hermitian structure is not invariant, then H will fail to be self-adjoint w.r.t. the given Hermitian structure. Remark 2
A family of privileged (actually global) charts for H , all unitarilyrelated to each other, is provided by the choice of any O.N. basis { | k i} ni , h h | k i = δ hk . In any such basis: h ( φ, ψ ) =: h φ | ψ i = h ij φ i ψ j with: h ij = δ ij , and allthe above statements (in particular Eqn.(1.25)) are self-evident. However, thestatements of the previous Remark have a tensorial meaning. As such, they willremain true also under (possible) non-linear changes of coordinates. Remark 3
We can decompose the Hermitian structure into real and imaginaryparts as: h ( ., . ) = g ( ., . ) + iω ( ., . ) (1.26) where: g ( φ, ψ ) = 12 [ h φ | ψ i + h ψ | φ i ] (1.27) and: ω ( φ, ψ ) = 12 i [ h φ | ψ i − h ψ | φ i ] (1.28) According to Eqn.(1.23) we may consider h as an Hermitian tensor. It is clearthat both g and ω are (0 , tensors, and that g is symmetric, while ω is skew-symmetric, hence a two-form. Eqn.(1.25) implies then that both tensors are(separately) invariant under Γ H . Notice that: ω ( φ, iψ ) = g ( φ, ψ ) . Hence, non-degeneracy of h entails separately that of ω and of g . Remark 4
The non-degenerate two-form ω will be represented, in any one ofthe privileged charts, by a constant (and unitarily invariant) matrix. Hence itwill be closed: dω = 0 (1.29) But, again, we stress that an equation like Eqn.(1.29) has a tensorial meaning.Hence, ω will be a symplectic form, while g will be a ( non-degenerate andconstant in any privileged chart) metric tensor. H be a vector field of the form (1.19). Then, a little algebra showsthat: ( i Γ H ω ) ( ψ ) = ω (cid:18) − i ~ Hφ, ψ (cid:19) = 12 ~ [ h Hφ | ψ i + h ψ | Hφ i ] (1.30)On the other hand, if we define the quadratic function: f H ( φ ) = 12 ~ h φ | Hφ i (1.31)we can define its differential as the one-form: df H ( φ ) = 12 [ h . | Hφ i + h φ | H. i ] = 12 [ h . | Hφ i + h Hφ | . i ] (1.32)the last passage following from H being self-adjoint. Therefore: ( i Γ H ω ) ( ψ ) = df H ( φ ) ( ψ ) ∀ ψ , and hence: i Γ H ω = df H (1.33)i.e. Γ H is Hamiltonian w.r.t. the symplectic structure with the quadratic Hamil-tonian f H .As a further remark, we recall that H is endowed with a natural complexstructure J defined simply by : J : φ → iφ (1.34)Then: J = − I (the identity on H ) and: ω ( φ, Jψ ) = g ( φ, ψ ) (1.35)Therefore the complex structure J is compatible [160] with the pair ( g, ω ) andwe can reconstruct the Hermitian structure as: h ( φ, ψ ) = ω ( φ, Jψ ) + iω ( φ, ψ ) (1.36)or equivalently, as: h ( φ, ψ ) = g ( φ, ψ ) − ig ( φ, Jψ ) (1.37)Notice also that: ω ( Jφ, Jψ ) = ω ( φ, ψ ) (1.38)as well as: g ( Jφ, Jψ ) = g ( φ, ψ ) (1.39)We can summarize what has been proved up to now by saying that H isa K¨ahler manifold[40, 41, 224], and that h is the associated Hermitian metric,while g is the Riemannian metric and ω the fundamental two-form. As ω isclosed, g is also [224] a K¨ahler metric.Choosing an O.N. basis {| k i} n , h h | k i = δ hk , the Hermitian product can bewritten as: h ( φ, ψ ) = δ ij φ i ψ j (1.40) Of course the best choice would be a basis in which the Hamiltonian is diagonal. | φ i = φ k | k i , and similarly for ψ .Writing: φ = φ + iφ , φ , ∈ R n , we can realify [4, 82] C n to R n via: C n ∋ φ → (cid:12)(cid:12)(cid:12)(cid:12) φ φ (cid:12)(cid:12)(cid:12)(cid:12) (1.41)In this way: g ( φ, ψ ) = Re n δ ij φ i ψ j o = (cid:12)(cid:12) φ φ (cid:12)(cid:12) G (cid:12)(cid:12)(cid:12)(cid:12) ψ ψ (cid:12)(cid:12)(cid:12)(cid:12) (1.42)where G is the matrix: G = I n ≡ (cid:12)(cid:12)(cid:12)(cid:12) I n n n I n (cid:12)(cid:12)(cid:12)(cid:12) (1.43)the I ’s being the identity matrices. Quite similarly, we find that ω has therepresentative matrix Ω given by:Ω = (cid:12)(cid:12)(cid:12)(cid:12) n I n − I n n (cid:12)(cid:12)(cid:12)(cid:12) (1.44)in R n , and J is represented by the matrix: J = (cid:12)(cid:12)(cid:12)(cid:12) n − I n I n n (cid:12)(cid:12)(cid:12)(cid:12) = − Ω = Ω − (1.45)consistently with Eqn.(1.35) which implies,in terms of the representative matri-ces: J = Ω − G (1.46)Notice, however, that while G and Ω are representatives of (0 ,
2) tensors, J isthe representative of a (1 , tensor. Explicitly, denoting with (cid:13)(cid:13) Ω ij (cid:13)(cid:13) the inverseof Ω (i.e. a (2 ,
0) tensor): Ω ij Ω jk = δ i k (1.47)then: J i j = Ω ik G kj (1.48)Let us turn now to the Schr¨odinger equation (1.4). Written in components,it reads : ddt ψ h = − i ~ h h | H | k i ψ k (1.49)Writing then, as before, ψ = ψ + iψ , ψ , ∈ R n and introducing the realcolumn vector: (cid:12)(cid:12)(cid:12)(cid:12) ψ ψ (cid:12)(cid:12)(cid:12)(cid:12) ∈ R n (1.50) It is clear that the matrix elements of the Hamiltonian have to be viewed as those of a(1 ,
1) tensor.
14e find (separating real and imaginary parts) the equation: ddt (cid:12)(cid:12)(cid:12)(cid:12) ψ ψ (cid:12)(cid:12)(cid:12)(cid:12) = A (cid:12)(cid:12)(cid:12)(cid:12) ψ ψ (cid:12)(cid:12)(cid:12)(cid:12) (1.51)where A is the skew-symmetric matrix: A =: 1 ~ (cid:12)(cid:12)(cid:12)(cid:12) Im H Re H − Re H Im H (cid:12)(cid:12)(cid:12)(cid:12) (1.52)and Im H and Re H are the n × n matrices:(Im H ) h k = Im h h | H | k i , (Re H ) h k = Re h h | H | k i (1.53)Just as before, Im H will be skew-symmetric and Re H symmetric. Remark 5
If we write the representative matrix of the Hamiltonian as: H =Re H + i Im H , then the ”realified” version of it is [4] the symmetric matrix: R H = (cid:12)(cid:12)(cid:12)(cid:12) Re H − Im H Im H Re H (cid:12)(cid:12)(cid:12)(cid:12) (1.54) Then it is easy to check that: A = − J ◦ ( R H/ ~ ) (1.55) This completes the identification of the Schr¨odinger equation as a real dynamicalsystem on a real space of dimension n . Taking a further time derivative, we obtain: d dt (cid:12)(cid:12)(cid:12)(cid:12) ψ ψ (cid:12)(cid:12)(cid:12)(cid:12) = A (cid:12)(cid:12)(cid:12)(cid:12) ψ ψ (cid:12)(cid:12)(cid:12)(cid:12) (1.56)and a simple calculation shows that: A = − (cid:18) R H ~ (cid:19) (1.57)Actually this result follows simply from the fact that the complex structureand the realified form of H commute, i.e.: J ◦ R H = R H ◦ J (1.58)and from: J = − I . 15s already remarked, things simplify if the basis in C n is chosen as the basisof the eigenvectors of H itself: H | k i = E k | k i . Then it is immediate to see that: A = 1 ~ (cid:12)(cid:12)(cid:12)(cid:12) H − H (cid:12)(cid:12)(cid:12)(cid:12) (1.59)where H is now the diagonal n × n matrix: H = diag { E , ..., E n } (1.60)Then we obtain the equations of motion: ddt ψ = Hψ , ddt ψ = − Hψ (1.61)or: d dt ψ i + (cid:18) H ~ (cid:19) ψ i = 0 , i = 1 , d dt ψ ki + (cid:18) E k ~ (cid:19) ψ ki = 0 , k = 1 , ..., n, i = 1 , ψ and ψ behaves as a simple harmonic oscillator with frequency ν k = E k / ~ . Let now K be a (strictly) positive linear operator on H , and consider thebilinear (sesquilinear) functional: h φ | Kψ i ≡ h ( φ, Kψ ) , φ, ψ ∈ T H (1.64)It is immediate to check that this functional enjoys all the three propertieslisted after Eqn.(1.22). Hence it defines a new Hermitian structure that we willdenote as h K ( ., . ) or as: h . | . i K : h ( φ, Kψ ) =: h K ( φ, ψ ) =: h φ | ψ i K (1.65)It is easy to show now that, as a consequence of the Hermiticity of H : L Γ H ( h K ( φ, ψ )) = i ~ h ( φ, [ H, K ] ψ ) (1.66)Invariance of the new Hermitian structure w.r.t. the dynamics requires thenthat K be a ”constant of the motion” for H :[ H, K ] = 0 (1.67)16 K will now be given explicitly as: h K ( φ, ψ ) = ( h K ) ij φ i ψ j , ( h K ) ij = h i | K | j i = S ij + iA ij , with S, A n × n real matrices. Hermiticity implies then: e S = S and e A = − A , i.e. that S be symmetric and A skew-symmetric. Proceeding asbefore, it is not difficult to see that the new metric tensor, symplectic form andcomplex structure g k , ω k and J K would be represented in the previous basis bythe matrices: G K = (cid:12)(cid:12)(cid:12)(cid:12) S A − A S (cid:12)(cid:12)(cid:12)(cid:12) , Ω K = (cid:12)(cid:12)(cid:12)(cid:12) A S − S A (cid:12)(cid:12)(cid:12)(cid:12) (1.68)with J K being given again by Eqn.(1.46).The above results have been derived by considering ”time” (i.e. Hamilto-nian) evolution of vectors in the Hilbert space, i.e. in the framework of theSchr¨odinger picture.It is not hard to show that similar results can be achieved in the context ofthe Heisenberg picture. Indeed, the new scalar product (1.65) induces a newassociative product among linear operators, namely : A, B → A · ( K ) B =: AKB (1.69)and a new commutator:[
A, B ] ( K ) =: A · ( K ) B − B · ( K ) A = AKB − BKA (1.70)that will fulfill the Jacobi identity in view of the associativity of the product(1.69).Now, if we want to represent the same dynamics in terms of the new com-mutator bracket, we will have to define a new Hamiltonian H ′ such that: i ~ dAdt = [ H ′ , A ] ( K ) = [ H, A ] (1.71)As A is generic,this requires: H ′ K = KH ′ = H , and hence: H ′ = HK − (1.72)as well as: [ H, K ] = 0 (1.73)as before. Notice that this will ensure that ”time” evolution will be a derivationon the new product algebra, i.e. that: ddt (cid:18) A · ( K ) B (cid:19) = dAdt · ( K ) B + A · ( K ) dBdt (1.74)for all A, B .Let us summarize at this point what we have found starting from theSchr¨odinger equation (1.4): See also Ref. [206] for the Abelian case. Eqn.(1.4) defines a real, linear Hamiltonian vector field on the realificationof the complex (and finite-dimensional, for the time being) Hilbert space H . • On this space, Eqn.(1.4) defines a Killing vector field for the Euclideanmetric tensor associated with the real part of the Hermitian scalar product. • Eqn.(1.4) decomposes into n non-interacting harmonic oscillators withproper frequencies E k / ~ and is therefore [54, 134] (see also next Chap-ter) a completely integrable system. Finally: • Eqn.(1.4) preserves alternative Hermitian structures associated with pos-itive linear operators K which commute with H . Therefore, Γ H is alsoKilling for the new metric tensor and Hamiltonian for the new symplecticstructure. We turn now to the infinite-dimensional case, concentrating on a quantum sys-tem described, in the Schr¨odinger picture, on the Hilbert space L (cid:0) R d , C (cid:1) , d ≥
1, of complex, square-integrable functions. Defining real variables q and p via: L (cid:0) R d , C (cid:1) ∋ ψ ( r , t ) =: q ( r , t ) + ip ( r , t ) , r ∈ R d (1.75) q and p will be functions in L (cid:0) R d , R (cid:1) .With a Schr¨odinger operator of the form: H = − ~ m ∇ + U ( r ) (1.76)(with U ( r ) a potential), the (time-dependent) Schr¨odinger equation will be: i ~ dψdt = H ψ (1.77)In a natural way, we will have to deal here with (real) functionals instead offunctions. We will consider functionals such that the functional differential δF of any one of them, F = F [ q, p ] ( R d r ... =: R d d r... ): δF = Z d r (cid:26) δFδq ( r ) δq ( r ) + δFδp ( r ) δp ( r ) (cid:27) (1.78)is well defined, and this will require both the ”differentials” (i.e. the variations) δq and δp and the functional derivatives δF/δq and δF/δp to be (real) square-integrable functions. With respect to the Lebesgue measure. One can also identify [180] L ( R d , C ) with the cotangent bundle of L ( R d , R ). H [ q, p ] as: H [ q, p ] = 12 Z d r (cid:26) ~ m h ( ∇ q ) + ( ∇ p ) i + U ( r ) (cid:0) q + p (cid:1)(cid:27) (1.79)or (integrating by parts): H [ q, p ] = 12 {h q, H q i + h p, H p i} (1.80)with h ., . i denoting the (real) scalar product in L (cid:0) R d , R (cid:1) , we have, takingfunctional derivatives: δH δq ( r ) = H q ( r ) , δH δp ( r ) = H p ( r ) (1.81)and the Schr¨odinger equation (1.77) can be rewritten as the (infinite-dimensional)Hamiltonian system: ddt (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) = 1 ~ J (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δH δpδH δq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1.82)where: J = (cid:12)(cid:12)(cid:12)(cid:12) −
11 0 (cid:12)(cid:12)(cid:12)(cid:12) (1.83)As: J (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) − qp (cid:12)(cid:12)(cid:12)(cid:12) (1.84)the tensor J is the realified [4] version of the standard complex structure J on L (cid:0) R d , C (cid:1) defined by: J : ψ → iψ (1.85)Explicitly: J = Z d r (cid:18) δp ( r ) ⊗ δδq ( r ) − δq ( r ) ⊗ δδp ( r ) (cid:19) (1.86)The Schr¨odinger equation (1.82) can be rewritten as: ddt (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) { p, H } { q, H } (cid:12)(cid:12)(cid:12)(cid:12) (1.87)where the Poisson bracket { ., . } and the associated Poisson tensor Λ ( ., . ) aredefined, for any two functionals F [ q, p ] and G [ q, p ], as:Λ ( δF, δG ) =: { F, G } = 1 ~ Z d r (cid:26) δFδq ( r ) δGδp ( r ) − δFδp ( r ) δGδq ( r ) (cid:27) (1.88)or: { F, G } = 1 ~ Z d r (cid:26)(cid:12)(cid:12) δF/δp δF/δq (cid:12)(cid:12) J (cid:12)(cid:12)(cid:12)(cid:12) δG/δpδG/δq (cid:12)(cid:12)(cid:12)(cid:12)(cid:27) (1.89)19he corresponding symplectic structure ω is given by: ω = ~ Z d r ( δq ∧ δp ) (1.90)or: ω = ~ Z d r (cid:12)(cid:12) δp δq (cid:12)(cid:12) ⊗ J (cid:12)(cid:12)(cid:12)(cid:12) δpδq (cid:12)(cid:12)(cid:12)(cid:12) (1.91)and the composition of the symplectic and the complex structures gives rise[160, 180] to the metric tensor: g =: J ◦ ω = ~ Z d r ( δp ( r ) ⊗ δp ( r ) + δq ( r ) ⊗ δq ( r )) (1.92)Given any functional F = F [ q, p ], the Hamiltonian vector field X F associ-ated with F via: i X F ω = δF (1.93)is easily seen to be: X F = 1 ~ Z d r (cid:26) δFδp ( r ) δδq ( r ) − δFδq ( r ) δδp ( r ) (cid:27) (1.94)In particular: X H = 1 ~ Z d r (cid:26) H p ( r ) δδq ( r ) − H q ( r ) δδp ( r ) (cid:27) (1.95)The Poisson bracket (1.89) can then be written also as: { F, G } = ω ( X G , X F ) (1.96) Digression.
Things acquire a more familiar (and manageable) form if we introduce a(real) complete orthonormal set of functions : { ψ n ( r ) } ∞ ; h ψ n , ψ m i = δ nm ; X n Z d r ψ n ( r ) ψ n ( r ′ ) = δ ( r − r ′ ) (1.97)in L (cid:0) R d , R (cid:1) . Then, defining: δq ( r ) = X n ψ n ( r ) dq n , dq n =: h ψ n , δq i (1.98)and similarly for δp , the functional differential (1.78) becomes: δF = X n (cid:26) ∂F∂q n dq n + ∂F∂p n dp n (cid:27) (1.99) A non-degenerate two-form which is closed, being constant in the (global) ( q, p ) chart. They could be, e.g., the eigenfunctions of a d -dimensional isotropic harmonic oscillator. ∂F∂q n =: (cid:28) ψ n , δFδq (cid:29) (1.100)(and similarly for ∂F/∂p n ) . In other words: δF = (cid:28) δFδq , δq (cid:29) + (cid:28) δFδp , δp (cid:29) (1.101)Proceeding in a similar way, it is easy to check that the Poisson tensor(3.196), the symplectic form (1.90) and the Hamiltonian vector field (1.94) as-sociated with F can be written in this basis as:Λ = 1 ~ X n ∂∂p n ∧ ∂∂q n (1.102) ω = ~ X n dq n ∧ dp n (1.103)and: X F = 1 ~ X n (cid:26) ∂F∂p n ∂∂q n − ∂F∂q n ∂∂p n (cid:27) (1.104) Let’s assume now the Schr¨odinger operator (1.76) to be positive or, moregenerally, invertible, and let, for simplicity, the ψ n ’s be the associated eigen-functions: H ψ n = E n ψ n , E n > ∀ n (1.105)Then, defining [180] a new Poisson tensor and Poisson bracket as:Λ ( δF, δG ) =: { F, G } = 1 ~ Z d r (cid:26) δFδq ( r ) H δGδp ( r ) − δFδp ( r ) H δGδq ( r ) (cid:27) (1.106)the same Schr¨odinger equation can be written also as: ddt (cid:12)(cid:12)(cid:12)(cid:12) pq (cid:12)(cid:12)(cid:12)(cid:12) = 1 ~ (cid:12)(cid:12)(cid:12)(cid:12) −HH (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δH δpδH δq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1.107)or: ddt (cid:12)(cid:12)(cid:12)(cid:12) p ( r ) q ( r ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) { p ( r ) , H } { q ( r ) , H } (cid:12)(cid:12)(cid:12)(cid:12) (1.108)where: H [ q, p ] = 12 Z d r (cid:0) q + p (cid:1) (1.109) Note that, under the stated assumptions, the series on the r.h.s. of Eqn.(1.99) will beconvergent. It could be, e.g., the Schr¨odinger operator for the isotropic harmonic oscillator: H = − ℏ m ∇ + U ( r ) with: U ( r ) = mω r /
21s a sort of ”universal” Hamiltonian functional.In the basis of the eigenfunctions of H the Poisson bracket (1.106) can bewritten as: { F, G } = 1 ~ X n E n (cid:26) ∂F∂q n ∂G∂p n − ∂F∂p n ∂G∂q n (cid:27) (1.110)and the associated symplectic form will be given by: ω = ~ X n E − n dq n ∧ dp n (1.111)or, in a basis-free notation: ω = ~ Z d r (cid:0) H − δq ∧ δp (cid:1) (1.112)Moreover, the Hamiltonian vector field associated, via ω now, with the func-tional F = F [ q, p ] is given by: X F = 1 ~ X n ǫ n (cid:26) ∂F∂p n ∂∂q n − ∂F∂q n ∂∂p n (cid:27) (1.113)or, in basis-independent form: X F = 1 ~ Z d r (cid:26) H δFδp ( r ) δδq ( r ) − H δFδq ( r ) δδp ( r ) (cid:27) (1.114)In particular: X H = 1 ~ Z d r (cid:26) H p ( r ) δδq ( r ) − H q ( r ) δδp ( r ) (cid:27) (1.115)which coincides with the Hamiltonian vector field (1.95). Remark 6
One could have also rewritten ω as: ω = ~ Z d r (cid:0) δq ∧ H − δp (cid:1) (1.116) but the two forms of course coincide, in view of the fact that H is self-adjoint. What has been proved up to here is that the same vector field, namely:Γ = 1 ~ Z d r (cid:26) H p ( r ) δδq ( r ) − H q ( r ) δδp ( r ) (cid:27) (1.117)is Hamiltonian w.r.t. two different Poisson brackets ( { ., . } and { ., . } ) andHamiltonian functionals ( H and H ), i.e. that it is bi-Hamiltonian . It turns I.e.: Γ = {H , . } = {H , . } . H n [ q, p ] = 12 {h q, H n q i + h p, H n p i} , n ≥ H [ q, p ] = H [ q, p ] (1.118)with associated symplectic forms: ω n = ~ Z d r (cid:0) H n − δq ∧ δp (cid:1) (1.119)and Poisson tensors:Λ n ( δF, δG ) = { F, G } n = 1 ~ Z d r (cid:26) δFδq ( r ) H − n δGδp ( r ) − δFδp ( r ) H − n δGδq ( r ) (cid:27) (1.120)such that: i Γ ω n = δH n ∀ n (1.121)where Γ is the vector field (1.117) and that the Hamiltonian functionals H n arepairwise in involution w.r.t. all the Poisson brackets, i.e.: { H n , H m } k = 0 ∀ n, m, k (1.122)In other words, the Schr¨odinger equation admits of infinitely many constantsof the motion pairwise in involution, which is another hallmark [54, 134] ofcomplete integrability. Having established this, as well as the fact that theSchr¨odinger equation admits of infinitely many Hamiltonian descriptions, andthat it can be considered as an infinite-dimensional Hamiltonian system on someinfinite-dimensional space, it will be appropriate to devote the next Chapter tothe study of completely-integrable dynamical systems and of their alternativeHamiltonian descriptions. 23 Completely Integrable Systems and Bi-HamiltonianDescriptions
In order to avoid reducing the generality of our treatment, and for future ref-erence, when the carrier space of a quantum system may be a manifold (likethe complex projective Hilbert space (see below Sect.4.2.2)) instead of a vectorspace, we will work here in the framework of symplectic manifolds and Hamil-tonian systems. So, let ( M , ω ) be a symplectic manifold (dim M =2 n for some n and ω a symplectic form). A dynamical system, i.e. a vector field Γ ∈ T M is ω -Hamiltonian or, for short, Hamiltonian iff: i Γ ω = d H (2.1)for some H ∈ F ( M ). A Hamiltonian dynamical system is said to be completelyintegrable if it has n constants of the motion f , ..., f n that are: i ) functionally independent: df ∧ ... ∧ df n = 0 (2.2)and: ii ) pairwise in involution, i.e.: { f i , f j } = 0 ∀ i, j (2.3)where { ., . } is the Poisson bracket associated with the symplectic form ω . The Arnold-Liouville theorem [4] states then that the level sets: M c = f − ( c ) , c ∈ R n , dim M c = n (2.4)provide a foliation of M whose leaves are invariant manifolds for the Hamilto-nian flow (2.1). Moreover, if the leaves of the foliation (2.4) are compact andconnected, then they are diffeomorphic to n -dimensional tori, i.e.: M c ≈ T n = S × ... × S | {z } n times = (cid:8) φ ≡ (cid:0) φ , ..., φ n (cid:1) mod 2 π (cid:9) (2.5)and one can find a set of frequencies: ν ≡ ( ν , ..., ν n ), ν = ν ( f ) such that theHamiltonian flow on the torus is given by : dφ i dt = ν i ⇒ φ i ( t ) = φ i (0) + ν i t (2.6)and Hamilton’s equations of motion are integrable by quadratures. Such motions are called quasi-periodic or conditionally periodic . action-angle variables.Calling X i the Hamiltonian vector field associated with f i , i = 1 , ..., n ,Eqn.(2.3) leads at once to: { f i , f j } ≡ L X j f i ≡ ω ( X j , X i ) = 0 (2.7)Moreover, as: i [ X,Y ] = L X · i Y − i Y · L X (2.8)we obtain : i [ X i ,X j ] ω = L X · ( i X j ω ) − i X j · ( L X i ω ) ≡ d ( L X i f j ) = 0 (2.9)the final result following from Eqn.(2.7). Therefore, the X i ’s commute pairwise.Moreover, it follows again from Eqn.(2.7) that the invariant leaves (2.4) of thefoliation are Lagrangian submanifolds. Defining the immersion: i c : M c ֒ → M ,we have therefore: i ∗ c ω = 0 (2.10)Therefore, if we denote by θ the Cartan one-form ( ω = − dθ ), its pull-back i ∗ c θ will be closed : di ∗ c θ = i ∗ c dθ = 0 (2.11)It need not be exact, though, as the invariant tori are not contractible. Cycleson the torus need not be boundaries, and therefore the integral of i ∗ c θ alonga one-dimensional cycle need not vanish. We can select a basis ( γ , ..., γ n ) ofloops, i.e. n one-dimensional cycles each one of which winds around the torusexactly once and none of which is homologous [2] to any other one (nor to thetrivial loop), and define the action variables I i as: I i = 12 π I γ i i ∗ c θ, i = 1 , ..., n (2.12)Of course: I i = I i ( f ) depends only on the homology class [2] of γ i and, providedthe jacobian of the transformation does not vanish or, equivalently: dI ∧ dI ∧ ... ∧ dI n = 0 (2.13)invariant tori can be uniquely labelled by the set I = ( I , ..., I n ) of the values ofthe action variables. Defining then: S = S ( I, q ) = q Z q i ∗ c θ (2.14)the integral being along a path γ on the invariant torus labelled by I joininga fiducial point q to the point q , S will depend only on the homology class of As X j is Hamiltonian, L X j ω = 0. q to q to which γ belongs . Switching to a different homology classmultiplying γ by, say, a loop γ i in the basis will change S by a fixed amount: S → S + ∆ S i ; ∆ S i = 2 πI i (2.15)We can then use S as the generator of a time-independent canonical trans-formation: ( q, p ) → ( φ, I ) (2.16)with the I ’s playing the rˆole of the new momenta, via : p i = ∂S∂q i , φ i = ∂S∂I i (2.17)and with the new Hamiltonian: K = H . Now, as n is the maximum allowednumber of independent constants of the motion pairwise in involution , eitherthe Hamiltonian is one of the f i ’s or is a function thereof: H = H ( f ) andtherefore it is ultimately a function of the action variables alone. Hamilton’sequations become then: ddt I i = 0 , ddt φ i = ν i ; ν i =: ∂ H ∂I i = ν i ( I ) (2.18)and we recover Eqn.(2.6). In the new coordinates the dynamical vector fieldwill be given by: Γ = n X i =1 ν i ∂∂φ i (2.19)and the symplectic structure will be: ω = n X i =1 dφ i ∧ dI i (2.20)We notice that in these coordinates the dynamics is nilpotent of index two,i.e.: dφ i dt = ν i ; dν i dt = 0 (2.21)Moreover, in these coordinates the system is linear and associated with anilpotent matrix. It should be remarked that the transformation (2.16) is notlinear. Therefore, even if the system is linear in the ( q, p ) coordinates, thetransformation need not be isospectral, i.e. it may take us from a semisimplematrix to a nilpotent one. This approach goes back to a paper [64] by A.Einstein of 1917. The ambiguity expressed by Eqn.(2.15) tells us that the φ ’s are actually defined ”modulo”2 π , i.e. that they are indeed angles. If ω is non-degenerate, n is the maximum allowed dimension for an isotropic subspace. .2 From Invariant Structures to Integrability In the case of Eqn.(2.18), if we are in the so-called non-resonant case, i.e. if: dν ∧ dν ∧ ... ∧ dν n = 0 (2.22)we can choose the ν i ’s as new momenta (the transformation will be in general not canonical, however!). In the new coordinates the dynamical system will becompletely separated into n independent systems, while the Hamiltonian andsymplectic structure will become respectively: H = 12 n X i =1 ν i (2.23)and: ω = n X i =1 dφ i ∧ dν i (2.24) Separability of a dynamical system into a family of non-interacting subsystemsappears therefore to be intimately connected with integrability . It is also well-known that a way to achieve (if possible) integrability via separability occursin the Hamilton-Jacobi theory [16, 17, 18, 107, 158], a subject that we will notdiscuss here, though. Notice also that, in general, the two notions of separabilitydo not in general coincide.In this Subsection we will discuss a way to achieve separability (and even-tually integrability) with the aid of additional invariant structures [53, 54]. Wewill not make reference, for the time being, to symplectic structures and thelike. What we are going to say generalizes to vector fields, and hence also tonon-linear situations, the familiar block-diagonal form of matrices.Let then M be a smooth manifold and let Γ ∈ X ( M ) be a vector field. Γwill be said to be separable into dynamics of lower dimension on an open set U ⊆ M if a holonomic frame (cid:8) e ( i,k ) (cid:9) can be found for the tangent bundle T U ,with dual forms (cid:8) θ ( i,k ) (cid:9) , such that: L e ( i,k ) D θ ( j,h ) | Γ E = 0 ⇔ i = j (2.25)This implies of course, in local coordinates, that we can choose coordinates x ( i,k ) ( e ( i,k ) = ∂/∂x ( i,k ) ) in such a way that:Γ = Γ ( i,k ) ∂∂x ( i,k ) (2.26)and: Γ ( i,k ) = Γ ( i,k ) (cid:0) x i (cid:1) ; x i =: (cid:16) x ( i, , x ( i, , ..., x ( i,k ) , ... (cid:17) (2.27) See also Refs.[72, 73] for a similar discussion in the Lagrangian context. separable if we can choose U = M or, at least, U to be an open dense set in M .Let us review briefly how one can achieve separation of the dynamics in thepresence of an invariant diagonalizable (1 , tensor field T ∈ F ( M ) with atleast two distinct eigenvalues and vanishing Nijenhuis torsion .Recall that, given a (1 ,
1) tensor T , the Nijenhuis torsion [78, 152, 194]associated with T is the (0 ,
2) tensor N T defined by: N T ( α, X, Y ) =: h α |H T ( X, Y ) i ; α ∈ X ∗ ( M ) , X, Y ∈ X ( M ) (2.28)where: X ( M ) ∋ H T ( X, Y ) =: [
T X, T Y ] + T [ X, Y ] − T [ T X, Y ] − T [ X, T Y ] (2.29)Let’s remark that, if T is diagonalizable: T e i = λ i e i (2.30)the eigenvectors e i are (locally at least) a basis of vector fields , and we willdenote as S λ i the eigenspace of the eigenvalue λ i . The e i ’ being a basis implies:[ e i , e j ] = X k c kij e k ; c kij = − c kji (2.31)for some set of ”structure constants” (actually in principle functions) c kij . Thedual cobasis (cid:8) θ i (cid:9) , defined as usual via: (cid:10) θ i | e j (cid:11) = δ ij (2.32)will be also a basis of eigenforms: e T θ i = λ i θ i (2.33)where e T denotes the transpose action of T on forms ( h θ | T X i =: D e
T θ | X E ).Using then the identity [41]: dθ ( X, Y ) = L X ( θ ( Y )) − L Y ( θ ( X )) − h θ | [ X, Y ] i (2.34)it is easy to prove that: dθ k ( e i , e j ) = − c kij (2.35)i.e. that: dθ k = − X ij c kij θ i ∧ θ j (2.36) More properties of Nijenhuis torsions and tensors are briefly reviewed in App. A . In fact, they are not only a vector space, but have in addition the structure of an F ( M )-module. H T ( e i , e j ) = ( T − λ i ) ( T − λ j ) [ e i , e j ] + ( λ i − λ j ) (cid:8) ( L e i λ j ) e j + (cid:0) L e j λ i (cid:1) e i (cid:9) (2.37)Let us remark first that:( T − λ i ) ( T − λ j ) [ e i , e j ] = X k ( λ k − λ i ) ( λ k − λ j ) c kij e k (2.38)has no components in S λ i ⊕ S λ j . If the Nijenhuis torsion vanishes , then thecondition H T ( e i , e j ) = 0 separates into:( T − λ i ) ( T − λ j ) [ e i , e j ] = 0 (2.39)and: ( λ i − λ j ) L e i λ j ≡ ( λ i − λ j ) dλ i ( e j ) = 0 (2.40)Contracting the first of the above equations with θ k we obtain:( λ k − λ i ) ( λ k − λ j ) (cid:10) θ k | [ e i , e j ] (cid:11) = 0 (2.41)which implies: (cid:10) θ k | [ e i , e j ] (cid:11) = 0 for λ k = λ i , λ j , i.e.:[ e i , e j ] ∈ S λ i ⊕ S λ j (2.42)and hence: c kij = 0 when λ k = λ i , λ j (2.43)At this point we can somehow sharpen the analysis and make it a bit moreprecise. If the eigenspaces are not one-dimensional (i.e. the eigenvalues of T have degeneracy), denoting by (cid:8) e ( i,r ) (cid:9) , r = 1 , , ..., d i , d i being the dimensionof the i -th eigenspace, a basis of eigenvectors in S λ i , it is not difficult to provethat Eqn.(2.41) generalizes to:( λ k − λ i ) ( λ k − λ j ) D θ ( k,r ) | (cid:2) e ( i,p ) , e ( j,q ) (cid:3)E = 0 ∀ r, p, q (2.44)which holds in particular for i = j , thus leading to the conclusion that: (cid:2) e ( i,p ) , e ( i,q ) (cid:3) ∈ S λ i (2.45)i.e. that if T is diagonalizable and has vanishing Nijenhuis torsion, the eigen-vectors belonging to every eigenspace are an involutive distribution . As such, thedistribution will be integrable by Frobenius’ theorem [167], and we can speak(locally at least) of eigenmanifolds .We can also reach the same conclusion in dual form as follows. Eqn.(2.43)implies that in Eqn.(2.36) at least one of the one-forms on the r.h.s. must be inthe (dual) eigenspace of the eigenvalue λ k . To be more specific, if we denote by i.e. T is (see App. A ) a Nijenhuis tensor ( k,r ) , r = 1 , , ... the eigenforms belonging to the eigenvalue λ k and by c ( k,r )( i,s )( j,p ) the ”structure constants”, Eqns.(2.36) and (2.43) imply: dθ ( k,r ) = − X ( i,p ) ,s c ( k,s )( i,p )( k,s ) θ ( i,p ) ∧ θ ( k,s ) (2.46)But this is equivalent to the statement that: dθ ( k,r ) ^ s θ ( k,s ) = 0 (2.47)which is again [54] a statement of Frobenius’ theorem.The main conclusion is then that, under the stated assumptions, one canalways find a holonomic frame (and coframe) that diagonalizes T in the form: T = X i λ i e i ⊗ θ i (2.48)Let us turn now to the consequences of the invariance of T under the dy-namics. First of all, an invariant (1 ,
1) tensor T will generate an algebra A ofvector fields all commuting with Γ given by: A = (cid:8) Γ , T Γ , T Γ , .., T k Γ , ... (cid:9) (2.49)If L Γ T = 0, it can be proved [54] that: (cid:2) T k Γ , T k + h Γ (cid:3) = X α + β + γ =2 k + h − α,β ≥ ,γ ≥ k T α H T (cid:0) T β Γ , T γ Γ (cid:1) (2.50)hence, if H T = 0, A will be an abelian algebra of vectors fields all commutingwith Γ, i.e. an abelian algebra of symmetries [167].Consider next the eigenvalue equation for T . Let e and θ be an eigenvectorand an eigenform belonging to the same eigenvalue λ : T e = λe, e T θ = λθ (2.51)We can assume, without loss of generality: h θ | e i = 1.If T is invariant under the dynamics, L Γ T = 0, then: T ( L Γ e ) = L Γ ( T e ) = L Γ ( λe ) = ( L Γ λ ) e + λ ( L Γ e ) (2.52)On the other hand: D L Γ e | e T θ E = h T ( L Γ e ) | θ i = L Γ λ + λ hL Γ e | θ i ≡ L Γ λ + D L Γ e | e T θ E (2.53)and hence: L Γ λ ≡ i Γ dλ = 0 (2.54)i.e., if T is invariant under the dynamics, so are the eigenvalues of T .30otice that, by Cartan’s identity [167]: L e i θ j = (cid:10) e i | dθ j (cid:11) + d (cid:10) e i | θ j (cid:11) (2.55)and hence, if the (co)basis is holonomic, dθ j = 0 (together with (cid:10) e i | θ j (cid:11) = δ ji )leads to: L e i θ j = 0 ∀ i, j (2.56)Then, for i = j we obtain:( λ i − λ j ) L e i (cid:10) Γ | θ j (cid:11) = λ i (cid:10) L e i Γ | θ j (cid:11) − D L e i Γ | e T θ j E = (2.57)= λ i (cid:10) L e i Γ | θ j (cid:11) − (cid:10) T ( L e i Γ) | θ j (cid:11) == λ i (cid:10) L e i Γ | θ j (cid:11) − ( L i Γ λ i ) (cid:10) e i | θ j (cid:11) − λ i (cid:10) L e i Γ | θ j (cid:11) = 0Hence: L e i (cid:10) Γ | θ j (cid:11) = 0 , i = j (2.58)and (cfr. Eqn.(2.25)) this proves separability of Γ. To be more explicit, we canwrite T as: T = n X i =1 λ i d i X k =1 ∂∂x ( i,k ) ⊗ dx ( i,k ) (2.59)where n is the number of distinct eigenvalues and d i is the degeneracy of the i -th eigenvalue. Finally, Γ will be of the form already given in Eqns.(2.26) and(2.27). On the eigenspaces of T that are one-dimensional integrability of Γ willbe then essentially trivial, and this case will not be considered further.Proceeding further we obtain from Eqn.(2.40):0 = ( λ i − λ j ) h e i | dλ j i = h T e i | dλ j i−h e i | λ j dλ j i = D e i | e T dλ j E −h e i | λ j dλ j i (2.60)and hence: e T dλ j = λ j dλ j (2.61)i.e. dλ j is an eigenform belonging to the eigenvalue λ j . Let us now assume theeigenvalues of Γ to be doubly degenerate and functionally independent. Thisimplies: dim ( M ) = 2 n and: dλ ∧ dλ ∧ ... ∧ dλ n = 0 (2.62)Then the dλ i ’s can be taken as half of the cobasis, and we can write T as: T = n X i =1 λ i (cid:0) e i ⊗ θ i + e n + i ⊗ dλ i (cid:1) (2.63)With this choice, Eqn.(2.54) tells us that Γ has no components ”along” the dλ i ’s, and that it is therefore of the form:Γ = n X i =1 Γ i e i (2.64)31roceeding further, closure of the θ i ’s allow us to write: θ i = dφ i , and hence: e i = ∂/∂φ i for i = 1 , ..., n . The φ i ’s are in general only locally defined (whilethe λ i ’s are globally defined), and can be allowed to be angles. Hence we canrewrite T as: T = n X i =1 λ i (cid:18) ∂∂φ i ⊗ dφ i + ∂∂λ i ⊗ dλ i (cid:19) (2.65)and, in view of Eqn.(2.64), Γ will be of the form: n X i =1 Γ i (cid:0) λ i , φ i (cid:1) ∂∂φ i (2.66)The associated equations of motion will be: ddt φ i = Γ iddt λ i = 0 ; i = 1 , .., n (2.67)Now, it is easy to show that the dynamical system (2.67) can be madeHamiltonian with respect to a large family of symplectic structures. Indeed,let’s assume that no one of the Γ i ’s vanishes identically . Then, with any setof (smooth) functions g i = g i (cid:0) λ i (cid:1) we can associate the symplectic form: ω = n X i =1 f i (cid:0) λ i , φ i (cid:1) dφ i ∧ dλ i (2.68)where: f i (cid:0) λ i , φ i (cid:1) =: g i (cid:0) λ i (cid:1) Γ i ( λ i , φ i ) (2.69)and Γ will be Hamiltonian: i Γ ω = d H (2.70)with: d H = n X i =1 g i (cid:0) λ i (cid:1) dλ i (2.71)Therefore, under the assumption that there exists a (1 ,
1) diagonalizable tensorfield T invariant under the dynamics, with vanishing Nijenhuis torsion andat most doubly degenerate and functionally independent eigenvalues, what hasbeen proved up to now is that the dynamical vector field Γ is separable, integrableand, on the eigenspaces of doubly degenerate eigenvalues, Hamiltonian. (cid:4) The equation L Γ T = 0 expresses the invariance of the tensor T in intrinsicterms. It may be useful to write down the same condition in the language of If they have isolated zeros, the closed set of the zeros, which is an invariant subset, canbe excluded from the manifold. The case in which some component of Γ vanishes has beendiscussed in Ref.[54]. (cid:0) x i , ..., x m (cid:1) , m = (dim ( M )) are local coordinates, and T and Γare given by: T = T i j dx j ⊗ ∂∂x i ; Γ = Γ i ∂∂x i (2.72)then: L Γ T = (cid:26) L Γ T i j − ∂ Γ i ∂x k T k j + T i k ∂ Γ k ∂x j (cid:27) dx j ⊗ ∂∂x i (2.73)and hence invariance under Γ implies the matrix equation: L Γ T =: ddt T = [ C, T ] (2.74)where, with abuse of notation, we have denoted by T the m × m matrix: T = (cid:13)(cid:13) T i j (cid:13)(cid:13) and: C = (cid:13)(cid:13) C i j (cid:13)(cid:13) ; C i j =: ∂ Γ i ∂x j (2.75)while [ ., . ] denotes the usual commutator among matrices. Whenever two ma-trices C and T satisfy Eqn.(2.74) they are said to form a Lax pair [122, 123,124, 152, 221] .Whenever we may define a map µ from M to a space of matrices suchthat the dynamics is µ -related to a dynamics on the matrix space of the formof Eq.(2.74), we say that the original dynamics can be given a ”Lax form”.This is what might be called also a ”Heisenberg” form, and has many generalproperties. For instance, the evolution of T ruled by the ”Hamiltonian” C isclearly isospectral.Whenever it is possible to find a map from our carrier space to a space oflinear operators such that the dynamics on the carrier space may be casted intothe Heisenberg form we will say that our dynamics may be put into the Laxform. As a matter of fact, by using the momentum map associated with thesymplectic action of the unitary group on the Hilbert space or on the complexprojective space (see below, Sect.4.2), we may relate the Schr¨odinger picturewith the Heisenberg picture on the space of observables. Reversing somehow our path, let’s start by considering a dynamical systemΓ that is Hamiltonian and completely integrable ”a’ la” Liouville. Hence:dim ( M ) = n . Introducing action-angle variables (cid:0) I , .., I n ; φ , .., φ n (cid:1) in theneighborhood of an Arnold- Liouville torus T n , we will have: dI ∧ dI ∧ ... ∧ dI n = 0 (2.76) We should notice that Eq.(2.74) depends on the coordinate system we are using, andtherefore has no intrinsic meaning. H be a function of the action variablesalone can be written as: d H ∧ dI ∧ ... ∧ dI n = 0 (2.77)The symplectic form can be written as:Ω = X k dφ k ∧ dI k (2.78)and the vector field Γ in action-angle variables will be given by:Γ = X k ω k ∂∂φ k ; ω k =: ∂ H ∂I k (2.79)Assume first that the Hamiltonian is separable: H = X k H k ( I k ) (2.80)Then the class of (1 ,
1) tensor fields defined by: T = X k λ k ( I k ) (cid:26) dI k ⊗ ∂∂I k + dφ k ⊗ ∂∂φ k (cid:27) (2.81)with the λ k ’s arbitrary functions with nowhere vanishing differential has all therequired properties. Indeed: • It is invariant under the dynamics; • It has doubly degenerate eigenvalues and: • It has vanishing Nijenhuis torsion.This last property can be checked directly by testing Eqn.(2.29) on: (
X, Y ) =( ∂/∂I h , ∂/∂I k ) , (cid:0) ∂/∂I h , ∂/∂φ k (cid:1) and (cid:0) ∂/∂φ h , ∂/∂φ k (cid:1) (cid:4) .A second case in which an invariant (1 ,
1) tensor can be constructed is the”non-resonant” case, i.e. when the Hamiltonian has a non-vanishing Hessian:det (cid:13)(cid:13)(cid:13)(cid:13) ∂ H ∂I h ∂I k (cid:13)(cid:13)(cid:13)(cid:13) = 0 (2.82)This means, of course: dω ∧ dω ∧ ... ∧ dω n = 0 (2.83)Solving then for the I ’s as functions of the ω ’s, we can use the ω ’s as newcoordinates and introduce a new symplectic structure: e Ω = X k dω k ∧ dφ k = X hk ∂ H ∂I h ∂I k dI h ∧ dφ k (2.84) This change of variables need not be a canonical transformation. H = 12 X k (cid:0) ω k (cid:1) (2.85)The class of (1 ,
1) tensor fields will be given now by: T = X k λ k (cid:0) ω k (cid:1) (cid:26) dω k ⊗ ∂∂ω k + dφ k ⊗ ∂∂φ k (cid:27) (2.86)Complete integrability is also known to be related to the existence of re-cursion operators [53, 120, 179, 239]. A brief account of the latter is given inAppendix B . 35 Alternative Structures for Classical Systems
After having examined briefly in the previous Chapter the problem of the inte-grability of a classical dynamical system, and before turning to the main topicof this review, i.e. quantum systems , we restate here in a very cursory waywhat is known in the literature as the ”Inverse Problem of Classical Dynamics”.Let then Γ be a vector field on a (smooth) manifold M . In a nutshell, theInverse Problem ( IP ) can be formulated in (at least ) three different, and oftenrelated, contexts, namely: • IP
1: Lagrangian context [98, 99, 186]. Let then M be the tangent bundleof a smooth manifold Q , i.e. M = T Q equipped with tangent bundlecoordinates (cid:0) q i , v i (cid:1) such that Γ ∈ X ( T Q ) is a second-order vector field[184], i.e.: Γ = v i ∂∂q i + F i ( q, v ) ∂∂v i (3.1)The Lagrangian IP amounts then to the following: find all the smoothfunctions L = L ( q, v ) ∈ F ( T Q ) such that: ∂ L ∂v i ∂v j F j = ∂ L ∂q i − ∂ L ∂v i ∂q j v j , i = 1 , ..., n = dim Q (3.2)It follows that if the Lagrangian L is regular, i.e.:det (cid:13)(cid:13)(cid:13)(cid:13) ∂ L ∂v i ∂v j (cid:13)(cid:13)(cid:13)(cid:13) = 0 (3.3)then the Euler-Lagrange equations can be put in normal form and, viaa Legendre transformation [4, 167] one can go over to a Hamiltonian de-scription of the dynamical system on the cotangent bundle T ∗ Q . We willnot discuss this setting of the IP any further, and refer for a full accountof it to the literature [184]. • IP
2: Hamiltonian context. Let instead M = T ∗ Q for some smooth man-ifold Q and Γ ∈ X ( T ∗ Q ). The Hamiltonian IP amounts then to findingall pairs ( ω, H ) with ω a symplectic form (a closed and non-degeneratetwo-form) and H ∈ F ( T ∗ Q ) such that: i Γ ω = d H (3.4)At a local level, the problem reduces to finding all the closed and non-degenerate two-forms ω such that: L Γ ω = 0 (3.5) What we mean exactly by a ”quantum” system will be specified in the next Chapter. We will not consider here the Hamilton-Jacobi form of Classical Dynamics, but see [158] L Γ denoting the Lie derivative w.r.t. Γ, which is a system of coupled P DE ′ s in (cid:18) n (cid:19) = 2 n − n unknowns . As a simple example, ina neighborhood U ⊆ M in which Γ = 0 and defines a flow-box (the”straightening-up-of-the-flux” theorem [4] holds) we can find coordinates( x , x , ..., x n − ) such that Γ = ∂/∂x and hence the problem has infinitesolutions: ω = dx ∧ df + a ij df i ∧ df j (3.6)with: a ij = − a ji ∈ R , det k a ij k 6 = 0 and: ∂f /∂x = ∂f i /∂x = 0 ,dx ∧ df ∧ df ∧ ... ∧ df n − = 0, and any such f will be an acceptableHamiltonian ( i Γ ω = df ). • IP
3: Poisson context [35, 60]. M is assumed here to be a Poisson man-ifold [167]. In local coordinates x i , i = 1 , ..., dim M , and the IP inthis context amounts to finding all pairs ( { ., } , H ) with { ., } a (possiblydegenerate ) Poisson bracket and H ∈ F{M} such that: (cid:8) x i , H (cid:9) = dx i dt ; (cid:8)(cid:8) x i , H (cid:9) , H (cid:9) = F i (cid:0) x, (cid:8) x i , H (cid:9)(cid:1) (3.7) In view of the fact that what we are interested in this paper is a theory that isusually casted in a linear setting, i.e. Quantum Mechanics on Hilbert spaces, wewill review here[83] the Inverse Problem in the Hamiltonian context for linearvector fields, and we will assume: M = R n for some n . In the appropriatecoordinates, a linear vector field is then a vector field of the form:Γ = G i j x j ∂∂x i , G i j ∈ R (3.8)and the matrix (cid:13)(cid:13) G i j (cid:13)(cid:13) (which represents a (1 , A Digression on: ”Extracting the linear part” of a vector field.
Ingeneral, let M be a smooth manifold and Γ ∈ X ( M ) be a vector field with anisolated fixed point at m ∈ M : Γ ( m ) = 0. Considering then, for an arbitraryvector field Y ∈ X ( M ) and function f ∈ F ( M ) the quantity L Y ( L Γ f ) ( m ),it is not hard to see that it is linear in Y and , by virtue of Γ ( m ) = 0, in df . itdefines then a (1 ,
1) tensor T Γ at m : L Y ( L Γ f ) ( m ) = T Γ ( df, Y ) ( m ) (3.9) Notice that, in this as well as in the previous case, M has obviously to be an even -dimensional manifold. Which will be certainly the case if M is odd -dimensional. Not a tensor field, in general. linear part of Γ at m , Γ , will be defined as:Γ = T Γ (∆) (3.10)with ∆ the Liouville field.Indeed, in the domain of a chart (cid:0) x , ..., x n (cid:1) ( n = dim M ) with the originat m and: Γ = Γ i ∂/∂x i , Y = Y i ∂/∂x i : L Y ( L Γ f ) = Y i ∂ (cid:0) Γ j ∂f /∂x j (cid:1) /∂x i = Y i (cid:0) ∂ Γ j /∂x i (cid:1) (cid:0) ∂f /∂x j (cid:1) + Y i Γ j (cid:0) ∂ f /∂x i ∂x j (cid:1) T. But the second term vanishesat m = 0, and hence: T Γ = T j i dx i ⊗ ∂∂x j ; T j i = ∂ Γ j ∂x i | m (3.11)and: Γ = T Γ (∆) = (cid:18) ∂ Γ j ∂x i | m (cid:19) x i ∂∂x j (3.12)This is of course what one would have guessed on much more elementarygrounds. The advantage of the definition (3.9) is that it provides a tensorialcharacterization of the linear part of a vector field at a critical point.In a shorthand notation we can write Γ as:Γ = (cid:16) f G x, ∂/∂x (cid:17) (3.13)where: ( G x ) i = G i j x j , ” ∼ ” stands for the transpose: ∂∂x = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂/∂x ...∂/∂x n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.14)and: ( a, b ) =: a i b i .A symplectic form can be written as: ω = 12 Ω ij dx i ∧ dx j (3.15)and the matrix: Ω= k Ω ij k will be (pointwise) skew-symmetric and non-degenerate. ω will be said to be a constant symplectic form iff the Ω ij ’s are constant. If:Ω = (cid:12)(cid:12)(cid:12)(cid:12) n × n I n × n − I n × n n × n (cid:12)(cid:12)(cid:12)(cid:12) (3.16) ω will be said to be in the canonical (or Darboux) form. If Γ is linear andHamiltonian w.r.t. a constant symplectic form, then the Hamiltonian is forcedto be a quadratic function, i.e.: H = 12 H ij x i x j , H ij ∈ R (3.17)38 emark 7 The above is clearly a coordinate-dependent definition of a quadraticfunction. A coordinate (and dimension)-free characterization of quadratic func-tions, and one that is more suitable in the case of (infinite-dimensional) Hilbertspaces, can be given as follows. A mapping: H : V → V ′ with V , V ′ vectorspaces (over a field K , with K = R or C ) is quadratic(a quadratic function if V ′ = R or C ) if: • H ( λx ) = λ H ( x ) , ∀ x ∈ V , λ ∈ K (3.18) and: • b ( x, y ) =: H ( x + y ) − H ( x ) − H ( y ) (3.19) is a bilinear mapping for all x, y ∈ V . Remark 8
Notice that, while G is a (1 , -type tensor (it ”maps vectors tovectors”) Ω and H = k H ij k are (0 , -type tensors (they ”map vectors to cov-ectors” (and viceversa in both cases)). This difference manifests itself in thetransformation under a general change of coordinates. If: x i = T i j y , j , then: G → G ′ = T − GT (3.20) while ( e T standing for the transpose of T ): Ω → Ω ′ = e T Ω T , H → H ′ = e THT (3.21) (the difference is not apparent when T − = e T , i.e. T is an orthogonal transfor-mation, T ∈ O (2 n ) ). Restricting from now on to linear vector fields and constant symplectic struc-tures, and omitting the superscripts and suffixes ”0”, if Λ is the Poisson tensor( (cid:8) x i , x j (cid:9) = Λ ij , Λ ij Ω jk = δ i k ), then if Γ = G i j x j ∂/∂x i is Hamiltonian w.r.t. ω = (1 /
2) Ω ij dx i ∧ dx j , this implies:Ω G = − H (3.22)and, equivalently: G = − Λ H (3.23)Hence: Looking for a Hamiltonian description w.r.t. a constant symplecticstructure for a linear vector field Γ is therefore equivalent to looking for thedecomposition of the representative matrix G into the product of an invertible skew-symmetric matrix Λ and a symmetric matrix H . The former will providea (non-degenerate) Poisson structure, the latter a Hamiltonian adapted to thegiven Poisson structure . Λ will be a (2 , → Λ ′ = T − Λ ^ ( T − ) .
39t this point we can make contact with the discussion of Ch.1, where wedealt with linear Hamiltonian vector fields on a finite-dimensional Hilbert space.There it was shown that the Hermitian structure gives rise to both a metrictensor and a symplectic form, and that the two are compatible in the sense thatthey are connected to one another by a third structure, the complex structure J .Here too we can reconstruct a (compatible ) complex structure starting fromthe tensors Λ and H , at least in the case when H is positive-definite. If such isthe case, we can find, as already discussed elsewhere, a system of coordinatesin which the vector field Γ is given explicitly as a sum of independent harmonicoscillators with proper frequencies ν , .., ν n (possibly not all distinct):Γ = n X i =1 ν i Γ (0) i ; Γ (0) i = x i + n ∂∂x i − x i ∂∂x i + n , i = 1 , ..., n (3.24)i.e.: G = (cid:12)(cid:12)(cid:12)(cid:12) n × n ν − ν n × n (cid:12)(cid:12)(cid:12)(cid:12) (3.25)where: ν = diag ( ν , .., ν n ), with the standard Poisson tensor:Λ = 12 Λ ij ∂∂x i ∧ ∂∂x j (3.26)whose representative matrix will be:Λ = | Λ ij | = (cid:12)(cid:12)(cid:12)(cid:12) n × n − I n × n I n × n n × n (cid:12)(cid:12)(cid:12)(cid:12) (3.27)and Hamiltonian: H = (1 / P i ν i (cid:0) x i + x i + n (cid:1) . It is now clear that the vectorfield: Γ (0) = n X i =1 Γ (0) i (3.28)will be Hamiltonian with a new Hamiltonian: H ′ = (1 / P i (cid:0) x i + x i + n (cid:1) andthat, in terms of the representative matrices:(Λ H ′ ) = − I (3.29)i.e. that the (1 ,
1) tensor Λ H ′ (whose representative matrix will coincide withthe matrix (3.27)) will provide the required complex structure.Some (necessary) consequences of Γ being Hamiltonian have been drawn inRef.[83], namely:1. As e G = H Λ = Λ (cid:0) Λ − H Λ (cid:1) , e G is a representative of a vector field which isHamiltonian w.r.t. the same Poisson structure with Hamiltonian: − Λ − H Λ.Indeed, in the basis in which Λ has the standard form, i.e.:Λ = (cid:12)(cid:12)(cid:12)(cid:12) n × n − I n × n I n × n n × n (cid:12)(cid:12)(cid:12)(cid:12) (3.30) See Ref.[160] and the following Ch.4. − = − Λ. Hence: − Λ − H Λ = Λ H Λ, which is symmetric. Notice,however, that in general e G and G will not commute, nor will then theassociated vector fields.2. G = − Λ H Λ H Λ H = Λ ( H Λ) H ( − Λ H ) = Λ (cid:16) e GHG (cid:17) . More generally, G k +1 can be written as: G k +1 = − Λ H... Λ H | {z } k +1 = − Λ H Λ ...H Λ | {z } k H Λ H... Λ H | {z } k (3.31)i.e.: G k +1 = − ( − ) k Λ (cid:16) e G k HG k (cid:17) (3.32)Hence: G k +1 will represent a Hamiltonian vector field Γ k with the Hamil-tonian: H k = 12 ( − ) k (cid:16) e G k HG k (cid:17) ij x i x j ; H = H (3.33) w.r.t. the same Poisson structure. As the correspondence between ma-trices and linear vector fields is a Lie algebra homomorphism, all theseHamiltonian vector fields will commute pairwise. As the correspondencebetween linear vector fields and Hamiltonian functions is a Lie algebra an-tihomomorphism , in the linear case H k will be a constant of the motionfor Γ k ′ ∀ k, k ′ , and they will be pairwise in involution . Remark 9 If G is generic (and Hamiltonian), we will generate in this way alsoa maximal set of (i.e. n ) constants of the motion pairwise in involution, and Γ will be completely integrable a’ la Liouville. iii ) As: e G = H Λ = Λ − (Λ H )Λ = Λ − ( − G )Λ ⇒ T rG = 0 it follows that:
T rG k +1 = 0 ∀ k (3.34) Notes . a ) That this is a necessary condition for the representative matrix of aHamiltonian vector field is pretty obvious. Indeed, for any vector field Γ ona symplectic 2 n -dimensional manifold, the divergence of Γ is defined by: L Γ ω n =: ( div Γ) ω n (3.35)where ω is the symplectic form and ω n the symplectic volume. If the flowassociated with Γ is Hamiltonian, it must be volume-preserving (Liouville’s The Lie algebra on functions being defined by the Poisson bracket. Recall that: { f, g } = i X g i X f ω = L X g f = − L X f g , with X f , X g the associated Hamiltonian vector fields, and that,for any two vector fields X and Y : i [ X,Y ] = L X i Y − i X L Y . Therefore: i [ X f ,X g ] ω = − d { f, g } . Notice that, in general (see the previous footnote): i [ X f ,X g ] ω = − d { f, g } and that,therefore: (cid:2) X f , X g (cid:3) = 0 only implies in general: { f, g } = const. For linear vector fields,however, both f and g will be quadratic functions. The Poisson bracket { f, g } will be quadraticas well, and it will be constant iff it vanishes. div Γ = 0. But it is easy to prove that, fora linear vector field and for a constant symplectic structure: div
Γ =
T rG . b ) The vanishing of the trace of odd powers of G implies that the charac-teristic polynomial P ( λ ) will contain only even powers of λ (i.e. P ( λ ) will beactually a polynomial in λ of degree n ). Real roots will appear then in pairs( λ, − λ ) and (the coefficients of P ( λ ) being real) complex roots will appear inquadruples (cid:0) λ, λ, − λ, − λ (cid:1) . c ) If T is an invertible matrix: T − GT = − T − (Λ H ) T = − (cid:16) T − Λ ^ ( T − ) (cid:17) (cid:16) e T HT (cid:17) (3.36)Then, if T is in the commutant of G ([ T, G ] = 0) we find a new Hamilto-nian description ( H ′ = e T HT ) with a new Poisson structure (Λ ′ = T − Λ ^ ( T − ))provided: T − Λ ^ ( T − ) = Λ. This implies that T be not a canonical transfor-mation. Any ”non-canonical” matrix T in the commutant of G will provide anew Hamiltonian description for the same vector field. Powers of G are of course in the commutant of G . From: Ω G = − H weobtain ( H being symmetric and Ω skew-symmetric): e G Ω = H and hence: e G Ω = − Ω G (3.37)It is then easy to prove that, in general: e G h Ω = ( − ) h Ω G h (3.38)Indeed, this holds for h = 1. By induction: e G h +1 Ω = ( − ) h e G Ω G h = ( − ) h ( − Ω G ) G h =( − ) h +1 Ω G h +1 . As (Ω being skew-symmetric):Ω G h = − ^ (cid:16) e G h Ω (cid:17) (3.39)this result implies: e G h Ω = ( − ) h +1 ^ (cid:16) e G h Ω (cid:17) (3.40)and hence e G h Ω will be symmetric for h odd (and, indeed, for h = 2 k + 1, e G k +1 Ω = −H k ) and skew -symmetric for even h = 2 k . Moreover: e G ( e G k Ω) = − ( e G k Ω) G (3.41)i.e. e G k Ω will be an admissible skew-symmetric factor in the decomposition of G . A slightly different way [135] to exploit even powers of G to generate alter-native Hamiltonian descriptions is as follows (basically, we are reverting from a42nite to an infinitesimal description). Let, e.g., Γ (2) be the linear vector fieldassociated with G , i.e.: Γ (2) = (cid:0) G (cid:1) i j x j ∂∂x i (3.42)If the Poisson structure is given by:Λ = 12 Λ hk ∂∂x h ∧ ∂∂x k (3.43)then: L Γ (2) Λ = − (cid:0) G Λ (cid:1) ij ∂∂x i ∧ ∂∂x j = − (Λ H Λ H Λ) ij ∂∂x i ∧ ∂∂x j (3.44)Notice that G Λ = Λ H Λ H Λ is manifestly skew-symmetric. Therefore L Γ (2) Λ,if it does not vanish, defines a new Poisson structure:Λ (2) = 12 (Λ H Λ H Λ) ij ∂∂x i ∧ ∂∂x j (3.45)and Poisson brackets: { f, g } (2) = (Λ H Λ H Λ) ij ∂f∂x i ∂g∂x j (3.46)The new Poisson structure will be non-degenerate iff both Λ and H areinvertible, i.e., as G = − Λ H , iff G is invertible. Requiring then that there existsa new Hamiltonian H (2) s.t. Γ is again Hamiltonian w.r.t. the new Poissonstructure, i.e.: (cid:8) x i , H (2) (cid:9) = (cid:0) Λ (2) (cid:1) ij ∂ H (2) ∂x j = G i j x j (3.47)together with G = − Λ H leads to: H (2) = 12 H (2) ij x i x j ; H (2) = (Λ H Λ) − (3.48)If G is not invertible, then one can proceed by exponentiation [83, 135]. Example 10
We have seen in Ch. how the dynamics of a quantum systemseparates into that of a set of non-interacting harmonic oscillators. All finite-level quantum systems can be written as a family of harmonic oscillators withfrequencies related to the eigenvalues of the Hamiltonian. It is therefore appro-priate to consider here again the harmonic oscillator. For this system the aboveprocedure (i.e. taking Lie derivatives of the Poisson structure) provides alterna-tive Hamiltonian descriptions. Proceeding instead as in the previous discussionwith T = G and: G = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) /m
00 0 0 1 /m − m Ω − m Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.49)43 nd: Λ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.50) one finds: G = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − Ω − Ω − Ω
00 0 0 − Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.51) and: G − Λ e G − = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) / Ω
00 0 0 1 / Ω − / Ω − / Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.52) So, but for the isotropic case Ω = Ω in which G − Λ e G − and Λ H Λ H Λ becomeproportional, the two approaches appear to be genuinely different. Example 11
As a last (almost trivial but explanatory) example let us take themost general linear vector field in R = { ( x, y ) } : Γ = ( ax + by ) ∂∂x + ( cx + dy ) ∂∂y (3.53) corresponding to the matrix G = (cid:12)(cid:12)(cid:12)(cid:12) a bc d (cid:12)(cid:12)(cid:12)(cid:12) , a, b, c, d ∈ R . (3.54) with T rG k +1 = 0 if and only if a = − d . Given then the constant symplecticstructure Ω = αdx ∧ dy ( α ∈ R ): Ω = α (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) . (3.55)Γ will be Hamiltonian with Hamiltonian: H = αaxy + α (cid:0) by − cx (cid:1) / ,corresponding to: H = − Ω G .Three situations are possible:1. The eigenvalues of G are ± λ ; λ ≡ √ a + bc ∈ R . Then there exist coordi-nates ( x, y ) such that the matrix (3.54) is of the form G = (cid:12)(cid:12)(cid:12)(cid:12) λλ (cid:12)(cid:12)(cid:12)(cid:12) . (3.56) If we set: x = A cosh Φ , y = A sinh Φ (3.57)44 hen: Ω = dH ∧ d Φ H = λα A . (3.58)
2. The eigenvalues of G are ± iλ ; λ ≡ p | a + bc | ∈ R . Then G may be putin the form: G = (cid:12)(cid:12)(cid:12)(cid:12) λ − λ (cid:12)(cid:12)(cid:12)(cid:12) . (3.59) We can now define x = A cos Φ , y = A sin Φ (3.60) that allow to write the symplectic form and the hamiltonian as in (3.58).3. Finally we consider the case a + bc = 0 , when there exist coordinates ( x, y ) such that G assumes the form G = (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) . (3.61) Now H = αy / . Returning now to the general case, we have seen that a necessary conditionfor a linear vector field Γ with representative matrix G to be Hamiltonian isthat the traces of odd powers of G vanish. Whether or not this is also sufficientrequires a rather long analysis of the decomposition of G into Jordan blocks [22],for whose details we refer to the literature, and whose main result is containedin the following [83]: Proposition 12
A linear vector field Γ is Hamiltonian iff the representativematrix G satisfies T rG k +1 = 0 and:i) no further condition if the eigenvalues are non-degenerate or purely imag-inary,ii) for degenerate real or genuinely complex (i.e not purely imaginary) eigen-values the Jordan block belonging to a given eigenvalue λ has the same structureas the block belonging to − λ , this meaning that the Jordan block associated withthe eigenvalue λ can be brought to the form: G { λ } = (cid:13)(cid:13)(cid:13)(cid:13) J − e J (cid:13)(cid:13)(cid:13)(cid:13) (3.62) iii) zero eigenvalues have even multiplicity. This solves the problem of under which conditions a linear vector field isHamiltonian, but does not tell us how many genuinely different Hamiltonians(and symplectic structures) are permissible for a given vector field. A morestringent result has also been proved in Ref.[83] and precisely that:45 roposition 13 If Γ has non-complex (i.e. either real or purely imaginary)non-degenerate eigenvalues, then it has a minimal family ( a ”pencil” [81, 107])of equivalent admissible symplectic forms parametrized by a number of param-eters equal to the number of couples ( λ, − λ ) of eigenvalues minus one (i.e. a ( n − -parameter family). The case in which Γ has (only) purely imaginary eigenvalues is of particularinterest for the analysis of (finite-dimensional, for the time being) quantumsystems. Indeed, we can remark that: • If the eigenvalues are purely imaginary, then all the motions of the sys-tem will be stable [4, 5]. Considering the decomposition: G = − Λ H ofEq.(3.23), if H is positive, it will define an Euclidean metric and, afterpossibly a rescaling that will be discussed in the next Chapter, Λ willdefine the Poisson tensor and G will become the complex structure. Thesystem will become what we will call a quantum system , and that becausethe evolution is unitary with respect to the Hermitian structure associ-ated with G and Λ. In this sense, as we will see shortly, the analysis ofthis Chapter provides also a way to classify the possible, and alternative,Hamiltonian descriptions for quantum systems. • With reference in particular to Ch.1, if the (quantum) Hamiltonian H hasa real spectrum, then (cfr. Eq.(1.19)) (the realified of) − iH/ ~ will turnout to have purely imaginary eigenvalues. Even if H is not Hermitianw.r.t. the given Hermitian structure, one can always find [14, 175, 220] amodified scalar product (see again Ch.1) w.r.t. which H turns out to beHermitian.All this material will be expanded and put into use in the next Chapter. In this section we discuss some methods to obtain inequivalent descriptions fora given classical system defined by a dynamical vector field Γ, not necessarily alinear one.
As explained in Appendix A, given any 1-1 tensor T , we can define an an-tiderivation d T which acts on functions as d T f ≡ T ( df ) . (3.63) Or a pseudo-Euclidean one if it is non-degenerate but not necessarily positive.
46n the sequel we will use extensively this construction with T = J , the complexstructure. Suppose now that the function F be a constant of motion and thatthe tensor T be invariant under the action of Γ so that L Γ F = 0 , L Γ T = 0 . (3.64)Then we can define a closed two-form ω F ≡ d ( d T F ) (3.65)which is invariant under action of Γ since L Γ ω F = d ( L Γ d T F ) and L Γ d T F = 0because of (3.64). Assuming that ω F be non-degenerate, it will define a newinvariant symplectic structure. To obtain the alternative Hamiltonian functionH associated to ω F it is sufficient to notice that:0 = L Γ d T F = i Γ d ( d T F ) + di Γ ( d T F ) = i Γ ω F + dF ( T (Γ)) = i Γ ω F + d ( L T (Γ) F ) . (3.66)Hence: H = − L T (Γ) F = − ( d T F ) (Γ) (3.67) Remark 14
The above construction may turn out to be empty if the function F is in the kernel of dd T : dd T F = 0 . For example, in R ≈ C with (real)coordinates ( q, p ) , take T to be the complex structure : J = dp ⊗ ∂∂q − dq ⊗ ∂∂p (3.68) which is invariant under the dynamics of the D harmonic oscillator. Then, itis immediate to check that: dd J F = (cid:18) ∂ F∂q + ∂ F∂p (cid:19) dq ∧ dp (3.69) and hence all the harmonic functions in the plane will be in the kernel of dd J . Remark 15
Suppose now that
Γ = G i j x j ∂∂x i be a linear vector field and T = T i j dx j ⊗ ∂∂x i a constant invariant 1-1 tensor. Then it is not difficult to checkthat ω F is constant if and only if F is a quadratic function: F = 12 F ij x i x j , F ij = F ji . (3.70) In this case, using the matrix notation of sect. 3.2, we have: H = 12 H ij x i x j , H ij = H ji = − ( F T G ) ij − ( F T G ) ji , (3.71) ω F = 12 Ω ij dx i ∧ dx j , Ω ij = − Ω ji = ( F T ) ij − ( F T ) ji . (3.72)47 sing the fact that Eqs. (3.64) are equivalent to the conditions: ( F G ) ij = − ( F G ) ji and ( GT ) i j = ( T G ) i j , one can show that, as it should be, the relation Ω G = − H is trivially satisfied. As an example, let us consider the two-dimensional isotropic harmonic os-cillator whose dynamics is described by the vector fieldΓ = p a ∂∂q a − q a ∂∂p a , (3.73)where the summed-over index a assumes the values: a = 1 ,
2. We will take for T the complex structure of the phase space R , i.e: T = J = dp a ⊗ ∂∂q a − dq a ⊗ ∂∂p a . (3.74)Thus the representative matrices will be: G = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − −
11 0 0 00 1 0 0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , T = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (3.75)With T = J we have: − J (Γ) = ∆ ≡ q a ∂∂q a + p a ∂∂p a (3.76)with ∆ the dilation (Liouville) field associated with the standard linear structureon R and: d J F = ∂F∂q a dp a − ∂F∂p a dq a ⇒ ω F = dd J F = (cid:18) ∂ F∂q a ∂q b + ∂ F∂p a ∂p b (cid:19) dq a ∧ dp b (3.77)as well as: H = − L J (Γ) F = L ∆ F (3.78)for any function F = F ( q , p ).It is well known that a basis of constants of motion is given, for example, bythe four independent functions F = (cid:2) ( p ) + ( q ) + ( p ) + ( q ) (cid:3) , F = (cid:2) ( p ) + ( q ) − ( p ) − ( q ) (cid:3) ,F = (cid:2) p p + q q (cid:3) , F = (cid:2) q p − q p (cid:3) . (3.79)All four functions being quadratic , the above construction yields then thefollowing four alternative hamiltonian descriptions: H = (cid:2) ( p ) + ( q ) + ( p ) + ( q ) (cid:3) , ω = dq ∧ dp + dq ∧ dp ; H = (cid:2) ( p ) + ( q ) − ( p ) − ( q ) (cid:3) , ω = dq ∧ dp − dq ∧ dp ; H = p p + q q , ω = dq ∧ dp + dq ∧ dp ; H = q p − q p , ω = − dq ∧ dq + dp ∧ dp . (3.80) L ∆ F i = 2 F i for i = 0 , , , J are associated withthe standard linear structure on R . As we will now see, this observation may beexploited to obtain alternative Hamiltonian descriptions by defining inequivalentlinear structures on phase space. We recall here [66] some known facts about the possibility of defining alternative(i.e. not linearly related) linear structures on a vector space and/or of using thelinear structure of a vector space to endow with a linear structure manifoldsthat are related to the given vector space.Let E be a (real or complex) linear vector space with addition + and mul-tiplication by scalars · , and a nonlinear diffeomorphism: φ : E ↔ E. (3.81)We can define a new linear structure if we define: • Addition of u, v ∈ M as: u + ( φ ) v =: φ ( φ − ( u ) + φ − ( v )) . (3.82) • Multiplication by a scalar λ ∈ R or C of u ∈ M as: λ · ( φ ) u =: φ (cid:0) λφ − ( u ) (cid:1) . (3.83)Obviously, the two linear spaces ( E, + , · ) and ( E, + ( φ ) , · ( φ ) ) are finite dimen-sional vector spaces of the same dimension and hence are isomorphic. However,the change of coordinates defined by φ that we are using to “deform” the lin-ear structure is a nonlinear diffeomorphism. In other words, we are using twodifferent (diffeomorphic but not linearly related) global charts to describe thesame manifold space E Within the framework of the new linear structure, it makes sense to considerthe mapping: Ψ : M × R → M ,
Ψ ( u, t ) =: e t · ( φ ) u =: u ( t ) , (3.84)that defines a one-parameter group as it can be easily checked. Its infinitesimalgenerator, the dilation (Liouville) field, is given by∆ ( u ) = (cid:20) ddt u ( t ) (cid:21) t =0 = (cid:20) ddt φ (cid:0) e t φ − ( u ) (cid:1)(cid:21) t =0 . (3.85)As an example consider T ∗ R with coordinates ( q, p ) and linear structuredefined by the dilation field: ∆ = q ∂∂q + p ∂∂p , (3.86) More examples may be found in Ref.[66]. i ∆ ω = qdp − pdq with respect to the standard symplecticform ω = dq ∧ dp .As it is well known the dynamics of the 1 D harmonic oscillator is described, inappropriate units, by the vector field:Γ = p ∂∂q − q ∂∂p , (3.87)which is ω -Hamiltonian: i Γ ω = dH with Hamiltonian: H = (cid:0) q + p (cid:1) /
2. Wecan also define the complex structure: J = dp ⊗ ∂∂q − dq ⊗ ∂∂p (3.88)which is such that: J = − I , J (∆) = Γ , J (Γ) = − ∆ (3.89)The composition of the symplectic and the complex structures gives rise to acompatible [160] metric tensor g : ω ◦ J =: − g, g = dq ⊗ dq + dp ⊗ dp (3.90)Notice also that the complex structure and the Hamiltonian are connectedby: ω = 12 dd J H (3.91)Let us consider now the nonlinear change of coordinates on T ∗ R [174]:( q, p ) → ( Q, P ) with: Q = q (1 + f ( H )) (3.92) P = p (1 + f ( H )) . (3.93)Under very mild assumptions on the function f ( H ) the mapping (3.93) willbe smooth and invertible with a smooth inverse. One might assume, e.g., that f ( · ) be nonnegative and monotonically increasing for positive argument.Withthe dynamics given by Eq.(3.87), it is immediate to check that: L Γ Q = P, L Γ P = − Q (3.94)Hence, although the two coordinates ystem are not linearly related, the vectorfield Γ will be given, in the new coordinate system, by:Γ = P ∂∂Q − Q ∂∂P (3.95)which will be again Hamiltonian with respect to the symplectic form ω ′ = dQ ∧ dP with H ′ = (cid:0) Q + P (cid:1) / H (1 + f ( H )) as Hamiltonian. Now thenew Liouville field ∆ ′ , defined via i ∆ ′ ω ′ = QdP − P dQ , is given by:∆ ′ = Q ∂∂Q + P ∂∂P , (3.96)50otice also that we can define a new 1-1 tensor (the new complex structure): J ′ = dP ⊗ ∂∂Q − dQ ⊗ ∂∂P , (3.97)which is again such that J ′ (Γ) = − ∆ ′ . J ′ and ω ′ will generate then the newmetric tensor: g ′ = dQ ⊗ dQ + dP ⊗ dP . Thus, following the construction outlinedin the previous section, we might have obtained this alternative description ofthe dynamics of the one-dimensional harmonic oscillator also by setting: T = J ′ , (3.98) ω ′ = 12 dd J ′ H ′ . (3.99)One obtains in this way a new linear structure, which is in some sense ”adapted”to the chosen Hamiltonian description.Finally, we observe that the above construction to obtain alternative de-scriptions may be easily generalized to the n-dimensional harmonic oscillator bydefining ω F ≡ α a d (cid:18) ∂F∂p a (cid:19) ∧ d (cid:18) ∂F∂q a (cid:19) (3.100)and H F ≡ α a "(cid:18) ∂F∂p a (cid:19) + (cid:18) ∂F∂q a (cid:19) , (3.101)where F is a constant of the motion such that ω F is non-degenerate. Switching now to the Lagrangian framework, we recall [186] that a regularLagrangian L will define the symplectic structure on T Q : ω L = dθ L = d (cid:18) ∂ L ∂u i (cid:19) ∧ dq i ; θ L = (cid:18) ∂ L ∂u i (cid:19) dq i . (3.102)We look now [155] for Hamiltonian vector fields X j , Y j such that: i X j ω L = − d (cid:18) ∂ L ∂u j (cid:19) , i Y j ω L = dq j (3.103)Explicitly this implies: L X j q i = δ ij , L X j ∂ L ∂u i = 0 , (3.104) L Y j q i = 0 , L Y j ∂ L ∂u i = δ ji . (3.105)51sing then the identity i [ Z,W ] = L Z ◦ i W − i W ◦ L Z , and the fact that the Liederivative of the Hamiltonian of every field of the set (3.103) with respect toany other of the fields is either zero or a constant (actually unity), one can showthat: i [ Z,W ] ω L = 0 whenever [ Z, W ] = [ X i , X j ] , (cid:2) X i , Y j (cid:3) , (cid:2) Y i , Y j (cid:3) , (3.106)which proves that: [ X i , X j ] = (cid:2) X i , Y j (cid:3) = (cid:2) Y i , Y j (cid:3) = 0 . (3.107)This defines an infinitesimal action of an Abelian Lie group on T Q . If thisintegrates to an action of the group R n (dim Q = n ) that is free and transi-tive, this will define a new vector space structure on T Q that is ”adapted” tothe Lagrangian two-form ω L . More explicitly, defining dual forms (cid:0) α i , β i (cid:1) via: α i ( X j ) = δ ij , α i (cid:0) Y j (cid:1) = 0; β i (cid:0) Y j (cid:1) = δ ji , β i ( X j ) = 0, it is immediate to seethat: α i = dq i (3.108) β i = d (cid:18) ∂ L ∂u i (cid:19) (3.109)and that the symplectic form can be written as: ω L = β i ∧ α i . (3.110)Basically, what this means is that, to the extent that the definition of vectorfields and dual forms is global, we have found in this way a global Darbouxchart.As an example of this construction, we may consider a particle in a (time-independent) magnetic field B = ∇ × A . The corresponding second-ordervector field is given by ( e = m = c = 1):Γ = u i ∂∂q i + δ is ǫ ijk u j B k ∂∂u s . (3.111)The Lagrangian is given in turn by : L = 12 δ ij u i u j + u i A i . (3.112)while the symplectic form is: ω L = δ ij dq i ∧ du j − ε ijk B i dq j ∧ dq k . (3.113)The field Γ is hamiltonian, the Hamiltonian being given by: H = 12 δ ij u i u j . (3.114)52ow it is easy to see that: X j = ∂∂q j − δ ik ∂A k ∂q j ∂∂u i , (3.115) Y j = δ jk ∂∂u k . (3.116)The dual forms α i , β i , i = 1 , ..., n = dim Q are given by: α i = dq i , (3.117) β i = δ ij d ( u j + δ jk A k ) . (3.118)Therefore the mapping Q i = q i (3.119) U i = u i + δ ik A k , (3.120)provides us with a symplectomorphism that reduces ω L to the canonical form ω L = dq i ∧ dπ i , (3.121)where π i = δ ij U j . We may say that the chart ( Q, U ) is a Darboux chart”adapted” to the vector potential −→ A .The Liouville field will be then:∆ = Q i ∂∂Q i + (cid:20) U i + δ ik (cid:18) Q j ∂A k ∂Q j − A k (cid:19)(cid:21) ∂∂U i . (3.122)Denoting collectively the old and new coordinates as ( q, u ) and ( Q, U ) respec-tively, Eq. (3.120) defines a mapping:( q, u ) φ → ( Q, U ) . (3.123)It is then a straightforward application of the definitions (3.82) and (3.83) toshow that the rules of addition and multiplication by a constant become, in thisspecific case:( Q, U ) + ( φ ) ( Q ′ , U ′ ) = ( Q + Q ′ , U + U ′ + [ A ( Q + Q ′ ) − ( A ( Q ) + A ( Q ′ ))])(3.124)and: λ · ( φ ) ( Q, U ) = ( λQ, λU + [ A ( λQ ) − λA ( Q )]) . (3.125) We notice that ∆ depends on the gauge choice. The symplectic form will be howevergauge-independent .4 Symmetries and Constants of the Motion for SystemsAdmitting of Alternative Descriptions In our setting, according to which the primitive (or the more physically relevant[167] ) object is the vector field Γ describing the dynamics on some carrier space M , a symmetry will be defined as a one-parameter group of diffeomorphisms ofthe carrier space that maps solutions (i.e. integral curves of Γ) into solutions.At the infinitesimal level, if X ∈ X ( M ) is the associated infinitesimal generatorof the one-parameter group, this means [167] that it must commute with Γ, i.e.:[ X, Γ] = 0 (3.126)It is a straightforward consequence of the Jacobi identity on the commutatorbracket that : [ X , Γ] = 0 , [ X , Γ] = 0 ⇒ [[ X , X ] , Γ] = 0 (3.127)(but not viceversa, of course). Hence:
All the vector fields satisfying the condi-tion (3.126) for a given dynamical vector field Γ close on a Lie algebra, the Liealgebra of (infinitesimal) symmetries of Γ.On the other hand, constants of the motion are, as is well known, functions f ∈ F ( M ) that are invariant under the flow of Γ, i.e.: L Γ f = 0 (3.128)where L Γ is the Lie derivative. A considerable effort is usually devoted intextbooks (both in point-particle Mechanics and/or in Field Theory, both ele-mentary and more advanced) to try and define a clear-cut procedure allowingto associate constants of the motion (i.e. conserved quantities) with symmetries(and the other way around). This goes usually through the use of N¨other’sTheorem , that, for completeness, we will revisit briefly here both in the La-grangian and Hamiltonian formulations of point-particle Mechanics. Lagrangian Formalism.
In this case M = T Q , with Q a base manifoldwith (local) coordinates q , ..., q n , n = dim ( Q ). Before proceeding, werecall how vector fields on the base manifold can be lifted to vector fieldson T Q . Given: X = X i ∂∂q i ∈ X ( Q ) , X i ∈ F ( Q ) (3.129) This is very much reminiscent of Poisson’s theorem of Hamiltonian Mechanics. See however, e.g., Ref.[186] for the discussion of different approaches. tangent lift (sometimes called also the complete lift ) X c of X is definedas : X c = X i ∂∂q i + ( L Γ X i ) ∂∂u i ∈ X ( T Q ) (3.130)where the u i ’s are coordinates along the fibers and Γ is any second-ordervector field.If L is a Lagrangian appropriate for the description, via the Euler-Lagrangeequations, of the dynamics associated with a given second-order vectorfield Γ, a N¨other symmetry [186] is, by definition, a tangent lift X c thatis a symmetry for Γ, i.e. such that:[Γ , X c ] = 0 (3.131)and such that: L X c L = L Γ h (3.132)where : h = π ∗ g, g ∈ F ( Q ) and: π : T Q → Q is the canonical projection.The Lagrangian will be said to be strictly invariant if h = 0 (i.e. g = 0) , quasi-invariant [161] if g = 0 N¨other’s theorem states then that: F X c =: i X c θ L − h (3.133)is a constant of the motion. Here: θ L = ∂ L ∂u i dq i (3.134)is the Lagrangian one-form associated with L . In local coordinates: F X c = X i ∂ L ∂u i − h (3.135)2. Hamiltonian Formalism.
In this case M = T ∗ Q , the cotangent bundle ofthe base manifold, with local coordinates (cid:0) q i , p i (cid:1) , i = 1 , ..., n , equippedwith the Cartan form: θ = p i dq i (3.136)and the symplectic structure: ω = − dθ = dq i ∧ dp i (3.137)Here too there is a standard procedure for lifting vector fields from X ( Q )to X ( T ∗ Q ). namely, given a vector field X ∈ X ( Q ) of the form (3.129), Here, with abuse of notation, we write X i for what should be instead π ∗ X i , with: π : T Q → Q the canonical projection. Of course this is nothing but the familiar statement that, under the action of X c , theLagrangian changes by the total time derivative of a function of the q ’s alone. Barring the trivial case h (i.e. g )= const. , a second-order vector field does not admit ofconstants of the motion that are functions of the q ’s alone. cotangent lift (sometimes called the natural lift ) X ∗ of X is givenby: X ∗ = X i ∂∂q i − (cid:18) ∂X j ∂q i (cid:19) p j ∂∂p i ∈ X ( T ∗ Q ) (3.138)and it is easy to show that it is the unique vector field that projects downto X on the base manifold and that leaves the Cartan form invariant, i.e.such that: L X ∗ θ = 0 (3.139) Remark 16
In a more intrinsic way, both lifts can be defined [167] as theinfinitesimal generators of the tangent or, respectively, cotangent lift of the one-parameter group of diffeomorphisms of Q that has X as its infinitesimal gener-ator. Remark 17
Symmetries for the dynamics that are (tangent or cotangent) liftsof vector fields on the base manifold are also called point symmetries.
A vector field Γ ∈ X ( T ∗ Q ) is Hamiltonian if there exists a (Hamiltonian)function H ∈ F ( T ∗ Q ) such that: i Γ ω = dH (3.140)Given then a function F ∈ F ( T Q ), let X F be the associated Hamiltonian vectorfield (not necessarily a cotangent lift), i.e.: i X F ω = dF . Then: L X F H = i X F dH = i X F i Γ ω = − i Γ i X F ω = − i Γ dF = − L Γ F (3.141)Hence: L Γ F = 0 ⇔ L X F H = 0 (3.142)Therefore, if X F is a symmetry for the Hamiltonian (i.e.: L X F H = 0), then F will be a constant of the motion and viceversa. Moreover, using the identity[186]: i [ X,Y ] = i X ◦ L Y − L Y ◦ i X (3.143)valid for any pair of vector fields, it follows that, if X is at least locally Hamil-tonian (i.e.: L X ω = 0), then: dL X H = L X i Γ ω = − i [ X, Γ] ω (3.144)Hence, if X is a symmetry for the Hamiltonian, and as ω is non-degenerate: L X ω = 0 and L X H = 0 ⇒ [ X, Γ] = 0 (3.145)i.e. X is also a symmetry for the dynamics. The converse however is not true[167]: from: [ X, Γ] = 0 one can only infer that: L X H = const. , i.e. X need notbe a symmetry for the Hamiltonian. 56o far for the standard derivation of the N¨other Theorem. As a simpleexample, considering, e.g., the 3 D harmonic oscillator with the standard La-grangian: L = (1 / P i =1 h(cid:0) u i (cid:1) − (cid:0) q i (cid:1) i or the corresponding Hamiltonianleads to the well-known association of (strict) rotational invariance (of the La-grangian and/or of the Hamiltonian) with the conservation of angular momen-tum.The motivation for having gone here to some length through essentially stan-dard material has been to emphasize the crucial rˆole that ”intermediate” struc-tures such as the Lagrangian or the Hamiltonian, as well as the symplectic struc-ture, play along the way that leads to the association of symmetries with con-stants of the motion. When these ”intermediate” structures are not unique, asit happens when more non-equivalent (Lagrangian (on T Q ) or Hamiltonian (on T ∗ Q )) descriptions are available [36, 47, 130, 154, 155, 156, 159, 164, 186, 200],the connection becomes more ambiguous, and different (non-equivalent) de-scriptions of the same dynamical system may lead to the association of differentconstants of the motion with the same group of symmetries, or of the sameconstants of the motion with different groups of symmetry or to no associationat all, as we shall discuss now. We will consider here some simple examples:1. Let Q = R , and let Γ be the dynamics of an isotropic harmonic oscillator(with unit mass and frequency for simplicity). Then it is immediate toshow all the Lagrangians of the form: L B = 12 B ij (cid:0) u i u j − q i q j (cid:1) (3.146)where: B = k B ij k is a real and (necessarily) symmetric matrix are ad-missible Lagrangians for the isotropic harmonic oscillator, and, moreover,regular ones iff the matrix B is non-singular. By ”admissible” we meanobviously that the Euler-Lagrange equations associated with any one ofthe Lagrangians (3.146) reproduce the dynamics of the isotropic harmonicoscillator. As we can always diagonalize B with the aid of an orthogo-nal transformation, we can limit ourselves to considering only either thestandard Lagrangian: L = L + L + L ; L i = 12 h(cid:0) u i (cid:1) − (cid:0) q i (cid:1) i , i = 1 , , L ′ = L + L − L (3.148)57ow, it is obvious that the Lagrangian (3.147) is (strictly) invariant un-der the (lifted) action of O (3), while the Lagrangian (3.148) is (again,strictly) invariant under the (lifted) action of O (2 , n ofdimensions, the most general group of point symmetries for the dynamicsof the isotropic harmonic oscillator is GL ( n, R ), the above two groups aregroups of N¨other symmetries. While invariance under O (3) associates, viaN¨other’s theorem, the three components of the angular momentum withthe three generators of the group if the Lagrangian (3.147) is chosen asthe Lagrangian of the system, in the case in which one chooses L ′ as theLagrangian the situation is different. The three generators of O (2 ,
1) aregiven by the tangent lifts of the vector fields: X = q ∂∂q + q ∂∂q , X = q ∂∂q + q ∂∂q , J = q ∂∂q − q ∂∂q (3.149)While X and X correspond to ”boosts” in the q and q directions, J represents ordinary rotations in the (cid:0) q − q (cid:1) plane. They close on theLie algebra o (2 , X , X ] = J, [ X , J ] = X , [ J, X ] = X (3.150)and the same will hold true for the tangent lifts X c , X c and J c .Applying now N¨other’s theorem we find the following constants of themotion: F =: i X c θ L ′ = q u − q u ; F =: i X c θ L ′ = q u − q u (3.151)while, as before: i J c θ L ′ = q u − q u . Therefore, we find that the angularmomentum is the (vector) constant of the motion associated not with therotation group but instead with the Lorentz group O (2 , B ). We will consider here only linearvector fields that are generators of point symmetries, i.e. vector fields ofthe form : X = A i j (cid:18) q j ∂∂q i + u j ∂∂u i (cid:19) (3.152)for some matrix A = (cid:13)(cid:13) A i j (cid:13)(cid:13) ∈ End ( Q ). Then: L X L B = ( BA ) jk (cid:0) u j u k − q j q k (cid:1) (3.153)and strict invariance requires: ( BA ) jk + ( BA ) kj = 0, i.e.(as B is symmet-ric): A t B + BA = 0 (3.154) This is the case of the symmetries (3.149). AB has to be antisymmetric. For example,with the Lagrangian (3.148): B = diag (1 , , −
1) and, e.g. for the firstsymmetry X of Eq.(6.22): A = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (3.155)( A = A t ) and it is easy to check that the condition (3.154) is indeedsatisfied.By assumption, the matrix B in Eq.(3.146) can be diagonalized with theaid of an orthogonal transformation: B = OB ′ O t with B ′ diagonal and: OO t = O t O = Id . Then it is easy to see that Eq.(3.154) becomes: A ′ t B ′ + B ′ A ′ = 0 (3.156)with: A ′ = O t AO (3.157)defining the transformed infinitesimal symmetry in the new coordinatesystem.3. Consider, as a further example, the (isotropic) harmonic oscillator in 2 D .Apart from the standard Lagrangian ( L = L + L in the notation ofEq.(3.147)) we may consider the (regular) Lagrangian: L ′ = u u − q q (3.158)This Lagrangian is (strictly) invariant under the ”squeeze” transformation,i.e. the tangent lift of the one-parameter group: (cid:0) q , q (cid:1) (cid:0) q e t , q e − t (cid:1) ; t ∈ R (3.159)whose infinitesimal generator is: S = q ∂∂q − q ∂∂q (3.160)that lifts to: S c = q ∂∂q − q ∂∂q + u ∂∂u − u ∂∂u (3.161)N¨other’s theorem yields then the constant of the motion: F =: i s c θ L ′ = q u − q u (3.162)Hence: with the Lagrangian L ′ angular momentum is associated with in-variance under squeeze. In the notation of the previous example, here: B = (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) = σ and: A = (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) = σ , and, again, they satisfy the condition (3.154).59. The Lagrangian (3.158) can be diagonalized via a rotation of π/ Q = q + q √ , Q = q − q √ L → L − L (3.164)Now, the ”squeeze” transformation (3.159) becomes: (cid:12)(cid:12)(cid:12)(cid:12) Q Q (cid:12)(cid:12)(cid:12)(cid:12) −→ (cid:12)(cid:12)(cid:12)(cid:12) cosh t sinh t sinh t cosh t (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12) Q Q (cid:12)(cid:12)(cid:12)(cid:12) (3.165)whose infinitesimal generator is: X = Q ∂∂Q + Q ∂∂Q (3.166)(corresponding to the matrix: A = σ ) i.e., as expected, a (the unique)Lorentz boost with the parameter t playing the rˆole of the rapidity of theboost. Restricting ourselves for simplicity to dynamical systems described by regu-lar Lagrangians, we recall [167],[186] that the Euler-Lagrange equations for thesecond-order vector field Γ associated with a regular Lagrangian L can be writ-ten, in intrinsic terms, as: L Γ θ L − d L = 0 (3.167)where: θ L =: ∂ L ∂u i dq i (3.168)is the Lagrangian one-form or in the equivalent, ”Hamiltonian” form: i Γ Ω L = dE L (3.169)where: Ω L =: − dθ L (3.170)is the ”Lagrangian two-form”, which is symplectic if L is regular, and: E L =: i Γ θ L − L (3.171)is known [167],[186] as the ”energy function” associated with the Lagrangian L .60he transition to the Hamiltonian formulation on T ∗ Q is accomplished, asis well known[167], with the aid of the ”fiber derivative” (or ”Legendre map”): F L : T Q → T ∗ Q that is defined by: F L : (cid:0) q i , u i (cid:1) (cid:0) q i , p i = ∂ L /∂u i (cid:1) (3.172)If, as assumed here, the Lagrangian is regular, the fiber derivative is invert-ible and has the following properties (see Ref.[186] for details): • ( F L ) ∗ θ L = θ and: ( F L ) ∗ Ω L = ω where ( F L ) ∗ denotes the ”push-forward” associated with the fiber derivative,i.e.: ( F L ) ∗ = (cid:16) ( F L ) − (cid:17) ∗ ; • Via push-forward, the vector field Γ is mapped onto a vector field e Γ ∈ X ( T ∗ Q ) that is Hamiltonian with respect to the canonical symplecticform ω with an Hamiltonian H given by: H =: ( F L ) ∗ E L = E L ◦ ( F L ) − (3.173) • Explicitly (and locally): e Γ = ∂H∂p i ∂∂q i − ∂H∂q i ∂∂p i (3.174)All this can be summarized in the following scheme:(Γ , Ω L , dE L ) F L −→ (cid:16)e Γ , ω , dH (cid:17) (3.175) It is clear that the transition to T ∗ Q summarized in the scheme (3.175) will benon-ambiguous and unique if and only if, apart from trivial equivalencies, theLagrangian is unique.When more than one Lagrangian description is available, the situation canbecome more involved. To be more specific, let, say, L (1) and L (2) be two alter-native Lagrangians for the same dynamical system, Γ, on T Q . Each one definingits own fiber derivative, we can obtain different Hamiltonian descriptions on T ∗ Q with different vector fields and Hamiltonians but the same symplectic structure(i.e. ω ) using alternatively the two fiber derivatives according to the scheme:Ω L (1) , dE L (1) ր Γ ց Ω L (2) , dE L (2) | {z } T Q F L (1) −→ F L (2) −→ e Γ (1) dH (2) ց ր ω ր ց e Γ (2) dH (1) | {z } T ∗ Q (3.176)61lthough the vector fields e Γ (1) and e Γ (2) may look different, it is worth stressingthat nonetheless they offer different descriptions of the same dynamical system.Indeed, in both cases their trajectories in T ∗ Q project down to the same set oftrajectories in the physical space Q . Stated otherwise, the two sets of first-orderdifferential equations on T ∗ Q associated with e Γ (1) and e Γ (2) give rise to the same set of second-order differential equations on Q . Example 18
Let Γ represent, as in Sect.3.4.3, the dynamics of the two-dimensionalisotropic harmonic oscillator: Γ = u ∂∂q + u ∂∂q − q ∂∂u − q ∂∂u (3.177) and let, again in the notation of Sect.3.4.3, the two Lagrangians be: L (1) = L = L + L (the standard Lagrangian) and: L (2) = L ′ (cfr. Eq.(3.158)). Then,omitting unnecessary details, H (1) has the standard form: H (1) = 12 h(cid:0) p (cid:1) + (cid:0) p (cid:1) + (cid:0) q (cid:1) + (cid:0) q (cid:1) i (3.178) and: e Γ (1) = p ∂∂q + p ∂∂q − q ∂∂p − q ∂∂p (3.179) As to L (2) , we find instead: F L (2) : (cid:0) q , q , u , u (cid:1) (cid:0) q , q , p , p (cid:1) (3.180) and: e Γ (2) = p ∂∂q + p ∂∂q − q ∂∂p − q ∂∂p (3.181) with the Hamiltonian: H (2) = p p + q q (3.182) Concerning symmetries, while the Hamiltonian (3.178) is rotationally-invariantand we obtain, via N¨other’s theorem, the usual association of the angular mo-mentum with rotations, The Hamiltonian (3.182) is squeeze-invariant, thesqueeze transformation being generated by the cotangent lift of the vector field(3.160), i.e. : S ∗ = S − p ∂∂p + p ∂∂p (3.183) Now: i S ∗ ω = dF (3.184) where now the (Hamiltonian) constant of the motion is: F = q p − q p (3.185)62 hich, although it doesn’t look such at first sight, is again the (only componentof the) angular momentum,as: (cid:16) F L (2) (cid:17) ∗ F = q u − q u (3.186)The scheme (3.176) outlined above is not the only possible one, though. Wemight decide instead to perform the Legendre map by using only one of the twofiber derivatives in both cases. If we select, e.g., F L (1) , we obtain the followingscheme for the transition from T Q to T ∗ Q :Ω L (1) , dE L (1) ր Γ ց Ω L (2) , dE L (2) | {z } T Q F L (1) −→ ցր F L (1) −→ ω (1 , , dH (1 , ր e Γ (1) ց ω , dH (1) | {z } T ∗ Q (3.187)where now: ω (1 , = (cid:16) F L (1) (cid:17) ∗ Ω L (2) = (cid:16) F L (1) (cid:17) ∗ (cid:16) F L (2) (cid:17) ∗ ω = (cid:18) F L (2) ◦ (cid:16) F L (1) (cid:17) − (cid:19) ∗ ω (3.188)and similarly for H (1 , . Remark 19
If we forget about the ”
T Q part” of the scheme (3.187) and retainonly the ” T ∗ Q part”, we see that this procedure exhibits an example of a givendynamical system ( e Γ (1) ) on T ∗ Q that is bihamiltonian. Example 20
For the same system as in Example 18 above, e Γ (1) is again givenby Eq.(3.179), but we find instead: ω (1 , = dq ∧ dp + dq ∧ dp (3.189) while: H (1 , = p p + q q (3.190) as in the previous example. However, now: i S ∗ ω (1 , = q dp − p dq + p dq − q dp (3.191) and: d (cid:16) i S ∗ ω (1 , (cid:17) = L S ∗ ω (1 , = 2 (cid:0) dq ∧ dp − dq ∧ dp (cid:1) = 0 (3.192) Therefore, although: L S ∗ H (1 , = 0 and hence S ∗ is a symmetry for the Hamil-tonian H (1 , , it is not Hamiltonian with respect to the symplectic form ω (1 , ,and ceases therefore to be the generator of a N¨other symmetry.
63o conclude this Section, we would like to ”re-visit”, in the Hamiltonianformalism, the consequences of the use, for the isotropic harmonic oscillator, ofone of the Lagrangians (3.146), parametrized by the family of symmetric andnonsingular matrices: B = k B ij k .Let us specialize here too to n = 3. The canonical momenta are defined by: p i = B ij u j ⇔ u i = A ij p j , i = 1 , , A = (cid:13)(cid:13) A ij (cid:13)(cid:13) is the matrix inverse of B : A ij B jk = δ ik . The Hamiltonianis therefore: H = 12 (cid:0) A ij p i p j + B ij q i q j (cid:1) (3.194)while the three components of the angular momentum: J i = ε ijk q j u k are given,in the canonical formalism on T ∗ R , by: J i = ε ijk A kl q j p l (3.195)The J i ’s are of course constants of the motion, i.e.: { J i , H } = 0 , i = 1 , , { ., . } is the canonical Poisson bracket on T ∗ R . Now, some long butstraightforward algebra [164] shows that the Poisson brackets among the J i ’sare given by: { J h , J k } = ε hkr A rs J s (3.197)Eq.(3.197) defines a Lie algebra whose derived algebra is spanned by thevectors of the form: J hk =: ε hkr A rs J s . As the Ricci tensor is antisymmetric,there are only three independent such vectors and, as the matrix A is symmetric,they are independent. Therefore, the derived algebra is three-dimensional, andthe Lie algebra can be only [113] (apart from a sign) that of O (3) or that of O (2 , . Denoting by X i and X hk the associated Hamiltonian vector fields,defined by: i X i ω = dJ i ; i X hk ω = dJ hk (3.198)( X hk = ε hkr A rs X s ) which implies, in particular: L X i ω = 0, Eq.(3.196) isequivalent to the statement that: L X i H = 0. Hence (see Sect.3.4.2), the X i ’sare also symmetries for the dynamics. Moreover, using the identity (3.143), onesees at once that: i [ X h ,X k ] ω = −L X h i X k ω = −L X h dJ k = − d L X h J k (3.199)i.e. that: i [ X h ,X k ] ω = d { J h , J k } = dJ hk = i X hk ω (3.200)which implies in turn, as ω is nondegenerate:[ X h , X k ] = ε hkr A rs X s (3.201)Hence, the X i ’s generate the same algebra of symmetries (that of O (3) or thatof O (2 , These are called su (2) and su (1 ,
1) in Ref.[113], but the Lie algebras are isomorphic. Geometry of Quantum Mechanics and Alter-native Structures
Alternative descriptions for both classical and quantum systems have been dis-cussed already all along the previous Chapters. In particular, in Sect. 1.2we have discussed how one can obtain alternative descriptions both in theSchr¨odinger and Heisenberg pictures either by modifying the Hermitian struc-ture using constants of the motion (Sect. 1.2.1) or, in the infinite-dimensionalcase (Sect. 1.2.3) by changing the symplectic structure (as well as the Hamilto-nian) using powers of the original Hamiltonian.The discussion was carried on systematically within the framework of thedescription of states as vectors on some (finite- or infinite-dimensional) complexHilbert space H (with the associated Hermitian structure h . | . i ) and of observ-ables as self-adjoint linear operators on H .Hilbert spaces were introduced and used in a systematic way first by Dirac[56] as a consequence of the fact that one needs a superposition rule (and hencea linear structure) in order to accommodate a consistent description of theinterference phenomena that are fundamental for Quantum Mechanics. Par-enthetically, we should note that a complex Hilbert space carries with it in anatural way a ”complex structure” (multiplication of vectors by the imaginaryunit). The rˆole of the latter was discussed in the early Forties by Reichenbach[202]. Later on St¨uckelberg [217] emphasized the rˆole of the complex structurein deducing in a consistent way the uncertainty relations of Quantum Mechanics(see also the discussion in Refs.[69] and [170]).However, it is well known that a ”complete” measurement in Quantum Me-chanics (a simultaneous measurement of a complete set of commuting observ-ables [56, 69, 183]) does not provide us with an uniquely defined vector insome Hilbert space, but rather with a ”ray”, i.e. an equivalence class of vec-tors differing by multiplication through a nonzero complex number. Even fixingthe normalization, an overall phase will remain unobservable. Quotient-ing w.r.t. both multiplications leads, for a finite-dimensional Hilbert space H (dim C H = n ), to the following double fibration: R + −→ H = H− { }↓ U (1) −→ S n − ↓ P ( H ) (4.1) We will not worry at this stage about the technical complications that can arise, in theinfinite-dimensional case, when the spectrum of some observable has a continuum part. Not a relative phase in a superposition of vectors, of course. projective Hilbert space P H , and it is clear that: P ( H ) ≃ C P n − = { [ | ψ i ] : | ψ i , | ψ ′ i ∈ [ | ψ i ] ⇔ | ψ i = λ | ψ ′ i}| ψ i , | ψ ′ i ∈ H− { } , λ ∈ C = C − { }} (4.2)where [ | ψ i ] denotes the equivalence class to which | ψ i ∈ H belongs under mul-tiplication by a non-zero complex number. Remark 21
Notice that in this way the Hilbert space H acquires the structureof a principal fiber bundle [104, 167, 215], with base P H and typical fiber C . The self-duality of H determined by the Hermitian structure allows for the(unique) association of every equivalence class [ | ψ i ] with the rank-one projector: ρ ψ = | ψ ih ψ |h ψ | ψ i (4.3)with the known properties: ρ † ψ = ρ ψ T rρ ψ = 1 ρ ψ = ρ ψ (4.4)It is clear by construction that the association depends on the Hermitian struc-ture we consider.The space of rank-one projectors is usually denoted [86] as D ( H ). It isthen clear that in this way we can identify it with the projective Hilbert space P H . Hence, what the best of measurements will yield will be always (no moreand not less than) a rank-one projector (also called a pure state [95]).Also, transition probabilities that, together with the expectation valuesof self-adjoint linear operators that represent dynamical variables, are amongthe only observable quantities one can think of, will be insensitive to overallphases, i.e. they will depend only on the (rank-one) projectors associated withthe states. If A = A † is any such observable, then the expectation value h A i ψ inthe state | ψ i will be given by: h A i ψ = h ψ | A | ψ ih ψ | ψ i ≡ T r { ρ ψ A } (4.5)Transition probabilities are in turn expressed via a binary product that canbe defined on pure states. Again, if | ψ i and | φ i are any two states, then the(normalized) transition probability from | ψ i to | φ i will be given by: | h φ | ψ i | h ψ | ψ i h φ | φ i = T r { ρ ψ ρ φ } (4.6)and the trace on the r.h.s. of Eq.(4.12) will define the binary product amongpure states (but more on this shortly below).66t appears therefore that the most natural setting for Quantum Mechanicsis not primarily the Hilbert space itself but rather the projective Hilbert space,or, equivalently, the space of rank-one projectors D ( H ), whose convex hull willprovide us with the set of all density states. [222, 223, 71].On the other hand, the superposition rule, which leads to interference phe-nomena, remains one of the fundamental building blocks of Quantum Mechanics,one that, among other things, lies at the very heart of the modern formulationof Quantum Mechanics in terms of path integrals [29, 74, 75, 85], an approachthat goes actually back to earlier suggestions by Dirac [56, 57].To begin with, if we consider, for simplicity, two orthonormal states: | ψ i , | ψ i ∈ H , h ψ i | ψ j i = δ ij , i, j = 1 , ρ = | ψ ih ψ | , ρ = | ψ ih ψ | (4.8)a linear superposition with (complex) coefficients c and c with: | c | + | c | = 1will yield the normalized vector: | ψ i = c | ψ i + c | ψ i (4.9)and the associated projector: ρ ψ = | ψ ih ψ | = | c | ρ + | c | ρ + ( c c ∗ ρ + h.c. ) (4.10)where: ρ =: | ψ ih ψ | , which cannot however be expressed directly in terms ofthe initial projectors.A procedure to overcome this difficulty by retaining at the same time theinformation concerning the relative phase of the coefficients can be summarizedas follows [44, 144, 145, 146, 148, 170].Considering a third, fiducial vector | ψ i with the only requirement that it benot orthogonal neither to | ψ i nor to | ψ i , it is possible to associate normalizedvectors | φ i i with the projectors ρ i ( i = 1 ,
2) by setting: | φ i i = ρ i | ψ i p T r ( ρ i ρ ) , i = 1 , Remark 22
Note that, as all the ρ ’s involved are rank-one projectors : • T r ( ρ i ρ ) T r ( ρ j ρ ) = T r ( ρ i ρ ρ j ρ ) ∀ i, j (4.12) and that: In terms of the associated rank-one projections, we require:
T r ( ρ i ρ ) = 0 , i = 1 , , with: ρ = | ψ ih ψ | . The proof of Eqs.(4.12) and (4.13)is elementary and will not be given here. | φ i ih φ i | = ρ i ρ ρ i p T r ( ρ i ρ ρ i ρ ) ≡ ρ i , i = 1 , | φ i = c | φ i + c | φ i and the asso-ciated projector: ρ = | φ ih φ | , one finds easily, using also Eqs.(4.12) and (4.13),that: ρ = | c | ρ + | c | ρ + c c ∗ ρ ρ ρ + h.c. p T r ( ρ ρ ρ ρ ) (4.14)which can be written in a compact form as: ρ = X i,j =1 c i c ∗ j ρ i ρ ρ j p T r ( ρ i ρ ρ j ρ ) (4.15)The results (4.14) and (4.15) are now written entirely in terms of rank-oneprojectors. Thus, a superposition of rank-one projectors which yields anotherrank-one projector is possible, but requires the arbitrary choice of the fiducialprojector ρ . This procedure is equivalent to the introduction of a connectionon the bundle, usually called the Pancharatnam connection [185, 197]. Remark 23
If the (normalized) probabilities | c | and | c | are given, Eq.(4.9)describes a one-parameter family of linear superposition of states, and the samewill be true in the case of Eq.(4.14). Both families will be parametrized by therelative phase of the coefficients. Remark 24
Comparison of Eqs.(4.10) and (4.14) shows that, while the firsttwo terms on the r.h.s. of both are identical, the last terms of the two differ byan extra (fixed) phase, namely that: ρ ρ ρ p T r ( ρ ρ ρ ρ ) = ρ exp { i [arg ( h ψ | ψ i − arg ( h ψ | ψ i ))] } (4.16) Remark 25
The result of Eq.(4.15) can be generalized in an obvious way tothe case of an arbitrary number, say n , of orthonormal states none of which isorthogonal to the fiducial state. The corresponding family of rank-one projectorswill be parametrized in this case by the ( n − relative phases. If, now, we are given two (rank-one) projectors and only the relative prob-abilities are given, we are led to conclude that the system is described by theconvex combination (a rank-two density matrix): ρ = | c | ρ + | c | ρ , whichis again Hermitian and of trace one, but now: ρ − ρ > Or more, with an obvious generalization. P H by noticing that, if A = A † is an observable, and considering from now on only normalized vectors, theexpectation value (4.5) associates with the observable A a (real) functional on P H . The standard variational principle of Quantum Mechanics [69, 183] canbe rephrased [31, 44] by saying that the critical points of this functional arethe eigenprojectors of A and that the critical values yield the correspondingeigenvalues.Unitary (and, as a matter of fact, also anti-unitary ) operators play also arelevant rˆole in Quantum Mechanics [69, 183]. In particular, self-adjoint oper-ators can act as infinitesimal generators of one-parameter groups of unitaries.Both unitary and anti-unitary operators share the property of leaving all tran-sition probabilities invariant. At the level of the projective Hilbert space theyrepresent then isometries of the binary product (4.6). The converse is alsotrue. Indeed, it was proved long ago by Wigner [227, 230] that bijective mapson P H that preserve transition probabilities (i.e., isometries of the projectiveHilbert space) are associated with unitary or anti-unitary transformations onthe original Hilbert space . For a recent version of this theorem, see Ref.[87].To summarize the content of this Section, we have argued that all the rel-evant building blocks of Quantum Mechanics can be re-formulated in termsof parent objects that ”live” in the projective Hilbert space P H . The latter,however, is no more a linear vector space. As will be discussed in the follow-ing Sections, it carries instead a rich manifold structure. In this context, thevery notion of linear transformations looses meaning, and we are led in a naturalway to consider a non-linear manifold and (non-linear) diffeomorphisms thereof.This given, only objects that have a tensorial character will be allowed. We willhave then, as a preliminary step, to proceed to, so-to-speak, ”tensorialize” allthe notions that have been established in the context of the linear Hilbert space.We will do that in the second part of this Chapter, where we will discuss the ge-ometry of Quantum Mechanics. In the last part of the Chapter, having achievedthis goal, we will re-discuss the problem of alternative structures in the contextof Quantum Mechanics. Think of the operation [69, 183] of time-reversal. The association being up to a phase, this may lead to the appearance of ”ray” (or ”pro-jective”) representations [11, 69, 95, 132, 133, 183, 207] of unitary groups on the Hilbert spaceinstead of ordinary ones, a problem that we will not discuss here, though. .2 The Geometry of Quantum Mechanics We recall here some basic notions, in order mainly to fix the language andnotations to be employed in what follows.1. Given an n -dimensional vector space H over the field C of the complexnumbers, the realified [5] H R of H is a real vector space that coincides with H as a group (abelian group under addition) but in which only multipli-cation by real scalars is allowed. The realified of H can be constructedas follows. Let ( e , ..., e n ) be a basis for H . Then, a basis for H R willbe provided by ( e , ..., e n , ie , ..., ie n ) and H R ≈ R n . Once a basis hasbeen chosen, H ≈ C n . If: x = x k e k , x k = u k + iv k ; u k , v k ∈ R (in short: x = u + iv ; u, v ∈ R n ), then the corresponding vector in H R is representedby (cid:0) u , ..., u n , v , ..., v n (cid:1) , or ( u, v ), again for short, and it is immediateto check that the group property is satisfied. Let now: A : H → H be a linear operator on H . The realified of A will be the linear opera-tor: A R : H R → H R that coincides with A pointwise, i.e., if: Ax = x ′ , x = u + iv, x ′ = u ′ + iv ′ , then: A R ( u, v ) = ( u ′ , v ′ ). In any given basis for H , A will be represented by a matrix of the form: A = α + iβ , with α, β real n × n matrices. Then it is also immediate to check that A R will berepresented by the 2 n × n real matrix: A R = (cid:12)(cid:12)(cid:12)(cid:12) α − ββ α (cid:12)(cid:12)(cid:12)(cid:12) (4.17)It is also immediate to check that: ( A + B ) R = A R + B R , as well asthat: ( AB ) R = A R B R , and hence the set of the linear operators that arerealifications of complex operators on H is both a subspace of the vectorspace of all linear operators on H R as well as a subalgebra of the associativealgebra gl (2 n, R ). In particular, multiplication in H by the imaginary unitwill be represented by the linear operator: J = (cid:12)(cid:12)(cid:12)(cid:12) n × n − I n × n I n × n n × n (cid:12)(cid:12)(cid:12)(cid:12) (4.18)(or: ( u, v ) → ( − v, u )) with the property: J = − I n × n (4.19)2. A complex manifold [40, 210] is a manifold Z that can be locally modeledon C n for some n , and for which the chart-compatibility conditions arerequired to be C ω diffeomorphisms. Then, on the tangent bundle T Z onecan define the complex structure J via: J : T Z → T Z ; J ( v ) =: iv, v ∈ T Z. (4.20)Clearly: J = − I . Also: 70. An almost complex manifold [186] is an even-dimensional real manifold M endowed with a (1 , J , called an almost complexstructure, satisfying: J = − I (4.21)It was proved in Ref.[195] that an almost complex manifold becomes acomplex one iff the almost complex structure J satisfies the Nijenhuiscondition N J = 0, where N J is the Nijenhuis torsion associated with J .4. Finally, let K be a real, even-dimensional, manifold with a complex struc-ture and a closed two-form satisfying the compatibility condition: ω ( x, Jy ) + ω ( Jx, y ) = 0; x, y ∈ T K (4.22)Notice that this implies that: g ( ., . ) =: ω ( ., J ( . )) ; ( x, y ) g ( x, y ) =: ω ( x, Jy ) (4.23)is symmetric ( g ( x, y ) = g ( y, x ) ∀ x, y ) and nondegenerate iff ω is, hence ametric. When g is positive, then K is a K¨ahler manifold [40, 210, 224] .Also, J = − I implies: ω ( Jx, Jy ) = ω ( x, y ) ; g ( Jx, Jy ) = g ( x, y ) ∀ x, y (4.24)Notice that Eq.(4.23) implies the analog of Eq.(4.22) for g , namely: g ( x, Jy ) + g ( Jx, y ) = 0 (4.25)A tensorial triple ( g, J, ω ), with g a metric, J a complex structure and ω a symplectic structure satisfying the conditions (4.22),(4.23) and (4.24) will becalled an admissible triple. Eq.(4.23) and the parent equation, obtained bysubstituting: y → Jy in it tell us also that: ω ( ., . ) = − g ( ., J ( . )) (4.26)Coming back now to the complex vector space H , let it be endowed alsowith an Hermitian structure h ( ., . ) = h . | . i , i.e. a positive-definite sesquilinearform,nondegenerate, linear in the second factor and antilinear in the first one.Then H will become a (finite-dimensional: dim C H = n ) Hilbert space. Wewill keep denoting vectors in H with Latin letters (i.e.: x, y etc.) and we willuse Dirac’s notation ( | x i , | y i etc.) only when convenient. Separating real andimaginary parts, we can write: h ( x, y ) = g ( x, y ) + iω ( x, y ) g ( x, y ) = Re h ( x, y ) ω ( x, y ) = Im h ( x, y ) (4.27) If not, then K is also called [186] a pseudo-K¨ahler manifold. is clearly symmetric, positive and nondegenerate, while ω is antisymmetricand nondegenerate too.Now we can consider H R together with its tangent bundle T H R ≈ H R × H R .Points in H R , i.e. in the first factor, will be again denoted by the same Latinletters , and we will use Greek letters for the second factor. Then, e.g., ( x, ψ )will denote a point in H R and a tangent vector at x : ψ ∈ T x H R ≈ H R . We canassociate with every point x ∈ H R the constant vector field: X ψ =: ( x, ψ ) (4.28)Then, we can ”promote” g and ω to (0 ,
2) tensor fields by defining: g ( x ) ( X ψ , X φ ) =: g ( ψ, φ ) (4.29)and similarly for ω . In this way, g becomes a Riemannian metric and ω asymplectic structure. Proceeding in a similar way, we define: J ( x ) ( X ψ ) = ( x, Jψ ) (4.30)where: Jψ = iψ (i.e.: J ( u, v ) = ( − v, u )) and in this way J too is ”promoted”to a (1 ,
1) tensor field. As all these tensors fields are translationally invariant,and hence the Nijenhuis condition for J is trivially satisfied, and as all thecompatibility conditions are also satisfied, H R becomes in this way a linear K¨ahler manifold, with J playing the rˆole of the complex structure. Explicitly,if ( e , ..., e n ) is an orthonormal basis for H , and: x = ( u, v ) , y = ( u ′ , v ′ ), then: g ( x, y ) = u · u ′ + v · v ′ ω ( x, y ) = u · v ′ − v · u ′ (4.31)It may be convenient to give explicit expressions by introducing real coordi-nates x , ..., x n on H R ≈ R n . Then, e.g., g and J will be explicitly representedas: g = g ij dx i ⊗ dx j (4.32)and : J = J ij dx j ⊗ ∂∂x i (4.33)Hence: J = − I ⇐⇒ J i k J k j = − δ i j (4.34) With reference to a basis, x = u + iv will stand (see item 1 above) for the (real) pair ( u, v ) Here: Jx = n ( Jx ) i o n ; ( Jx ) i = J i j x j . emark 26 With the given metric, orthogonal matrices will be those leavingthe scalar product invariant, and they will provide a representation of O (2 n ) which need not be the standard one. Eq.(4.24) tells us that J is what we mightcall a ” g -orthogonal” matrix. In this context, it is worth recalling that the adjoint A † w.r.t. g of any linear operator A (a (1 , tensor)is defined by: g ( x, A y ) = g (cid:0) A † x, y (cid:1) (4.35) In terms of matrices: A † = g − e A g (4.36) where e A stands for the transpose matrix and hence, for a generic metric tensor,(real) symmetric matrices need not be self-adjoint. Eq.(4.25) tells us then that J is skew-adjoint w.r.t. g , i.e. that: J † = − J , which implies, according toEq.(1.46): J † J = I (4.37) Remark 27 ii ) If we consider a one-parameter group { exp ( t A ) } t ∈ R of g -orthogonalmatrices, then: g (cid:0) e t A x, e t A y (cid:1) = g ( x, y ) implies, at the infinitesimal level: g ( A x, y ) + g ( x, A y ) = 0 (4.38) Hence, J acts at the same time as a generator of finite and infinitesimalorthogonal transformations (rotations). iii ) in terms of the representative matrices, the condition g ( Jx, y )+ g ( x, Jy ) = 0 can be written as: e J ◦ g + g ◦ J = 0 (4.39) i.e., as g is symmetric: ^ ( g ◦ J ) = − g ◦ J , i.e. g ◦ J must be a skew-symmetricmatrix. Using g and J we can construct, as discussed before, the skew-symmetrictensor ω (cfr Eq.(4.26)). ω will be nondegenerate iff g is, hence a symplecticform. In terms of matrices: ω = − g ◦ J (4.40)( ω ij = − g ik J k j ), Moreover. Eqs.(4.24) and (4.22), i.e.: ω ( Jx, Jy ) = ω ( x, y ) ∀ x, y (4.41)and: ω ( Jx, y ) + ω ( x, Jy ) = 0 ∀ x, y (4.42)tell us that J will generate (both finite and infinitesimal) symplectic transfor-mations as well. Notice that, for y = Jx : ω ( x, Jx ) = g ( x, x ) (4.43)and hence: ω ( x, Jx ) > g is positive-definite.73ne could start instead from the datum of a symplectic form and of a com-plex structure, requiring the admissibility condition ω ( Jx, y ) + ω ( x, Jy ) = 0(which implies ω ( Jx, Jy ) = ω ( x, y ) and viceversa), and define then: g ( x, y ) =: ω ( x, Jy ) (4.44)( g = ω ◦ J in terms of representative matrices), the only difference being that,although g will be still nondegenerate iff ω is, it need not be positive unless ω ( x, Jx ) > ∀ x .Finally, one could start from g and ω and require the admissibility conditionthat: J =: g − ◦ ω be a complex structure,i.e.: J = − I . In conclusion, a thirdtensor is determined whenever any other admissible two are given. Remark 28
We have already encountered examples of admissible triples ( g, ω, J ) in Sect. 3.3. E.g., for the isotropic two-dimensional harmonic oscillator we mayconsider ( H , ω , J ) or ( H , ω , J ) as given in Eqns. (3.76) and (3.80), whilefor the one-dimensional harmonic oscillator we may choose (see again Sect.3.3) ( H, ω, J ) or H ′ , ω ′ , J ′ ) , as long as the Hamiltonian is positive definite. Here and in the following we will exploit the already-discussed connection be-tween the space P ( H ) of rays and the space D ( H ) of density states of rank oneto see how it is possible to use symplectic methods to study quantum systems.This geometric approach is based on some observations that will be developedin the following.We have just proved that the realification H R of the Hilbert space H (thespace of states) is a linear K¨alher manifold, equipped with an admissible triple( J, g, ω ). Now, taking into account that P ( H ) is not a linear space, we willhave to use a tensorial description of these structures. Via a momentum mapon P ( H ) that we shall define shortly below, the space of Hermitian operators(the observables) will be identified with the dual u ∗ ( H ) of the Lie algebra ofthe unitary group U ( H ), which can be thought of as the intersection of the Liealgebras of the symplectic and orthogonal groups. By exploiting the fact thatthe action of the latter is Hamiltonian, we will use the momentum map to definecontravariant metric and Poisson tensors on u ∗ ( H ). Finally we will study howthese structures behave under the U ( H )-action on u ∗ ( H ) and see how D ( H )itself becomes a K¨alher manifold. We have seen how we can construct the tensor fields g, J and ω on T H R . The(0 , g and ω define maps from T H R to T ∗ H R . The two being bothnon-degenerate, we can also consider their inverses, i.e. the (2 ,
0) contravariant74ensors G (a metric tensor) and Λ (a Poisson tensor) mapping T ∗ H R to T H R and such that: G ◦ g = Λ ◦ ω = I T H R (4.45)i.e., in short: G = g − , Λ = ω − . G and Λ can be used together to define anHermitian product between any two α, β in the dual H ∗ R equipped with the dualcomplex structure J ∗ : h α, β i H ∗ R = G ( α, β ) + i Λ( α, β ) . (4.46)This induces two (non-associative) real brackets on smooth, real-valued func-tions on H R : • the (symmetric) Jordan bracket { f, h } g =: G ( df, dh ), and: • the (antisymmetric) Poisson bracket { f, h } ω =: Λ( df, dh ).By extending both these brackets to complex functions via complex linearitywe obtain eventually a complex bracket { ., . } H defined as: { f, h } H = h df, dh i H ∗ R =: { f, h } g + i { f, h } ω . (4.47)To make these structures more explicit, we may introduce an orthonormalbasis { e k } k =1 , ··· ,n in H and global coordinates ( q k , p k ) for k = 1 , · · · , n on H R defined as h e k , x i = ( q k + ip k )( x ) , ∀ x ∈ H . (4.48)Then : J = dp k ⊗ ∂∂q k − dq k ⊗ ∂∂p k (4.49) g =: dq k ⊗ dq k + dp k ⊗ dp k (4.50) ω =: dq k ⊗ dp k − dp k ⊗ dq k (4.51)as well as: G = ∂∂q k ⊗ ∂∂q k + ∂∂p k ⊗ ∂∂p k (4.52)Λ = ∂∂p k ⊗ ∂∂q k − ∂∂q k ⊗ ∂∂p k (4.53)and hence: { f, h } g = ∂f∂q k ∂h∂q k + ∂f∂p k ∂h∂p k (4.54) { f, h } ω = ∂f∂p k ∂h∂q k − ∂f∂q k ∂h∂p k (4.55)Introducing complex coordinates: z k =: q k + ip k , ¯ z k =: q k − ip k , we can alsowrite G + i · Λ = 4 ∂∂z k ⊗ ∂∂ ¯ z k , (4.56) Which will act (see Footnote 59) via the transpose matrix of J . Summation over repeated indices being understood here and in the rest of the Section. ∂∂z k =: 12 (cid:18) ∂∂q k − i ∂∂p k (cid:19) , ∂∂ ¯ z k =: 12 (cid:18) ∂∂q k + i ∂∂p k (cid:19) . (4.57)Complex coordinates are employed here and also elsewhere in this paper onlyas a convenient shorthand or as a stenographic notation. Their use does notmean at all that vector fields like those in Eq.(4.57) should operate on functionsthat are holomorphic (or anti-holomorphic) in the z k ’s. They must rather beseen as complex-valued vector fields that operate on (smooth) complex-valuedfunctions defined on a real differentiable manifold.With this in mind, we have : { f, h } H = 4 ∂f∂z k ∂h∂ ¯ z k , (4.58)or, in more detail: { f, h } g = 2 (cid:18) ∂f∂z k ∂h∂ ¯ z k + ∂h∂z k ∂f∂ ¯ z k (cid:19) ; { f, h } ω = 2 i (cid:18) ∂f∂z k ∂h∂ ¯ z k − ∂h∂z k ∂f∂ ¯ z k (cid:19) (4.59)Notice also that: J = − i (cid:18) dz k ⊗ ∂∂z k − d ¯ z k ⊗ ∂∂ ¯ z k (cid:19) (4.60)In particular, for any A ∈ gl ( H ) we can define the quadratic function: f A ( x ) = 12 h x, Ax i = 12 z † Az (4.61)where z is the column vector ( z , ..., z n ). It follows immediately from Eq.(4.59)that, for any A, B ∈ gl ( H ): { f A , f B } g = f AB + BA (4.62) { f A , f B } ω = f AB − BAi (4.63)So, the Jordan bracket of any two quadratic functions f A and f B is related tothe (commutative) Jordan bracket of A and B , [ A, B ] + , defined as:[ A, B ] + =: AB + BA (4.64)while their Poisson bracket is related to the commutator product ( the Liebracket) [ A, B ] − defined as:[ A, B ] − =: 1 i ( AB − BA ) (4.65) This is actually twice the Jordan Bracket as it is usually defined in the literature [65], butwe find here more convenient to employ this slightly different definition.
76n particular, if A and B are Hermitian, their Jordan product (4.64) and theirLie bracket will be Hermitian as well. Hence, the set of Hermitian operatorson H R , equipped with the binary operations (4.64) and (4.65), becomes a Lie-Jordan algebra [65, 108, 109], and the binary product [65]:(
A, B ) = 12 (cid:0) [ A, B ] + + i [ A, B ] − (cid:1) (4.66)is an associative product (Indeed: ( A, B ) ≡ AB ). We remark parentheticallythat all this extends without modifications [65] to the infinite-dimensional case,if we assume: A, B ∈ B sa ( H ), the set of bounded self-adjoint operators on theHilbert space H .Coming back to quadratic functions, it is not hard to check that: { f A , f B } H = 2 f AB , (4.67)which proves the associativity of the bracket (4.47) on quadratic functions, i.e.: {{ f A , f B } H , f C } H = { f A , { f B , f C } H } H = 4 f ABC , ∀ A, B, C ∈ gl ( H ) . (4.68)We look now at real, smooth functions on H R .First of all, it is clear that f A will be a real function iff A is Hermitian. TheJordan and Poisson brackets will define then a Lie-Jordan algebra structure onthe set of real, quadratic functions, and, according to Eq.(4.68), the bracket {· , ·} H will be an associative bracket.For any such f ∈ F ( H R ) we may define two vector fields, the gradient ∇ f of f and the Hamiltonian vector field X f associated with f , defined by: g ( · , ∇ f ) = dfω ( · , X f ) = df or G ( · , df ) = ∇ f, Λ( · , df ) = X f . (4.69)which allow us also to obtain the Jordan and the Poisson brackets as: { f, h } g = g ( ∇ f, ∇ h ) , (4.70) { f, h } ω = ω ( X f , X h ) . (4.71)Explicitly, in coordinates: ∇ f = ∂f∂q k ∂∂q k + ∂f∂p k ∂∂p k = 2 (cid:18) ∂f∂z k ∂∂ ¯ z k + ∂f∂ ¯ z k ∂∂z k (cid:19) (4.72) X f = ∂f∂p k ∂∂q k − ∂f∂q k ∂∂p k = 2 i (cid:18) ∂f∂z k ∂∂ ¯ z k − ∂f∂ ¯ z k ∂∂z k (cid:19) (4.73)which are such that J ( ∇ f ) = X f .Turning to linear operators, to any A : H → H we can associate:1. A quadratic function as in Eq. (4.61), and (cfr. also below, Sect.4.4),2. A vector field: X A : H → T H via: x ( x, Ax ) , and:77. A (1 ,
1) tensor field: T A : T x H ∋ ( x, y ) ( x, Ay ) ∈ T x H . Clearly, asalready remarked, f A is real if and only if A is Hermitian. In this case: ∇ f A = X A (4.74)and: X f A = J ( X A ) (4.75)Indeed, denoting with ( · , · ) the pairing between vectors and covectors,Eq.(4.74) holds because: g ( y, X A ( x )) = g ( y, Ax ) = 12 ( h y, Ax i H + h Ax, y i H ) == ( df A ( x ) , y ) (4.76)while Eq.(4.75) follows from the second expression in Eq.(4.23), i.e. from: g ( y, Ax ) = ω ( y, ( JX A )( x )) = ω ( y, iAx ). (cid:4) Thus, we will write: ∇ f A = A and : X f A = iA (4.77)In particular, if we consider the identity operator I , we obtain the dilation(or Liouville ) field (cfr. also Eq.(4.28)):∆ : x ( x, x ) (4.78)or, in real coordinates: ∆ = q k ∂∂q k + p k ∂∂p k (4.79)which is such that: X A = T A (∆) . (4.80)Finally we can also define the phase vector field :Γ = J (∆) = p k ∂∂q k − q k ∂∂p k (4.81)that will be considered in the next Section. We would like now to discuss in some detail the structure of the complex pro-jective Hilbert space P H , which, as we have already mentioned, represents theright context to describe a geometric formulation of Quantum Mechanics. In-deed, given any vector | x i ∈ H− { } , the corresponding element in P H maybe represented by the rank-one projector: b ρ x =: | x ih x | / h x | x i in D ( H ) (orsimply: b ρ x =: | x ih x | if the vector is already normalized), and this will encodeall the relevant physical information contained in | x i .78n more geometric terms, we can consider the distribution generated by thedilation field ∆ and the phase field Γ = J (∆), which is involutive as [∆ , J (∆)] =0. Going to the quotient with respect to the foliation associated with thisdistribution (cfr.Eq.(4.1)) will be a way of generating the ray space P H whichis independent on any Hermitian structure. Contravariant tensorial objects on H will ”pass to the quotient” (i.e. will be projectable) if and only if they areleft invariant by both ∆ and Γ, i.e. if they are homogeneous of degree zeroand invariant under multiplication of vectors by a phase. Typical quadraticfunctions that ”pass to the quotient” will be normalized expectation values ofthe form: ρ x ( A ) =: T r { b ρ x A } = h x | A | x ih x | x i (4.82)with A any linear operator and for any Hermitian structure on H . We noteparenthetically that the subalgebra of functions on H that are invariant underΓ and ∆ will define, via the construction of the Gel’fand-Kolmogoroff theorem[157], a manifold which can again be identified with P H .Concerning projectability of tensors, the complex structure J , being (cfr.,e.g., Eq.(4.60)) homogeneous of degree zero and phase-invariant, will be a pro-jectable tensor, while it is clear that the Jordan and Poisson tensors G and Λdefined respectively in Eq.(4.52) or, for that matter, the complex-valued tensorof Eq.(4.56) will not be projectable (as they are phase-invariant but homoge-neous of degree − θ ( z ) =: z † z , thus defining newtensors: e Λ ( z ) =: θ ( z ) Λ ( z ) (4.83)and similarly for G .Let us examine these structures directly on P H more closely . Recall that,in the finite dimensional case, P H is homeomorphic to CP n and it is thereforemade up of the equivalence classes of vectors Z = ( Z , Z , · · · , Z n ) ∈ C n +1 w.r.t. the equivalence relation Z ≈ λZ ; λ ∈ C − { } . The space CP n is a K¨ahlermanifold when endowed with the Fubini-Study metric [19, 105], whose pull-backto C n +1 is given by: g F S = 1( Z · ¯ Z ) (cid:2) ( Z · ¯ Z ) d Z ⊗ S d ¯ Z − ( d Z · ¯ Z ) ⊗ S ( Z · d ¯ Z ) (cid:3) (4.84)where Z · ¯ Z = Z a ¯ Z a , d Z · ¯ Z = dZ a ¯ Z a , d Z ⊗ S d ¯ Z = dZ a d ¯ Z a + d ¯ Z a dZ a , andso on (the sum over repeated indices has to be understood), together with thecompatible symplectic form: ω F S = i ( Z · ¯ Z ) (cid:2) ( Z · ¯ Z ) d Z ∧ d ¯ Z − ( d Z · ¯ Z ) ∧ ( Z · d ¯ Z ) (cid:3) = dθ F S (4.85)where: θ F S = 12 i Z d Z − Z d ZZ · Z (4.86) In the following of this Section, we will use the (0 , g, ω instead of their (inverse)(2 , G, Λ since calculations result to be more easily performed. Z a = iA ab Z b (4.87)where A = [ A ab ] is a Hermitian matrix. These are the equations for the flow ofa generic Killing vector field, which therefore has the form : X A = ˙ Z a ∂ Z a − ˙¯ Z a ∂ ¯ Z a = iA ab ( Z b ∂ Z a − ¯ Z a ∂ ¯ Z b ) (4.88)A straightforward calculation shows that: ω F S ( · , X A ) = 1 Z · ¯ Z [ d ¯ Z a A ab Z b + ¯ Z a A ab dZ b ] − ¯ Z a A ab Z b ( Z · ¯ Z ) [ dZ c ¯ Z c + Z c d ¯ Z c ] == d ( i X A θ F S ) (4.89)i.e. that X A is the Hamiltonian vector field X f A , ω F S ( · , X f A ) = df A associatedwith the (real) quadratic function: f A = ¯ Z · A Z b Z · ¯ Z = ¯ Z a A ab Z b Z c ¯ Z c = i X A θ F S (4.90)for the Hermitian matrix A . Also, some algebra shows that, given any two realquadratic functions f A , f B ( A, B being Hermitian matrices), their correspondingHamiltonian vector fields satisfy: ω F S ( X f A , X f B ) = X f A ( df B ) = f AB − BAi (4.91)Therefore, the Poisson brackets associated with the symplectic form: { f, g } ω FS := − ω ( X f , X g ) (4.92)are such that: { f A , f B } ω FS = f AB − BAi (4.93)In a similar way, one can prove that the gradient vector field ∇ f A , g F S ( · , ∇ f A ) = df A , of f A has the form: ∇ A = A ab ( Z b ∂ Z a + ¯ Z a ∂ ¯ Z b ) (4.94)so that g F S ( ∇ f A , ∇ f B ) = ∇ f A ( df B ) = f AB + BA − f A · f B (4.95)Given any two real quadratic functions f A , f B , we can therefore define a Jordanbracket by setting: { f A , f B } g := g F S ( ∇ f A , ∇ f B ) + f A · f B = f AB + BA (4.96) Notice that these are exactly the Killing vector fields of S n +1 . In particular, for A = I weobtain X k = Γ which is a vertical vector field w.r.t. the Hopf projection π H : S n +1 → CP n . P H is K¨ahlerian iff its Hamiltonianvector field is also Killing. Such functions represent quantum observables. Theabove calculations show that the space F ( P H ) of real quadratic functions on P H consists exactly of all K¨ahlerian functions. To extend this concept to thecomplex case, one says that a complex valued function on P H is K¨ahlerian iffare so its real and imaginary parts. Clearly, any such f is a quadratic function ofthe form (4.90) with now A ∈ B ( H ). Also, on the space, F C ( P H ), of K¨ahleriancomplex functions one can define both an Hermitian two-form: h ( · , · ) = g F S ( · , · ) + iω F S ( · , · ) (4.97)and and associative bilinear product (star-product) via: f ⋆ g := f · g + 12 h ( df, dg ) = 12 [ { f, g } g + i { f, g } ω ] + f · g (4.98)under which the space F C ( P H ) is closed since f A ⋆ f B = f AB , thus obtaining aparticular realization of the C ∗ -algebra of bounded operators B ( H ).Let us suppose now that ( M , ˜ h ) be a generic K¨ahler manifold. Also in thisgeneric case, given any two functions f, g in the space of K¨ahlerian (w.r.t. themetric ˜ g = Re (˜ h )) complex functions F C ( M ) one can define a ⋆ -product: f ⋆ g := f · g + 12 ˜ h ( df, dg ) (4.99)but now this product, although inner, will be not in general associative unlessthe functions are K¨ahlerian.The condition that F C ( M ) be closed puts veryrestrictive conditions on the K¨ahler structure of M which imply [43] that M bea projective Hilbert space P H . At the end of Sect. (4.3), after the discussion ofthe so called GNS construction, we will see how realizations of a C ∗ -algebra asbounded operators on a suitable Hilbert space are in one-to-one correspondencewith the action of the unitary group on the K¨ahler manifold. We shall consider now the action of the unitary group U ( H ) on H , which isthe group of linear transformations that preserve the triple ( g, ω, J ). In thefollowing, we will denote with u ( H ) the Lie algebra of U ( H ) of anti-Hermitianoperators and identify the space of all Hermitian operators with the dual u ∗ ( H )of u ( H ) via the pairing: h A, T i =: i T r ( AT ) , A ∈ u ∗ ( H ) , T ∈ u ( H ) (4.100)On u ∗ ( H ) we can define a Lie bracket (cfr.also Sect.4.2.3):[ A, B ] − =: 1 i ( AB − BA ) , (4.101)81ith respect to which it becomes a Lie algebra, and also a Jordan bracket:[ A, B ] + =: AB + BA. (4.102)with the two together giving u ∗ ( H ) the structure of a Lie-Jordan algebra [65].In addition, u ∗ ( H ) is equipped with the scalar product h A, B i u ∗ = 12 T r ( AB ) (4.103)which satisfies: h [ A, ξ ] − , B i u ∗ = 12 T r ([ A, ξ ] − B ) = 12 T r ( A, [ ξ, B ] − ) = h A, [ ξ, B ] − i u ∗ (4.104) h [ A, ξ ] + , B i u ∗ = 12 T r ([ A, ξ ] + B ) = 12 T r ( A, [ ξ, B ] + ) = h A, [ ξ, B ] + i u ∗ (4.105)With any A ∈ u ∗ ( H ), we can associate the fundamental vector field X A on the Hilbert space corresponding to the element i A ∈ u ( H ) defined by theformula: ddt e − ti A ( x ) | t =0 = iA ( x ) , ∀ x ∈ H (4.106)In other words, X A = iA . We already know from Sect. 4.2.3 that iA has f A asits Hamiltonian function: ω ( · , X A ) = df A . Thus, for any x ∈ H R we obtain a µ ( x ) ∈ u ∗ ( H ) such that: h µ ( x ) , i A i = f A ( x ) = 12 h x, Ax i H (4.107)In such a way we obtain a mapping: µ : H R → u ∗ ( H ) (4.108)which is called the momentum map [167].More explicitly, it follows from Eq.(4.100) that: h µ ( x ) , i A i = 12 T r ( µ ( x ) A ) (4.109)which, when compared with Eq.(4.107), yields: µ ( x ) = | x ih x | (4.110)We may therefore conclude that the unit sphere in H can be projected onto u ∗ ( H ) in an equivariant way with respect to the coadjoint action of U ( H ). Also,in finite dimensions, the unit sphere is odd dimensional and the orbit in u ∗ ( H )is symplectic.With every A ∈ u ∗ ( H ) we can associate, with the by now familiar identifi-cation (as with every other linear vector space) of the tangent space at everypoint of u ∗ ( H ) with u ∗ ( H ) itself, the linear function (hence a one-form) ˆ A :82 ∗ ( H ) → R defined as: ˆ A =: h A, ·i u ∗ . Then, we can define two contravarianttensors, a symmetric (Jordan) tensor: R ( ˆ A, ˆ B ) ( ξ ) =: h ξ, [ A, B ] + i u ∗ (4.111)and a Poisson (Konstant-Kirillov-Souriau [113, 114, 115, 214]) tensor: I ( ˆ A, ˆ B ) ( ξ ) = h ξ, [ A, B ] − i u ∗ (4.112)( A, B, ξ ∈ u ∗ ( H )). We notice that the quadratic function f A is the pull-back ofˆ A via the momentum map since, for all x ∈ H : µ ∗ ( ˆ A )( x ) = ˆ A ◦ µ ( x ) = h A, µ ( x ) i u ∗ = 12 h x, Ax i H = f A ( x ) (4.113)This means also that, if: ξ = µ ( x ):( µ ∗ G )( ˆ A, ˆ B ) ( ξ ) = G ( df A , df B ) ( x ) = { f A , f B } g ( x ) = f [ A,B ] + ( x ) = R ( ˆ A, ˆ B ) ( ξ )(4.114)where the last equality follows from Eq.(4.62), i.e.: µ ∗ G = R (4.115)Similarly, by using now Eq.(4.63), we find:( µ ∗ Λ)( ˆ A, ˆ B ) ( ξ ) = Λ( df A , df B ) ( x ) = { f A , f B } ω ( x ) = f [ A,B ] − ( x ) = I ( ˆ A, ˆ B ) ( ξ )(4.116)i.e.: µ ∗ Λ = I (4.117)Thus, the momentum map relates the contravariant metric tensor G and thePoisson tensor Λ with the corresponding contravariant tensors R and I . To-gether they form the complex tensor:( R + iI )( ˆ A, ˆ B ) ( ξ ) = 2 h ξ, AB i u ∗ (4.118)which is related to the Hermitian product on u ∗ ( H ). Example 29
Let H = C (the Hilbert space appropriate for a two-level system).We can write any A ∈ u ∗ ( C ) as: A = y I + y · σ (4.119) where I is the × identity, y · σ = y σ + y σ + y σ and: σ =( σ , σ , σ ) arethe Pauli matrices: σ = (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) , σ = (cid:12)(cid:12)(cid:12)(cid:12) − ii (cid:12)(cid:12)(cid:12)(cid:12) , σ = (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) (4.120)83 ith the well-known identities [183]: σ h σ k = δ hk I + iε hkl σ l (4.121)( h, k, l = 1 , , and: σ j σ k σ l = iε jkl I + σ j δ kl − σ k δ jl + σ l δ jk (4.122) Every A ∈ u ∗ ( C ) is then represented by the (real) ”four-vector” (cid:0) y A , y A (cid:1) , and: y A = 12 T r ( A ) ; y kA = 12 T r ( σ k A ) ; k = 1 , , or, in short: y µ ( A ) = h A | σ µ i , µ = 0 , , , , σ = I (4.124) Digression.
Rank-one projectors (cid:0) A = ρ, ρ † = ρ, T rρ = 1 , ρ = ρ (cid:1) can be parametrizedas [187]: ρ = ρ ( θ, φ ) = (cid:12)(cid:12)(cid:12)(cid:12) sin θ e iφ sin θ e − iφ sin θ cos θ (cid:12)(cid:12)(cid:12)(cid:12) ; 0 ≤ θ < π, ≤ φ < π (4.125)Then, they correspond to: y = 12 , y = 12 sin θ cos φ, y = −
12 sin θ sin φ, y = −
12 cos θ (4.126)(hence: y = 1 / A ≡ (cid:0) y A , y A (cid:1) the vector field: y ( A ) ∂ + y ( A ) ∂ + y ( A ) ∂ + y ( A ) ∂ (cid:0) ∂ = ∂/∂y and so on (cid:1) . Also (see the discussion immedi-ately above Eq.(4.111)), ˆ A = h A, ·i u ∗ will be represented by the one-form:ˆ A = y ( A ) dy + y ( A ) dy + y ( A ) dy + y ( A ) dy (4.127)Using then Eq.(4.119) one proves easily that: AB = (cid:0) y A y B + y A · y B (cid:1) I + (cid:0) y A y B + y B y A + i y A × y B (cid:1) · σ (4.128)(with ” × ” denoting the standard cross-product of three-vectors) and hence : h AB i u ∗ = 12 T r ( AB ) = y A y B + y A · y B (4.129)Moreover: In particular: h ρ ( θ, φ ) ρ ( θ ′ , φ ′ ) i u ∗ = { θ sin θ ′ cos ( φ − φ ′ ) + cos θ cos θ ′ } / A, B ] + = 2 (cid:8)(cid:0) y A y B + y A · y B (cid:1) I + ( y A y B + y B y A ) · σ (cid:9) (4.130)while: [ A, B ] − = 2 y A × y B · σ (4.131)Then: R ( ˆ A, ˆ B ) ( ξ ) = h ξ, [ A, B ] + i u ∗ = (cid:10) [ ξ, A ] + , B (cid:11) =2 ξ (cid:0) y A y B + y A · y B (cid:1) + 2 (cid:0) y A y B + y B y A (cid:1) · ξ =2 (cid:0) y A ξ + y A · ξ (cid:1) y B + 2 (cid:0) y A ξ + ξ y A (cid:1) · y B (4.132)and hence, explicitly [86]: R ( ξ ) = 2 ∂ ⊗ (cid:0) ξ ∂ + ξ ∂ + ξ ∂ (cid:1) + 2 (cid:0) ξ ∂ + ξ ∂ + ξ ∂ (cid:1) ⊗ ∂ +2 ξ ( ∂ ⊗ ∂ + ∂ ⊗ ∂ + ∂ ⊗ ∂ + ∂ ⊗ ∂ ) (4.133)Quite similarly, one finds: I ( ˆ A, ˆ B ) ( ξ ) = 2( ξ × y A ) · y B = 2( y A × y B ) · ξ (4.134)and: I ( ξ ) = 2 (cid:0) ξ ∂ ∧ ∂ + ξ ∂ ∧ ∂ + ξ ∂ ∧ ∂ (cid:1) (4.135)We thus find the following tensor: R + iI = 2 [ ∂ ⊗ y k ∂ k + y k ∂ k ⊗ ∂ + y ( ∂ ⊗ ∂ + ∂ k ⊗ ∂ k ) + iǫ hkl y h ∂ k ⊗ ∂ l (cid:3) (4.136)To conclude this Section, we define also two (1 ,
1) tensors, e R and e J : T u ∗ ( H ) → T u ∗ ( H ) that will be employed below in Sect.4.2.6 via: e R ξ ( A ) =: [ ξ, A ] + = R (cid:16) b A, . (cid:17) ( ξ ) (4.137)and: e J ξ ( A ) =: [ ξ, A ] − = I (cid:16) b A, . (cid:17) ( ξ ) (4.138)for any A ∈ T ξ u ∗ ( H ) ≈ u ∗ ( H ), the last passage in both equations followingfrom Eqns.(4.104) and (4.105).In the previous example ( H ≈ C ) we find explicitly, in coordinates: e R ξ ( A ) = 2 (cid:0) y A ξ + y A · ξ (cid:1) ∂ + 2 (cid:0) y A ξ i + ξ y iA (cid:1) ∂ i (4.139)or: e R ξ = 2 (cid:0) ξ dy + ξ · dy (cid:1) ⊗ ∂ + 2 (cid:0) ξ i dy + ξ dy i (cid:1) ⊗ ∂ i (4.140)and: e J ξ ( A ) = 2 ε ijk ξ i y jA ∂ k (4.141)or: e J ξ = 2 ε ijk ξ i dy j ⊗ ∂ k (4.142)85 .2.6 The space of density states We have seen in Sect. 4.2.4 that it is possible to obtain P ( H ) as a quotientof H − { } with respect to the involutive distribution associated with ∆ and J (∆). Eq. (4.110) shows that the image of H − { } under the momentum mapconsists of the set of all non-negative Hermitian operators of rank one, that willbe denoted as P ( H ), i.e. : P ( H ) = {| x ih x | ; x ∈ H , x = 0 } (4.143)On the other hand, the coadjoint action of U ( H ): ( U, ρ ) U ρU † ( ρ ∈P ( H ) , U ∈ U ( H )) foliates P ( H ) into the spaces D r ( H ) = {| x ih x | : h x, x i H = r } . In particular we have already denoted with D ( H ) the space of one-dimensionalprojection operators, which is the image via the momentum map of the sphere S H = { x ∈ H ; h x, x i H = 1 } and can be identified with the complex projectivespace P ( H ) via the identification:[ x ] ∈ P ( H ) ↔ | x ih x |h x, x i ∈ D ( H ) (4.144)We have also argued that P ( H ) is a K¨ahler manifold. In the following we willexamine this fact in more detail, by showing explicitly that D ( H ) is a K¨ahlermanifold.Let ξ ∈ u ∗ ( H ) be the image through the momentum map of a unit vector x ∈ S H , i.e. ξ = | x ih x | with h x | x i = 1, so that ξ = ξ . The tangent space of thecoadjoint U ( H )-orbit at ξ is generated by vectors of the form [ A, ξ ] − , for anyHermitian A . From Eq.(4.104), it follows that the Poisson tensor I defined in(4.112) satisfies: I ( ˆ A, ˆ B ) ( ξ ) = h ξ, [ A, B ] − i u ∗ = h [ ξ, A ] − , B i u ∗ (4.145)This defines an invertible map ˜ I that associates to any one-form ˆ A the tangentvector at ξ : ˜ I ( ˆ A ) =: I ( ˆ A, · ) = [ ξ, A ] − . We will denote with ˜ η ξ its inverse:˜ η ξ ([ ξ, A ] − ) = ˆ A . This allows us to define, on u ∗ ( H ), a canonical two-form whichis given by: η ξ ([ A, ξ ] − , [ B, ξ ] − ) =: (˜ η ξ ([ ξ, A ] − ) , [ B, ξ ] − ) = ( ˆ A, [ B, ξ ] − ) (4.146)for all [ A, ξ ] − , [ B, ξ ] − ∈ T ξ u ∗ ( H ).It is also easy to check that η satisfies the equalities: η ξ ([ A, ξ ] − , [ B, ξ ] − ) = − (cid:16) ˆ A, [ B, ξ ] − (cid:17) = −h A, [ B, ξ ] − i u ∗ = −h ξ, [ A, B ] − i u ∗ = h [ A, ξ ] − , B i u ∗ , for any A, B ∈ u ∗ ( H ).We can summarize these results in the following: Note that here the vectors are not necessarily normalized. heorem 30 The restriction of the two-form (4.146) to the U ( H ) -orbit D ( H ) defines a canonical symplectic form η characterized by the property η ξ ([ A, ξ ] − , [ B, ξ ] − ) = (cid:10) [ A, ξ ] − , B (cid:11) u ∗ = −h ξ, [ A, B ] − i u ∗ (4.147)In a very similar way, starting from the symmetric Jordan tensor R givenin (4.111) , one can construct a (1 ,
1) tensor ˜ R ( ˆ A ) =: R ( ˆ A, · ) = [ ξ, A ] + and itsinverse: ˜ σ ([ ξ, A ] + ) = ˆ A . Thus we obtain a covariant tensor σ such that: σ ξ ([ A, ξ ] + , [ B, ξ ] + ) = h [ A, ξ ] + , B i u ∗ = h ξ, [ A, B ] + i u ∗ . (4.148)Notice that, at this stage, σ ξ is only a partial tensor, being defined on vectors ofthe form [ A, ξ ] + , which belong to the image of the map ˜ R . However, on D ( H ),we have [ A, ξ ] − = [ A, ξ ] − = [[ A, ξ ] , ξ ] + , so that, after some algebra, one canalso prove that: σ ξ ([ A, ξ ] − , [ B, ξ ] − ) = σ ξ ([[ A, ξ ] − , ξ ] + , [[ B, ξ ] − , ξ ] + ) = h ξ, [[ A, ξ ] − , [ B, ξ ] − ] + i u ∗ == T r ( ξ [[ A, ξ ] − , [ B, ξ ] − ] + ) = T r ( ξ [ A, ξ ] − [ B, ξ ] − ) = h [ A, ξ ] − , [ B, ξ ] − i u ∗ .Therefore we have also the following: Theorem 31
On the U ( H ) -orbit D ( H ) we can define a symmetric covarianttensor σ such that: σ ξ ([ A, ξ ] − , [ B, ξ ] − ) = h [ A, ξ ] − , [ B, ξ ] − i u ∗ . (4.149) holds. Moreover, going back to the the (1 ,
1) tensor ˜ I given above, one has thefollowing result [86]: Theorem 32
When restricted to D ( H ) , the (1 , tensor ˜ I , which satisfies: ˜ I = − ˜ I (4.150) will become invertible. Hence: ˜ I = − I and therefore it will define a complexstructure such that: η ξ ([ A, ξ ] − , ξ ([ B, ξ ] − )) = σ ξ ([ A, ξ ] − , [ B, ξ ] − ) (4.151) η ξ ( ξ ([ A, ξ ] − ) , ξ ([ B, ξ ] − )) = η ξ ([ A, ξ ] − , [ B, ξ ] − ) (4.152)Eq. (4.150) follows from a direct calculation by taking into account that ξ = ξ . The last two expressions follow by combining Eqs.(4.147) and (4.149). Toprove that is a complex structure one has first to show that it defines an almostcomplex structure (which follows easily from the fact that [[[ A, ξ ] − , ξ ] − , ξ ] − = − [ A, ξ ] − ) and then that its Nijenhuis torsion vanishes. Detailed calculations ofthis can be found in Ref.[86].Putting everything together, we can now conclude that, as expected: Theorem 33 ( D ( H ) , , σ, η ) is a K¨ahler manifold.
87t last, we notice that there is an identification of the orthogonal complementof any unit vector x ∈ H with the tangent space of the U ( H )-orbit in u ∗ ( H ) at ξ = | x ih x | . Indeed, for any y perpendicular to x ( k x k = 1) the operators: P xy =: ( µ ∗ ) x ( y ) = | y ih x | + | x ih y | (4.153)can be written as P xy = [ A y , ξ ], where A y is a Hermitian operator such that A y x = iy , A y y = − i k y k x and A y z = 0 for any z perpendicular to both x and y , as it can be directly checked by applying both expressions to a genericvector in H which can be written as ax + by + cz with a, b, c ∈ C . Then, fromEqs.(4.147) and (4.149), it follows immediately that, for any y, y ′ orthogonal to x : η ξ ( P xy , P xy ′ ) = − T r ( ξ [ A y , A y ′ ] − ) = − i ( h y, y ′ i − h y ′ , y i ) = − ω ( y, y ′ ) (4.154) σ ξ ( P xy , P xy ′ ) = 12 T r ( ξ [ A y , A y ′ ] − ) = −
12 ( h y, y ′ i + h y ′ , y i ) = g ( y, y ′ ) (4.155)In conclusion, we have the following: Theorem 34
For any y, y ′ ∈ H , the vectors ( µ ∗ ) x ( y ) , ( µ ∗ ) x ( y ) are tangent tothe U ( H ) -orbit in u ∗ ( H ) at ξ = µ ( x ) and: σ ξ (( µ ∗ ) x ( y ) , ( µ ∗ ) x ( y )) = g ( y, y ′ ) (4.156) η ξ (( µ ∗ ) x ( y ) , ( µ ∗ ) x ( y )) = − ω ( y, y ′ ) (4.157) ξ ( µ ∗ ) x ( y )) = ( µ ∗ ) x ( Jy ) (4.158) where the last formula follows from Eq.(4.151). More generally, with minor changes, we can reconstruct similar structuresfor any D r ( H ), obtaining K¨ahler manifolds ( D r ( H ) , r , σ r , η r ). The analog ofabove theorem shows then that the latter can be obtained from a sort of “K¨ahlerreduction” starting from the original linear K¨ahler manifold ( H R , J, g, ω ). Example 35
Let us go back to the previous example of rank-one projectors on H = C . According to (4.126), the latter are described by three dimensionalvectors ξ = ( y , y , y ) such that ξ = 1 / ( y = 1 / always), which form a2-dimensional sphere of radius / . A generic tangent vector X A and a genericone form ˆ A at ξ are of the form X A = y A ∂ + y A ∂ + y A ∂ + y A ∂ and ˆ A = y A dy + y A dy + y A dy + y A dy with y A = 0 and y A · ξ = 0 .It is clear from (4.134) that the map ˜ I that associates to any one-form ˆ A thetangent vector at ξ : ˜ I ( ˆ A ) =: I ( ˆ A, · ) = [ A, ξ ] − is manifestly invariant and givenby: ˜ I ( ˆ A ) = 2( ξ × y A ) · ~∂ , where we have set ~∂ = ( ∂ , ∂ , ∂ ) . It follows that thetwo-form η ξ is such that: η ξ ([ A, ξ ] − , [ B, ξ ] − ) = 2 ξ · ( y A × y B ) (4.159)88 o that η ξ = 2 ǫ ijk y i dy j ∧ dy k (4.160) which is proportional by a factor (cid:0) y + y + y (cid:1) − to the symplectic two-formon a 2-dimensional sphere , when pulled back to the sphere.In a similar way, from (4.132), one can prove that ˜ R ( ˆ A ) =: R ( ˆ A, · ) =[ ξ, A ] + = 2( y A y + y A · ξ ) ∂ + 2( y A ξ + y y A ) · ~∂ . Thus, because of (4.149),we have: σ ξ ([ A, ξ ] − , [ B, ξ ] − ) = 4( ξ × y A ) · ( ξ × y B ) = y A · y B (4.161) where the last equality follows from the fact that ξ = 1 / and ξ is orthogonalto both y A and y B .Finally, starting for example from Eq. (4.151), it is not difficult to checkthat ξ ([ B, ξ ] − ) = y ′ B · ~∂ with : y ′ B = ξ × y B (4.162) A direct calculation shows that ξ = − ξ . GN S construction
In the previous Sections of this Chapter, we have worked out the geometricalstructures that naturally arise in the standard approach to quantum mechanics,which starts from the Hilbert space and identifies the space of physical stateswith the associated complex projective space. In this framework, algebraicnotions, such that of the C ∗ -algebra that contains observables as real elements,arises only as a derived concept.In this Section, we would like to see how geometrical structures emerge alsoin a more algebraic setting, where one starts from the very beginning withan abstract C ∗ -algebra containing the algebra of quantum observables as realelements to obtain the Hilbert space of states is a derived concept via the socalled Gelfand-Naimark-Segal ( GN S ) construction [26]. A detailed discussioncan be found in Ref. [42].
GN S construction
The algebraic approach known as the
GN S construction started with the workof Haag and Kastler [97], and is also at the basis of the mathematical approachto quantum field theory [95].The starting point of this construction is an abstract C ∗ -algebra A [26, 65]with unity, the latter being denoted as I . The elements a ∈ A such that: a = This is also the volume element of a 2-dimensional sphere of radius r = 1 /
2, as it shouldbe. ∗ constitute the set A re (a vector space over the reals) of the real elements of the algebra. In particular: I ∈A re . The obvious decomposition: a = a + ia ,with: a = a + a ∗ a = a − a ∗ i (4.163)means that, as a vector space, A is the direct sum of A re and of the set A im (alsoa vector space over the reals) of the imaginary elements , i.e. of the elementsof the form ia, a ∈ A re . A re can be given [42] the structure of a Lie-Jordanalgebra [65], where, using here the conventions of Sect.4.2.3, the Lie product isdefined as: [ a, b ] =: 12 i ( ab − ba ) (4.164)while the Jordan product is given by: a ◦ b = 12 ( ab + ba ) (4.165)for all a, b ∈ A re . The product in the algebra is then recovered as: ab = a ◦ b + i [ a, b ] (4.166) Remark 36
A typical example of a C ∗ -algebra is the algebra B ( H ) of thebounded operators on a Hilbert space H . In this case [65]: A re ≡ B sa ( H ) ,the set of the bounded self-adjoint operators on H . The space D ( A ) of the states over the C ∗ -algebra A is the space of thelinear functionals ω : A → C that are [95]: • real : ω ( a ∗ ) = ω ( a ) ∀ a ∈ A , • positive: ω ( a ∗ a ) ≥ ∀ a ∈ A and • normalized : ω ( I ) = 1Each functional ω defines a non-negative pairing h·|·i ω between any twoelements a, b ∈ A via: h a | b i ω := ω ( a ∗ b ) (4.167)Reality and positivity of the state guarantee that the pairing (4.167) satisfiesthe Schwartz inequality, i.e.: |h a | b i ω | ≤ q h a | a i ω q h b | b i ω (4.168)but the pairing might be degenerate. We are thus led to consider the ”Gelfandideal” [65, 95] I ω consisting of all elements j ∈ A such that ω ( j ∗ j ) = 0 and todefine the set A / I ω of equivalence classes:Ψ a =: [ a + I ω ] (4.169) Also called the observables .
90t is immediate to see that A / I ω is a pre-Hilbert space with respect to the scalarproduct : h Ψ a , Ψ b i = ω ( a ∗ b ) (4.170)Completing this space with respect to the topology defined by the scalarproduct, one obtains a Hilbert space H ω on which the original C ∗ -algebra A acts via the following representation : π ω ( a )Ψ b = Ψ ab (4.171)Clearly the equivalence class of the unit element in A , i.e. Ω = Ψ I , satisfies: k Ψ I k := p h Ψ I | Ψ I i = 1 and provides a cyclic vector for the representation π ω .Moreover: h Ω | π ω ( a ) | Ω i = ω ( a ) (4.172)This tells us that, if we consider that A acts by duality on D ( A ), theHilbert space corresponding to a given state ω is the orbit of A through ω itself.Notice that any other element b ∈ A such that the vector Ψ = π ω ( b )Ω is of unitnorm, defines a new state ω Ψ by: ω Ψ ( a ) = h Ψ | π ω ( a ) | Ψ i = ω ( b ∗ ab ) (4.173)These states are called vector states of the representation π ω , and are particularexamples of more general states of the form: ω ρ ( a ) = T r [ ρπ ω ( a )] (4.174)where ρ ∈ B ( H ω ) is a density operator [65, 95]. States of the form (4.174) arecalled a ”folium” of the representation π ω . Also, one says that a state ω is pureiff it cannot be written as a convex combination of other states in D ( A ), so thatthe set of pure states D ( A ) defines a set of extremal points in D ( A ).The universality and uniqueness of the GN S construction is guaranteed [26]by the following:
Theorem 37
1. If π ω is a cyclic representation of A on H , any vector representation ω Ψ for a normalized Ψ , see Eq.(4.174), ie equivalent to π ω .2. A GN S representation π ω of A is irreducible iff ω is a pure state. The Schwartz inequality (4.168) implies: h i | a i ω = h a | i i ω = 0 ∀ a ∈ A , i ∈ I ω , and hencethat the scalar product (4.170) does indeed depend only on the equivalence classes of a and b and not on the specific representatives chosen. Notice that if such a representation is faithful, i.e. the map π ω : a π ω ( a ) is anisomorphism, the operator norm of π ω ( a ) equals the C ∗ -norm of a [26]. We recall [95] that a vector Ω ∈ H ω is called cyclic if π ω ( A ) is dense in H ω . xample 38 The
GN S construction can be very simple for finite- dimensional C ∗ -algebras. Consider, e.g., the algebra A = B ( C n ) of linear operators on C n ,i.e. of the n × n matrices with complex entries. Any non-negative operator ω ∈ B ( C n ) defines a state by: ω ( A ) = T r [ ωA ] , ∀ A ∈ A (4.175) while we can define the scalar product in H ω as: h A | B i = ω ( A ∗ B ) = T r [ BωA ∗ ] (4.176) If ω is a rank-1 projector and { e k } is an orthonormal basis for which ω = | e ih e | , writing A km for the matrix elements of A in such a basis, the scalarproduct assumes the form: h A | B i = n X k =1 ¯ A k B k (4.177) while the Gelfand ideal I ω is given by: I ω = { X ∈ A : X k = 0 , k = 1 , · · · , n } (4.178) Thus H ω = A/I ω is nothing but C n itself and π ω is the defining representation.If ω is a rank- m density operator: ω = p | e ih e | + · · · + p m | e m ih e m | with p , · · · p m > and p + · · · p m = 1 , the scalar product is given by: h A | B i = n X k =1 m X j =1 p m ¯ A kj B kj (4.179) and the Gelfand ideal is given by: I ω = { X ∈ A : X kj = 0 , k = 1 , · · · , n ; j = 1 , · · · , m } (4.180) showing that H ω is the direct sum of m copies of C n . Now the representation π ω is no longer irreducible, decomposing into the direct sum of m copies of thedefining representation: π ω ( A ) = I m ⊗ A (4.181) where I m is the m × m identity matrix. C ∗ -algebra Let V be a vector space and V ∗ its dual. To any element v ∈ V , there is acorresponding element in the bi-dual ˆ v ∈ ( V ∗ ) ∗ given by:ˆ v ( α ) = α ( v ) , ∀ α ∈ V ∗ (4.182)Thus any multilinear function on V ∗ , f : V ∗ × · · · V ∗ → R defines, by restrictingit to the diagonal, a polynomial function ˜ f ∈ F ( V ∗ ), ˜ f ( α ) = f ( α, ..., α ) , which92an be obtained from the ”monomials of degree one”, ˆ v ∈ ( V ∗ ) ∗ , on which onehas defined the (commutative) product:(ˆ v · ˆ v )( α ) := ˆ v ( α ) ˆ v ( α ) (4.183)Suppose now that on V there is defined an additional bilinear operation: B : V × V → V (4.184)which induces a (in general noncommutative) product × B on V ⊂ F ( V ∗ ) by:ˆ v × B ˆ v = \ B ( v , v ) (4.185)Then we can define a 2-tensor τ B in F ( V ∗ ), at the point α , by the relation: τ B ( d ˆ v , d ˆ v )( α ) := α ( B ( v , v )) (4.186)which satisfies the Leibniz rule: τ B ( d ˆ v, d (ˆ v · ˆ v )) = τ B ( d ˆ v, ˆ v · d ˆ v + d ˆ v · ˆ v ) = ˆ v · τ B ( d ˆ v, ˆ v ) + τ B ( d ˆ v, ˆ v ) · ˆ v (4.187)Thus, τ B ( d ˆ v, · ) defines a derivation on V ⊂ F ( V ∗ ) with respect to the commu-tative product (4.183).In particular, suppose that B is a skew-symmetric bilinear operation whichsatisfies the Jacobi identity, so that g = ( V, B ) is a Lie algebra. The correspond-ing 2-tensor Λ := τ B : Λ( d ˆ v , d ˆ v ) = \ B ( v , v ) (4.188)is a Poisson tensor in F ( V ∗ ) and Λ( d ˆ v, · ) is a derivation with respect to thecommutative product (4.183). Moreover, Λ( d ˆ v, · ) is a derivation also with re-spect to the product (4.185). Indeed, by using the fact that B is antisymmetricand satisfies the Jacobi identity, one has:Λ( d ˆ v, d ( ˆ v · ˆ v )) = \ B ( v, B ( v , v )) = (4.189)= \ B ( v , B ( v, v )) + \ B ( B ( v, v ) , v ) == ˆ v · Λ( d ˆ v, d ˆ v ) + Λ( d ˆ v, d ˆ v ) · ˆ v Similarly, if on V one has a Jordan product B ′ , the corresponding 2-tensor G := τ B ′ is a metric tensor and G ( d ˆ v, · ) is a derivation with respect to thecommutative product (4.183), but not with respect to the product (4.185).If now V = A is a C ∗ -algebra, where we have defined both a Lie productand a Jordan product as: B ( a , a ) := [ a , a ] = 12 i ( a a − a a ) , ∀ a , a ∈ A (4.190)and a Jordan product B ′ ( a , a ) := a ◦ a = 12 ( a a + a a ) , ∀ a , a ∈ A (4.191)93n F ( A ∗ ) we have defined both a Poisson tensor Λ and a metric tensor G suchthat Λ( d ˆ a, · ) and G ( d ˆ a, · ) are both derivations with respect to the pointwisecommutative product, with the former being also a derivation with respect tothe Lie product. It is also not difficult to check that the subalgebra B ⊂ A composed of all real elements, when embedded in F ( A ∗ ), comes equipped withan antisymmetric and a symmetric product, denoted by [ · , · ] and ◦ respectively,such that:1. The Leibniz rule is satisfied: [ a, b ◦ c ] = [ a, b ] ◦ c + b ◦ [ a, c ],2. The Jacobi identity is satisfied: [ a, [ b, c ]] = [[ a, b ] , c ] + [ b, [ a, c ]], and3. The identity: ( a ◦ b ) ◦ c − a ◦ ( b ◦ c ) = [[ a, c ] , b ] holds.meaning that ( B , [ · , · ] , ◦ ) is a Lie-Jordan algebra [65]Finally, we notice that theHamiltonian vector fields: X ˆ a := Λ( · , d ˆ a ) = − [ˆ a, · ] (4.192)are derivations with respect to the Jordan product, since, by using the propertiesabove: X ˆ a ( d (ˆ a ◦ ˆ a )) = − [ˆ a, ˆ a ◦ ˆ a ] = − [ˆ a, ˆ a ] ◦ ˆ a + − ˆ a ◦ [ˆ a, ˆ a ]= X ˆ a ( d ˆ a ) ◦ ˆ a + ˆ a ◦ X ˆ a ( d ˆ a ) (4.193)Let us go back now to the GN S construction and consider first a pure state ω over A , which gives rise to the irreducible representation π ω in the Hilbertspace H ω . We have already seen (see Sect. 4.2.5) that self-adjoint operators,that correspond to the real elements of A , may be identified with the dual u ∗ ( H ω ) of the Lie algebra u ( H ω ) of the unitary group U ( H ω ) and how themomentum map µ ω : H ω → u ∗ ( H ω ) , µ ω ( ψ ) = | ψ ih ψ | (4.194)relates the Poisson tensors on u ∗ ( H ω ) with those on H ω , via the pull-back. Wewill say that a Poisson map Φ : S → M , with ( S, Ω) a Poisson manifold, is asymplectic realization of a Poisson manifold ( M, Λ). When S is a vector spacewe call Φ a classical Jordan-Schwinger map [149]; when S is a Hilbert space, asin the case we are considering, we say it is a Hermitian realization.We have also seen that the unit sphere in H ω − { } can be projected onto u ∗ ( H ω ) in an equivariant way, in such a way that the Poisson and the Riemanntensor in P ( H ω ) are both related to the same tensors defined on u ∗ ( H ω ) by usingthe Lie and the Jordan product that are defined on it. Thus the momentummap provides a symplectic realization, which we call a K¨ahlerian realizationwhere S is the complex projective space. R n Given now A ∈ gl (2 n, R ) ≡ End ( R n ), A = (cid:13)(cid:13) A i j (cid:13)(cid:13) we can make two distinctassociations, namely: 94 ) gl (2 n, R ) → (1 ,
1) tensor fields, via: A → T A = A i j dx j ⊗ ∂∂x i (4.195)The correspondence is an isomorphism of associative algebras, i.e.: T A ◦ T B = T AB (4.196)and T A is homogeneous of degree zero, i.e.: L ∆ T A = 0 (4.197)where ∆ is the dilation (Liouville) vector field associated with the linear struc-ture of R n : ∆ = x i ∂∂x i (4.198) ii ) gl (2 n, R ) → { linear vector fields } , via: A → X A = A i j x j ∂∂x i (4.199)The latter is only a Lie algebra (anti)isomorphism, i.e.:[ X A , X B ] = − X [ A , B ] (4.200) X A is also homogeneous of degree zero:[∆ , X A ] = 0 ∀ A (4.201) i ) and ii ) are connected by: T A (∆) = X A (4.202)Moreover, for any A , B ∈ gl (2 n, R ): L X A T B = − T [ A , B ] (4.203) Remark 39
Going back to the compatibility condition between, say, g and J ,and defining the linear vector field: X J = J i j x j (cid:0) ∂/∂x i (cid:1) , one checks easily thatthe compatibility condition e J ◦ g + g ◦ J = 0 is identical to requiring: L X J g = 0 (4.204) This clarifies also why J can be associated with infinitesimal g -orthogonal trans-formations. Given now a triple, a
Hermitian structure on R n will be a map: h : R n → R ; h ( x, y ) = ( g ( x, y ) , ω ( x, y )) ≡ ( g ( x, y ) , g ( x, Jy )) (4.205) R n can be given a complex vector space structure by defining, for z = α + iβ ∈ C : ( α + iβ ) · x =: αx + βJx (4.206)95 emark 40 Notice that, e.g., g ( x, Jx ) = 0 ∀ x , i.e. x, Jx ∈ R n are orthog-onal and hence R -linearly independent , but they are not linearly independentwhen linear combinations with complex coefficients are allowed, as: Jx =: ix .This means that the complex dimension is reduced from n to n , and R n ≈ C n as a complex vector space. One possible (non-canonical i.e. not unique) wayof ”mapping” R n onto C n is to choose a basis in R n , to pick up n vectors (cid:0) e , ..., e n (cid:1) of the basis and to construct C n by taking complex linear combina-tions thereof with the rule given above (i.e.: ze i =: αe i + βJe i ). Then, we can write: h ( x, y ) = g ( x, y ) + iω ( x, y ) ≡ g ( x, y ) + ig ( x, Jy ) (4.207)or: h ( x, y ) = ω ( Jx, y ) + iω ( x, y ) (4.208)and in this way h will be a Hermitian scalar product linear in the first factorand antilinear in the second factor .For the alternative descriptions obtained in the previous chapter, we get anew Hermitian scalar product by replacing ω in (4.207) with ω F .Let now an admissible triple ( g, J, ω ) be given on R n . First of all we canconstruct the quadratic function: g =: 12 g (∆ , ∆) (4.209)and the associated Hamiltonian vector field Γ via: i Γ ω = − d g (4.210)Explicit calculation shows that, with ω and g (admissible and) constant, Γ sforced to be a linear vector field:Γ = Γ i j x j ∂∂x i (4.211)and that: Γ i j = J i j (4.212)i.e. : Γ = J , for short. This can be written in coordinate-free language as:Γ = J (∆) and: ∆ = − J (Γ) (4.213)Notice that Γ is symplectic: L Γ ω = 0 (4.214) Indeed, if x = 0 and αx + βJx = 0 with α, β ∈ R , then: 0 = g ( αx + βJx, αx + βJx ) =( α + β ) g ( x, x ), implying α = β = 0. Had we been using: ω ( x, y ) = g ( Jx, y ) instead of ω ( x, y ) = g ( x, Jy ) we would haveobtained the opposite, which is the most common convention [56, 183] among physicists. In terms of representative matrices. g . Therefore:0 = L Γ g = 12 ( L Γ g ) (∆ , ∆) + g (∆ , [Γ , ∆]) (4.215)But [Γ , ∆] = 0, so Γ is also a Killing vector field: L Γ g = 0 (4.216)Thus Γ will preserve both the metric, the symplectic structure and (of course)the complex structure, i.e. all the tensors of the admissible triple. So, there willbe two linear vector fields ”canonically” associated with every admissible triple,one of them defining the linear structure.Of course: L Γ h = 0 (4.217)which is a complex condition equivalent to the two real ones: L Γ g = 0 and: L Γ J = 0. As the linear transformations that leave the Hermitian scalar productunchanged are those of the unitary group on C n , Γ will be an infinitesimaltransformation of this group, and the representative matrix (i.e. J ) will belongto its Lie algebra. All the vector fields with this property will be called quantumsystems. A quantum system will be therefore any linear vector field: X A = A i j x j ∂∂x i (4.218)such that: L X A h = 0 (4.219)In terms of the defining matrices. The matrix A belongs then both to the Liealgebra of the orthogonal ( g -orthogonal) group and to the Lie algebra of thesymplectic group, i.e. Eq.(4.219) splits into the two real conditions: L X A g = 0 and: L X A ω = 0 (4.220)The intersection of these algebras is the Lie algebra of the unitary group. Atthe finite level (i.e. by exponentiation) the one-parameter group exp { t A } willbelong to a real realization of the unitary group U ( n ) in R n . Notice also thatthe first of Eqs.(4.220) implies, together with Eq.(4.201), that: L X A g (∆ , ∆) = 0 (4.221) Example 41
Consider, e.g., SU (2) in the defining representation, i.e.: SU (2) ∋ U = (cid:12)(cid:12)(cid:12)(cid:12) α β − β α (cid:12)(cid:12)(cid:12)(cid:12) : C → C , | α | + | β | = 1 (4.222) (i.e. we are viewing U as a (1 , tensor). Writing: U = a + ib , with a and b real × matrices, the unitarity condition U † U = I becomes: e aa + e bb = I ; a e b − b e a = 0 (4.223)97 i.e. a e b must be a symmetric matrix). We can realify C onto R as ( z = x + iy etc.): z = (cid:12)(cid:12)(cid:12)(cid:12) z z (cid:12)(cid:12)(cid:12)(cid:12) → x = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x y y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (4.224) and U as the × real matrix: G = (cid:12)(cid:12)(cid:12)(cid:12) a − bb a (cid:12)(cid:12)(cid:12)(cid:12) (4.225) Assume for simplicity the metric to be the standard Euclidean metric. Then itcan be checked at once that the unitarity condition leads both to: e GG = I (4.226) and to: e G J G = J (4.227) where: J = (cid:12)(cid:12)(cid:12)(cid:12) − II (cid:12)(cid:12)(cid:12)(cid:12) (4.228) with I the × identity matrix, i.e. J is the realification of the multiplication bythe imaginary unit i in C . In this case, as matrices: ω = J (we stress howeverthat ω is a (0 , tensor, while J is a (1 , tensor), and one checks easily that: h ( x, x ′ ) = g ( x, x ′ ) + iω ( x, x ′ ) ⇔ zz ′ which is the Hermitian scalar product in C antilinear in the second factor. G provides then also a realization of both SO (4) and of Sp (4) , and hence of: SU (2) = SO (4) ∩ Sp (4) . Explicitly, thevector field associated with J will be: Γ = x ∂∂y − y ∂∂x + x ∂∂y − y ∂∂x (4.229) This is the dynamical vector field for the D harmonic oscillator. In C itcorresponds of course to: · z j = iz j , j = 1 , . Consider two different Hermitian structures, h and h , on R n , with associatedquadratic functions g a (∆ , ∆) and Hamiltonian vector fields Γ a (Γ a = X J a ), a = 1 , . The two structures will be called compatible iff: L Γ h = L Γ h = 0 (4.230) See, e.g., Ref.[5] Sect.18. bi Hamiltonian. In more detail,this implies: L Γ ω = L Γ g = 0 as well as: L Γ g = 0 (and similarly byinterchanging indices).As already recalled, given a symplectic form ω and/or a metric tensor g anda linear vector field X A , the following statements are equivalent: L X A ω = 0; ω ( A x, y ) + ω ( x, A y ) = 0; ω A = ] ( ω A ) (4.231)as well as: L X A g = 0; g ( A x, y ) + g ( x, A y ) = 0; g A = − ] ( g A ) (4.232)(remember that ω is skew-symmetric: e ω = − ω , while g is symmetric: e g = g ).So, X A will leave ω invariant iff ω A is symmetric , and it will leave g invariantiff g A is skew-symmetric.Now, as L Γ ω = 0 = L Γ g and: i Γ ω = − d g :0 = L Γ ( i Γ ω ) = L Γ ω (Γ , . ) = ω ([Γ , Γ ] , . ) (4.233)and, as the symplectic forms are non-degenerate:[Γ , Γ ] = 0 (4.234)which, in view of the fact that: Γ a = X J a , a = 1 , J , J ] = 0 (4.235)Given a symplectic form ω , the Poisson bracket of any two functions f and g is given by: { f, g } = ω ( X g , X f ) (4.236)where X f and X g are the Hamiltonian vector fields associated with f and g respectively. Hence, denoting with { ., . } a the Poisson bracket associated with ω a ( a = 1 ,
2) we have, e.g.: { g , g } = ω (Γ , Γ ) = − d g (Γ ) = −L Γ g = 0 (4.237)and similarly with the other Poisson bracket. All in all: { g , g } = { g , g } = 0 (4.238)Out of the metric tensors and symplectic structures one can form the (1 , G = g − ◦ g (4.239)(not to be confused with the (2 ,
0) tensor G introduced in Sect.4.2.3) and: T = ω − ◦ ω (4.240) Compare Ch.3.
99n intrinsic terms: G ( X ) = g − ( g ( X )), i.e.: G = G i j dx j ⊗ ∂∂x i ; G i j = ( g ) ik ( g ) kj (4.241)and similarly for T . The two are not independent, though. Indeed, using: J a = ( g a ) − ◦ ω a ( a = 1 ,
2) and: J − a = − J a : G = − J ◦ T ◦ J ⇔ T = − J ◦ G ◦ J (4.242)Having been built out of invariant tensors, it is clear that: L Γ a G = L Γ a T =0. In terms of the defining matrices, this implies (see the previous Section):[ G, J a ] = [ T, J a ] = 0 , a = 1 , GT = − J ◦ T ◦ J ◦ T = − T ◦ J ◦ J = T G , i.e.:[
G, T ] = 0 (4.244)By direct calculation, using the representative matrices and the symmetryof the metric tensors, one proves immediately that: g ( Gx, y ) = g ( x, y ) = g ( x, Gy ). Also, by direct calculation: g ( Gx, y ) = ( g ) − ( g ( x, . ) , g ( y, . )) = g ( x, Gy ). Hence, G is self-adjoint w.r.t. both metrics: g a ( Gx, y ) = g a ( x, Gy ) , a = 1 , L Γ ω = 0 . In terms of therepresentative matrices, this implies (see above): ω J = ^ ( ω J ). As: e ω = − ω and f J = − ω ◦ g − , we obtain: ω ◦ g − ◦ ω = ω ◦ g − ◦ ω . This implies: (cid:0) ω − ◦ ω (cid:1) ◦ g − ◦ ω = g − ◦ (cid:0) ω ◦ ω − (cid:1) ◦ ω or (multiplying on the right by ω − and remembering that: T = ω − ◦ ω ): T ◦ g − = g − ◦ e T . Rememberingthe definition of the adjoint of a (1 ,
1) tensor we have then: T = g − ◦ e T ◦ g ≡ (cid:0) T † (cid:1) (4.246)i.e., T is self-adjoint w.r.t. the metric g . Interchanging indices, one provesthat: (cid:0) T † (cid:1) = T as well. Finally, each J a ( a = 1 ,
2) is skew -adjoint w.r.t. therespective metric tensor: J a = − (cid:0) J † a (cid:1) a = − g − a ◦ f J a ◦ g a . On top of that we havealso, e.g.: (cid:16) J † (cid:17) = g − ◦ f J ◦ g = − g − ◦ g ◦ J ◦ g − ◦ g = − G − ◦ J ◦ G = − J ,as G and the J ’s commute. Interchanging indices, one proves a similar resultfor J . All in all: (cid:0) J † a (cid:1) b = − J a , a, b = 1 , G , T , J and J are a set of mutually commuting operators. G andT are self-adjoint, while J and J are skew-adjoint w.r.t. both metric tensors. G being self-adjoint, one can proceed to diagonalize it, and V = R n will splitinto an orthogonal sum of eigenspaces: V = M k =1 ,...,r V k where: G | V k = λ k I k The sum will be orthogonal w.r.t. both metrics. λ k ’s ( k = 1 , ..., r ≤ n ) are the distinct eigenvalues of G , and λ k > G = g − ◦ g , this implies: g | V k = λ k g | V k (4.248) T commutes with G and is self-adjoint as well. Then V k will decomposefurther into the (bi)orthogonal sum: V k = M α W k,α (4.249)where, denoting as µ k,α the distinct eigenvalues of T in V k (labeled by theindex α ), W k,α will be the eigenspace of the eigenvalue µ k,α . Once again: T | W k,α = µ k,α I k,α , and hence: ω | W k,α = µ k,α ω | W k,α (4.250)Notice that, neither symplectic form being degenerate by assumption, each W k,α will be necessarily even-dimensional. The dimension of each W k,α will be thenat least two.The complex structures J and J commute with both G and T . So, theywill leave the subspaces W k,α invariant. Reconstructing them from the g ’s and ω ’s we find: J | W k,α = µ k,α λ k J | W k,α (4.251)and, as: J = J = − I : ( µ k,α /λ k ) = 1, i.e.: µ k,α = ± λ k , implying: J | W k,α = ± J | W k,α (4.252)Therefore, the index α can assume only at most two values, corresponding to ± λ k , i.e.: V k = M α = ± W k,α at most , with W k, ± corresponding to the eigenvalues ± λ k respectively. The dimension of each eigenspace V k will be then at least twoif only one of the possible eigenvalues ± λ k of T is present, at least four if bothare present. Hence, the maximum number of distinct eigenvalues of G will be r ≤ n .In general, a (0 ,
2) and a (2 ,
0) tensors (such as, say, g and g − ) can becomposed to yield a (1 ,
1) tensor. They will be said to be ”in a generic position” iff the resulting (1 ,
1) tensor has eigenvalues of minimum degeneracy. In thepresent context, we will say that h and h are in a generic position iff theeigenvalues of both G and T have minimum degeneracy, which means double degeneracy. Then: r = n and we will have the (bi)orthogonal decomposition: V = M k =1 ,...,n E k (4.253)101here: dim E k = 2 and either E k = W k, + or E k = W k, − (only one can bepresent but not both, otherwise λ k would be fourfold degenerate). One canchoose in E k a g -orthogonal basis ( e , e ) in such a way that: g | E k = e ∗ ⊗ e ∗ + e ∗ ⊗ e ∗ (4.254)the e ∗ ’s being the dual basis: e ∗ i ( e j ) = δ ij . Then the condition: g ( x, J y ) + g ( J x, y ) = 0 will imply: J | E k = e ⊗ e ∗ − e ⊗ e ∗ (4.255)or the opposite (i.e.: J e = e , J e = − e ), and hence that: ω | E k = e ∗ ∧ e ∗ (4.256)Correspondingly, we will have: g | E k = λ k g | E k ; J | E k = ± J | E k ; ω | E k = ± λ k ω | E k (4.257)Coming now to the general problem of bihamiltonian fields, every linear vec-tor field Γ preserving both h and h will have a representative matrix commut-ing with those of G and T . Therefore, it will be block-diagonal in the commoneigenspaces of both tensors. In the generic (linear) case, the analysis can berestricted to the two-dimensional eigenspaces E k . On each one of these Γ willpreserve both a symplectic structure and a positive-definite metric. Thereforeit will be in sp (2) ∩ so (2) = u (1) and it will represent a harmonic oscillator,with a frequency possibly depending on E k .Using, say, Γ and T , one can construct the n vectors: Γ k +1 = T k Γ ,k = 0 , , ..., n −
1. First of all one sees immediately, by looking at the rep-resentative matrices, that, as that of Γ is J , which commutes with T , the Γ k ’swill commute pairwise, i.e.:[Γ r , Γ s ] = 0 ∀ r, s = 1 , , ..., n (4.258)Moreover, we have shown that T can be brought into the diagonal form: T = M k =1 ,...,n ρ k I k (4.259)with ρ k = ± λ k and ρ k = ρ r for k = r . If the Γ’s were linearly dependent, therewould exist a linear combination such that: n − X r =0 α r T r = 0 (4.260)But on each E k this would reduce to: n − X r =0 α r ( ρ k ) r = 0 , k = 1 , ...n (4.261)102he determinant of the coefficients of this system of linear equations beingthe Vandermonde determinant of the ρ ’s, it will be nonzero, and hence the α ’smust all vanish, which proves that the Γ’s are linearly independent, and hencea basis. As T is a constant tensor, its Nijenhuis torsion vanishes identically.Therefore, as discussed in Sect.B, T is a strong recursion operator . (cid:4) What has been proved up to now is the following. Given two admissibletriples: ( g , ω , J ) and ( g , ω , J ), on V ≈ R n , each triple defines a 2 n -dimensional real representation U r (2 n, g a , ω a ) , a = 1 ,
2, of the group that leavessimultaneously invariant both g a and ω a (and hence J a ), i.e. of the unitarygroup. The intersection: W r = U r (2 n, g , ω ) ∩ U r (2 n, g , ω ) (4.262)will be the common invariance group of both triples. As shown in a 2 D exam-ple in Ref.[160] and as emerges from the previous analysis, the compatibilitycondition implies that W r does not reduce to the identity alone. Any ”quan-tum” bihamiltonian (linear) vector field Γ, i.e. a field such that: L Γ ω a = 0 and L Γ g a = 0 will be in the Lie algebra of W r . In the generic case: W r = SO (2) × SO (2) × ... × SO (2) | {z } n times (4.263)otherwise: W r = U r (2 r ; g, ω ) × U r (2 r ; g, ω ) × ... × U r (2 r k ; g, ω ) (4.264)where ( g, ω ) is any one of the pairs ( g a , ω a ) and: r + ... + r k = n . Quite asimilar analysis can be done by complexifying V in two different ways using thetwo complex structures and reasoning in terms of the two Hermitian structures.In the generic case, then: W r = U (1) × U (1) × ... × U (1) | {z } n times (4.265)For further details, see Ref.[160].To end this Section, we will like to rephrase the previous results in a waymore suitable to be generalized to the infinite dimensional case.We first notice that, going back to the original complex n-dimensional Hilbertspace H , there exist two positive constants α and β , such that: α k x k ≤ k x k ≤ β k x k , ∀ x ∈ H (4.266)This implies, by Riesz’s theorem [112, 192, 204], that there exists a bounded positive and self-adjoint operator F such that: h ( x, y ) = h ( F x, y ) , ∀ x, y ∈ H (4.267) With respect to both Hermitian structures. h a = g a + iω a , a = 1 , F = h − ◦ h (4.268)and F replaces the previous G and T .Then [169, 171] a necessary and sufficient condition for h and h to be ingeneric position is that F be a cyclic operator, i.e. that there exists a vector x such that the vectors x , F x , · · · , F n − x span the whole Hilbert space.Indeed, when h and h are in generic position, F has n distinct eigenvalues, λ k . If we now denote with { f k } its eigenvector basis and with { µ ( k ) } a set of n nonzero complex numbers, we can construct the vectors F m x = X k µ ( k ) λ mk f k , m = 0 , , · · · , n − . (4.269)They are linear independent because the determinant of their components isgiven by ( Q k µ ( k ) ) V ( λ , · · · , λ n ), where the Vandermonde determinant V isnonzero, the eigenvalues λ k ’s being distinct. Clearly, the converse is also true.Also, it has been argued in Ref.[160], that ”bi-unitary” operators, i.e. oper-ators that are unitary w.r.t. both Hermitian structures , must commute with F (the proof is simple and we refer to the above reference for it), i.e. bi-unitaryoperators are in the commutant F ′ of F .The results of this discussion can be summarized in the following: Proposition 42
Two Hermitian forms are in a generic position iff the bicom-mutant of F coincides with the commutant: F ′′ = F ′ . It should be clear from our presentation that many results will carry overto the infinite-dimensional case, although new problems may arise because thealgebraic properties do not ”control” properties such as continuity and differen-tiability in infinite dimensions.
In the (genuinely) infinite-dimensional case of a Hilbert space H there arise twodifficulties, namely:i) Given two Hermitian structures, ( · , · ) and ( · , · ) on H defining two complexscalar products (both linear in, say, the second factor and antilinear in the first,but this is not a crucial point), they might define two non-equivalent topologieson H , and: Of course, any linear vector field that leaves both h ’s invariant will generate a one-parameter group of bi-unitary transformations. The commutant F ′ of F is the set of all operators that commute with F . It is of courseclosed under commutation because of the Jacobi identity, i.e. it is a Lie algebra. The bi-commutant F ′′ is the set of all operators that commute with all those in the commutant. Inparticular, they will commute with F itself, and hence: F ′′ ⊂ F ′ . Moreover, any two opera-tors in F ′′ must commute among themselves. F ′′ is therefore a (maximal) Abelian subalgebraof F ′ , i.e. F ′′ is the center of F ′ α and β , such that formula (4.266) holds. It followsthat we can define the operator F as in (4.267). But now, due to point ii ),we have to better specify what we mean, for example, by requiring F to havenondegenerate eigenvalues. On the other side, the definitions of the commutantand the bicommutant of F are of purely algebraic character and can thereforebe generalized to the infinite dimensional situation. Then, following Refs. [169]and [171], we will adopt the following definition: Definition.
Two Hermitian structures h and h are said to be in genericposition iff F ′′ = F ′ , F being their connecting operator. To proceed further in understanding the situation in which F has also a con-tinuous spectrum, one needs suitable mathematical tools such as the spectraltheory and the theory of rings of operators in Hilbert spaces [192]. We first ob-serve that F ′ and F ′′ ⊂ F ′ are both (weakly closed) rings of bounded operatorson H . Now, given any set S ∈ B ( H ), it can be proved [192] that the minimalweakly closed ring R ( S ) containing S contains only those elements A ∈ S ′′ suchthat E A = AE = A (4.270)where E is the so called principal identity of the set S , i.e. the projectionoperator on ( kerS ∩ kerS † ) ⊥ . If S = { F } , F being self-adjoint and positive, wehave that I ∈ R ( F ) and R ( F ) = F ′′ , which is therefore commutative.If we decompose now F in terms of its spectral family { P ( λ ) } : F = Z ∆ λ dP ( λ ) (4.271)where ∆ = [ a, b ] is a closed interval containing the spectrum of F , it is possibleto show that: a ) The weakly closed commutative ring R ( F ) corresponds to a decompositionof the Hilbert space H into the direct integral H = Z ∆ H λ dσ ( λ ) (4.272)where the measure σ ( λ ) is obtained from the spectral family { P ( λ ) } of F . b ) Any operator A ∈ F ′ can be represented as A = Z ∆ A ( λ ) dσ ( λ ) (4.273)where A ( λ ) is a bounded operator on H λ , for almost all λ . c ) Every B ∈ F ′′ = R ( F ) is a multiplication by a number b ( λ ) on H λ , for almostall λ .Moreover, since R ( F ) is a maximal commutative ring by itself, the family F ′ ( λ )105f all operators A ( λ ) corresponding to F ′ , for a fixed λ , is irreducible so thatwe can rewrite a, b ) above as: a ′ ) The spectrum ∆ of F is the union of a countable number of measurable sets∆ k such that, for λ ∈ ∆ k , the spaces H λ have the same dimension n k (finite orinfinite) and: H = M k Z ∆ k H λ dσ ( λ ) (4.274) b ′ ) Any A ∈ F ′ can be written as A = M k Z ∆ k A ( λ ) dσ ( λ ) (4.275)Now, going back to the two Hermitian structures h and h on H , since the con-necting operator F acts on each H λ as a multiplication by the number λ , we caneasily derive the following result generalizing the finite-dimensional situation. Proposition 43
There exists a decomposition of H as direct integral of Hilbertspaces H λ , of dimension n k such that in each H λ : h = λh . It follows that the elements of the unitary group that leave simultaneouslyinvariant h and h have the form (see Eq.(4.5.3)): U = M k Z ∆ k U k ( λ ) dσ ( λ ) (4.276)where U k ( λ ) is an element of the unitary group U ( n k ), for each λ ∈ ∆ k .Also, it is now immediate to prove that definition (1) is equivalent to: Definition.
Two Hermitian structures h and h are said to be in genericposition iff the spaces H λ are one-dimensional. Indeed, if h and h are in generic position, then R ( F ) = F ′′ = F ′ , sothat the latter is commutative and A ( λ ), for almost all λ ∈ ∆, acts on a one-dimensional Hilbert space H λ . Conversely, if R ( F ) = F ′′ = F ′ , F ′ is non-commutative and hence there is a subset ∆ ⊂ ∆ such that H λ has dimensiongreater than one for λ ∈ ∆ . (cid:4) Notice also that, in the generic case, the operators U k ( λ ) in (4.276) areone-dimensional and reduces to a multiplication by a phase factor exp[ iθ ( λ )].Finally, we may prove the following equivalence between the genericity con-dition and the cyclicity of the operator F : Definition. F is cyclic iff F ′′ = F ′ . This follows from the fact that, if F ′′ = F ′ , the latter is commutative andeach space H ( λ ), where F acts as a multiplication by λ , is one-dimensional. Sothe vector x = 1 /λ is a cyclic vector. Viceversa, if we suppose now that F iscyclic, each H ( λ ) is one-dimensional and any A ∈ F ′ acts as a multiplicationby a number. Hence F ′ = F ′′ = R ( F ). (cid:4) xample 44 A particle in a box.
We consider the operator F = 1 + X where X is the position operator which acts as multiplication by x on the Hilbertspace L ([ − α, α ] , dx ) . From the spectrum ∆ X = [ − α, α ] and the spectral family { P X ( λ ) = χ [ − α,λ ] } of X ( χ [ − α,λ ] being the characteristic function on [ − α, λ ] ),one easily sees that the spectrum of F is ∆ F = [1 , α ] while its spectral family { P F ( λ ) } is given by P F ( λ ) = P ( √ λ − − P ( −√ λ −
1) (4.277)
In fact, t is easy to check that: P F = P F ; P F (1) = 0; P F (cid:0) α (cid:1) = I (4.278) We can write F as: F = Z [ − α,α ] (1 + λ ) dP ( λ ) (4.279) If we now divide the interval as [ − α, α ] = [ − α, ∪ [0 , α ] and change variableby setting λ = −√ µ − or λ = √ µ − in the negative or positive parts of theinterval respectively, we get: F = Z [1 , α ] λ dP F ( λ ) (4.280) Now F has no cyclic vector on the whole L ([ − α, α ]) since G ′ , which containsboth X and the parity operator is not commutative. On the contrary, χ [ − α, iscyclic on L ([ − α, and, similarly, χ [0 ,α ] is so on L ([0 , α ]) . Thus the Hilbertspace splits in two F -cyclic spaces: L ([ − α, α ]) = L ([ − α, ⊕ L ([0 , α ]) andwe obtain the decomposition H = Z [1 , α ] H λ dσ ( λ ) (4.281) where the measure is obtained from: σ ( λ ) = P F ( λ ) χ [ − α, = P F ( λ ) χ [0 ,α ] = √ λ − Notice that the spaces H λ are one-dimensional if we work in the interval [0 , α ] or bidimensional if we consider [ − α, α ] . Also, the bi-unitary transformationsread, respectively, as: U = Z [1 , α ] e iφ ( λ ) dσ ( λ ) (4.283) U = Z [1 , α ] U ( λ ) dσ ( λ ) (4.284)107 From Finite to Infinite Dimensions. Weyl Sys-tems
A known theorem by A.Wintner [232] states that if, say, b q and b p are quantum-mechanical operators on an infinite-dimensional Hilbert space satisfying a com-mutation relation of the form: [ b q, b p ] = c b I (or, better: [ b q, b p ] ⊆ c b I ), with c aconstant and b I the identity operator, then at least one of them must be un-bounded.Motivated then by the need of formulating Quantum Mechanics withouthaving to do with unbounded operators, it was apparently H.Weyl [225] (seealso [218]) who proposed first a different scheme of quantization that goes asfollows:Let S be a (real) linear vector space endowed with a constant symplecticstructure ω . Weyl’s approach consists in the following: • It is a map W from S to the set of unitary operators on a ( so farunspecified ) Hilbert space H : W : S → U ( H ) (5.1)via: S ∋ z → c W ( z ) ∈ U ( H ) , c W ( z ) c W † ( z ) = c W † ( z ) c W ( z ) = b I (5.2)with the following specifications: • W is a strongly continuous map, and • For any z, z ′ ∈ S : c W ( z + z ′ ) = c W ( z ) c W ( z ′ ) exp {− iω ( z, z ′ ) / ~ } (5.3)with ~ the reduced Planck constant. It follows then that: c W ( z ) c W ( z ′ ) = c W ( z ′ ) c W ( z ) exp { iω ( z, z ′ ) / ~ } , ∀ z, z ′ (5.4)Moreover, setting z ′ = 0 in (5.3) we obtain: c W − ( z ) c W ( z ) = c W (0), andhence: c W (0) = b I , while setting z ′ = − z we obtain: c W − ( z ) = c W ( − z ), andhence: c W † ( z ) = c W ( − z ) (5.5) I.e. translationally-invariant. Hence, necessarily: dim ( S ) will be even , and: S ≈ R n for some n . That’s why the setting we are describing here has been defined as ”abstract”. a Weyl system is a projective unitary representation of the linearvector space S (thought of as the group manifold of the translation group) in theHilbert space H .As a running example we shall consider S = R with coordinates ( q, p ) andthe standard symplectic form: ω = dq ∧ dp , which is represented by the matrix: ω = (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) (5.6)Hence: ω (( q, p ) , ( q ′ , p ′ )) = (cid:12)(cid:12) q p (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) q ′ p ′ (cid:12)(cid:12)(cid:12)(cid:12) = qp ′ − q ′ p (5.7)and therefore: c W (( q, p ) + ( q ′ , p ′ )) = c W ( q, p ) c W ( q ′ , p ′ ) exp (cid:26) − i ~ ( qp ′ − q ′ p ) (cid:27) (5.8)In the general case, we can decompose S into the direct sum of two La-grangian subspaces: S = S ⊕S , and hence any vector z as: z = ( z , , z ), z ∈ S , z ∈ S . We can consider then the restrictions of W to the Lagrangiansubspaces, i.e.: U = W | S : S → H (5.9)and: V = W | S : S → H (5.10)As: ω | S = ω | S = 0, U and V are faithful representations of the correspondingLagrangian subspaces: b U ( z + z ′ ) = b U ( z ) b U ( z ′ ) ; z , z ′ ∈ S (5.11)and similarly for V . Moreover: b U ( z ) b V ( z ) = b V ( z ) b U ( z ) exp { iω (( z , , (0 , z )) / ~ } (5.12)Viceversa, we have the following: Proposition:
Given two faithful representations U and V of two transver-sal Lagrangian subspaces of a symplectic vector space S satisfying (5.12), themap: z −→ c W ( z ) = b U ( z ) b V ( z ) exp {− iω (( z , , (0 , z )) / ~ } (5.13) is a Weyl system. The proof that (5.13) does indeed satisfy the defining property (5.3) can bedone by direct calculation, and will be omitted here. (cid:4)
Consider now a one-dimensional subspace of H spanned by a fixed vector z .From (5.3) we have, with α, β real numbers: c W ( αz ) c W ( βz ) = c W (( α + β ) z ) (5.14)109herefore, nc W ( αz ) o α ∈ R is a strongly continuous one-parameter group of uni-taries and, by Stone’s theorem [201]: c W ( αz ) = exp n iα b G ( z ) / ~ o (5.15)with an infinitesimal generator b G ( z ) which is (essentially) self-adjoint. Further-more, nc W ( αβz ) o β ∈ R is also a strongly continuous one-parameter group, andtherefore: c W ( αβz ) = exp n iβ b G ( αz ) / ~ o (5.16)and, setting β = 1, we find: b G ( αz ) = α b G ( z ) (5.17)In terms of infinitesimal generators and setting: z → αz, z ′ → βz ′ , Eq. (5.4)reads: e iα b G ( z ) / ~ e iβ b G ( z ′ ) / ~ = e iαβω ( z,z ′ ) / ~ e iα b G ( z ) / ~ e iβ b G ( z ′ ) / ~ (5.18)and, for α and β infinitesimal, this yields, to the lowest nontrivial order: h b G ( z ) , b G ( z ′ ) i = − i ~ ω ( z, z ′ ) (5.19) What is lacking in the ”abstract” presentation of the previous Section is aconcrete realization of the Hilbert space H on which the mapping W shouldoperate.Before discussing von Neumann’s theorem, let us resume our running exam-ple on R ≈ T ∗ R . Writing ( q, p ) as: ( q, p ) = ( q, , p ), whence: ω (( q, , (0 , p )) = qp , our Weyl system becomes ( z = ( q, p ) , z = ( q, , z = (0 , p )) (see Eq.(5.8)): c W ( q, p ) = c W (( q,
0) + (0 , p )) = c W ( q, c W (0 , p ) exp {− iqp/ ~ } (5.20)while: c W ( q + q ′ ,
0) = c W ( q, c W ( q ′ ,
0) (5.21)and similarly for c W (0 , p ). Define then: c W ( q,
0) = exp n iq b P / ~ o ; c W (0 , p ) = exp n ip b Q/ ~ o (5.22)In other words, as: ( q,
0) = q (1 , , (0 , p ) = p (0 , b G (0 ,
1) = b Q, b G (1 ,
0) = b P (5.23)with (cfr. Eq. (5.19)): h b Q, b P i = i ~I (5.24)110oreover, using the truncated Baker-Campbell-Hausdorff [201] formula onefinds easily: c W ( q, p ) = exp n i (cid:16) q b P + p b Q (cid:17) / ~ o (5.25)Consider now L ( R , dx ) with the Lebesgue measure, and define the familiesof operators n b U ( q ) o q ∈ R and n b V ( p ) o p ∈ R via: (cid:16) b U ( q ) ψ (cid:17) ( x ) = ψ ( x + q ) (5.26)and: (cid:16) b V ( p ) ψ (cid:17) ( x ) = exp { ipx/ ~ } ψ ( x ) (5.27)for ψ ∈ L ( R , dx ). It is easy to show that both families are actually one-parameter, strongly continuous groups of unitaries, and that: (cid:16) b U ( q ) b V ( p ) ψ (cid:17) ( x ) = exp { iqp/ ~ } (cid:16) b V ( p ) b U ( q ) ψ (cid:17) ( x ) (5.28)Then: c W ( q, p ) = b U ( q ) b V ( p ) exp {− iqp/ ~ } (5.29)is a concrete realization of a Weyl system. Defining again: b U ( q ) = exp n iq b P / ~ o and: b V ( p ) = exp n ip b Q/ ~ o , we find both Eq.(5.25) and, at the infinitesimallevel : (cid:16) b Qψ (cid:17) ( x ) = xψ ( x ) , (cid:16) b P ψ (cid:17) ( x ) = − i ~ dψdx (5.30)Moreover: (cid:16)c W ( q, p ) ψ (cid:17) ( x ) = exp { ip [ x + q/ / ~ } ψ ( x + q ) (5.31)A generic matrix element of c W ( q, p ) will be given then by: D φ, c W ( q, p ) ψ E = exp { iqp/ ~ } + ∞ Z −∞ dxφ ( x ) exp { ipx/ ~ } ψ ( x + q ) (5.32) Remark 45
Viewed as a function on T ∗ Q , D φ, c W ( q, p ) ψ E is square-integrablefor all φ, ψ ∈ L ( R ) . Indeed, defining the Lebesgue measure on R as dqdp/ π ~ ,a direct calculation shows that: (cid:13)(cid:13)(cid:13)D φ, c W ( q, p ) ψ E(cid:13)(cid:13)(cid:13) =: Z Z dqdp π ~ (cid:12)(cid:12)(cid:12)D φ, c W ( q, p ) ψ E(cid:12)(cid:12)(cid:12) = k φ k k ψ k (5.33) e a + b = e a e b e − [ a,b ] / whenever: [ a, [ a, b ]] = [ b, [ a, b ]] = 0. And in the appropriate domains. φ ( x ) = (1 / √ π ) exp( ik ′ x ) , ψ ( x ) = (1 / √ π ) exp( ikx ) (5.34)and denoting as D k ′ | c W ( q, p ) | k E the matrix elements of c W ( q, p ) between thesestates, we obtain: D k ′ | c W ( q, p ) | k E = δ ( k − k ′ + p/ ~ ) exp ( iq ( k + k ′ ) /
2) (5.35)and, in particular: D k | c W ( q, p ) | k E = ~ δ ( p ) exp { ikq } (5.36)Integrating Eq.(5.36) over k , we obtain for the trace of W : T r nc W ( q, p ) o = 2 π ~ δ ( q ) δ ( p ) (5.37)Coming now to the general case, let’s assume that we are given a symplecticvector space ( S , ω ) and a decomposition of S as the direct sum: S = S ⊕ S (5.38)with S and S Lagrangian subspaces. Every vector z ∈ S can then be decom-posed in a unique way as: z = ( z ,
0) + (0 , z ) , z i ∈ S i , i = 1 ,
2. Let us remarkfirst of all that the symplectic structure allows each one of the two subspaces tobe identified with the dual of the other. Indeed, we can define a pairing: h ., . i : S × S → R (5.39)via: h z , z i : ω (( z , , (0 , z )) (5.40)The details of the proof that in this way S ≈ S ∗ (and viceversa, of course)can be found in Ref. [135].Assume now H to be a separable Hilbert space and let: U : S → H V : S → H (5.41)be unitary, irreducible and strongly continuous representations of S and S respectively on H , satisfying the additional condition that defines the ”Weylform” of the commutation relations: b U ( z ) b V ( z ) = b V ( z ) b U ( z ) exp { iω (( z , , (0 , z )) / ~ } (5.42) Actually, we can define the trace only if we admit distribution-valued traces. Strictlyspeaking [59], and as Eq.(5.33) shows, c W is bounded but not trace-class. c W ( z ) = b U ( z ) b V ( z ) exp {− iω (( z , , (0 , z )) / ~ } (5.43)which is a Weyl system. Let us denote z and z as ( q,
0) and (0 , p ) respectively,with q and p n -dimensional vectors ( n = dim S = dim S ). Correspondingly,we will denote b U ( z ) and b V ( z ) as b U ( q ) and b V ( p ) respectively.Von Neumann’s theorem [223] states then that there exists a unitary map: T : H → L ( R n , dµ ) (5.44)such that: (cid:16) T b U ( q ) T − ψ (cid:17) ( x ) = ψ ( x + q ) (5.45)and (cfr.Eqn.(5.40)): (cid:16) T b V ( p ) T − ψ (cid:17) ( x ) = e i h x,p i ψ ( x ) (5.46)This theorem proves that all the representations of the Weyl commutation rela-tions are unitarily equivalent to the Schr¨odinger representation, and hence areunitarily equivalent among themselves (but see below, Sect.7.3.1). Example 46
In the case of L ( R ) , setting ~ = 1 and using the Fourier trans-form: ψ ( x ) = ∞ Z −∞ dp √ π e ψ ( p ) exp { ipx } (5.47) one finds easily that: ^ (cid:16) exp (cid:16) ix b P (cid:17) ψ (cid:17) ( p ) = e ixp e ψ ( p ) (5.48) (i.e.: (cid:16) b P e ψ (cid:17) ( p ) = p e ψ ( p ) , and: ^ (cid:16) exp (cid:16) iπ b Q (cid:17) ψ (cid:17) ( p ) = e ψ ( p − π ) (5.49) ( (cid:16) b Q e ψ (cid:17) ( p ) = id e ψ ( p ) /dp ). Denoting by: F : L ( R ) → L ( R ) (5.50) the unitary operator defined by the Fourier transform, we can conclude that: F † b Q F = − b P (5.51) and: F † b P F = b Q (5.52)113 .3 Weyl Systems and Linear Transformations Let’s begin by considering linear transformations that preserve the symplecticstructure, i.e. linear maps: T : S → S such that: ω ( T z, T z ′ ) = ω ( z, z ′ ) ∀ z, z ′ ∈ S (5.53)In terms of matrices this means: e T ωT = ω (5.54)(where e T stands for the transpose of the matrix T ), and this defines a realizationof the symplectic group Sp (2 n, R ) associated with the symplectic structure ω .Then, we can define: c W T : S → H (5.55)via: c W T ( z ) =: c W ( T z ) (5.56)and, as: c W ( T ( z + z ′ )) = c W ( T z ) c W ( T z ′ ) exp {− iω ( T z, T z ′ ) / ~ } == c W ( T z ) c W ( T z ′ ) exp {− iω ( z, z ′ ) / ~ } (5.57)we find: : c W T ( z + z ′ ) = c W T ( z ) c W T ( z ′ ) exp {− iω ( z, z ′ ) / ~ } (5.58)i.e. c W T is also a Weyl system, and hence, by von Neumann’s theorem, it isunitarily equivalent to c W .As a simple example, consider, in R , the map:( q, p ) → ( − p, q ) (5.59)which is realized via the transformation : T = (cid:12)(cid:12)(cid:12)(cid:12) −
11 0 (cid:12)(cid:12)(cid:12)(cid:12) (5.60)Then it is clear that: b U ( q ) = c W (( q, → c W ((0 , − p )) = b V ( − p ) (5.61)and: b V ( p ) = c W ((0 , p )) → c W (( q, b U ( q ) (5.62)which is precisely (see the end of the previous Section) what the Fourier trans-form does. The matrix representing T is simply minus that of the complex structure. However, thetwo have different transformation properties (see Chapt.1). c W T is unitarily equivalent to c W , to the map T there is associated anautomorphism of the group U ( H ) of the unitary operators. As every automor-phism of U ( H ) is inner, there is a unitary operator b U T such that: c W T ( z ) = b U † T (cid:16)c W ( z ) (cid:17) b U T (5.63)More generally, we can consider a one-parameter group { T λ } λ ∈ R of linearsymplectic transformations. Calling Γ the linear vector field that is the in-finitesimal generator of the group, the condition: ω ( T λ z, T λ z ′ ) = ω ( z, z ′ ) ∀ z, z ′ ∈ S, ∀ λ ∈ R (5.64)becomes: L Γ ω = 0 (5.65)with L Γ the Lie derivative. There exists then (globally on a vector space) afunction g such that: i Γ ω = dg (5.66)and, for linear transformations, g will be a quadratic function of the coordinates.According to what has been said above, the family { T λ } defines a (stronglycontinuous) one-parameter group { U λ } λ ∈ R of unitary operators such that: c W ( z ( λ )) = b U † λ c W ( z ) b U λ (5.67)where: z ( λ ) = T λ ( z ). By Stone’s theorem, then: b U λ = exp n − iλ b G/ ~ o (5.68)with b G self-adjoint. The self-adjoint operator b G is the quantum counterpartof the quadratic function g . In this way we have achieved a way to quantizeall the quadratic functions: given G , we can define via Eq.(5.66) the associatedHamiltonian vector field. This in turns defines a one-parameter group of (linear)symplectic transformations, and the corresponding Weyl system allows us to findthe (self-adjoint) quantum operator to be associated with g .Let’s consider now a general linear transformation T ∈ GL (2 n, R ), notnecessarily a symplectic one. We will denote for clarity as ω a reference (com-parison) symplectic structure, written in a Darboux chart as: ω = (cid:12)(cid:12)(cid:12)(cid:12) I − I (cid:12)(cid:12)(cid:12)(cid:12) (5.69)We define then a new symplectic structure ω T via: ω T ( z, z ′ ) =: ω ( T z, T z ′ ) (5.70)That ω T is a symplectic structure is obvious. It is represented by the matrix: ω T = e T ω T (5.71)115ow, if we define again: c W T ( z ) =: c W ( T z ) (5.72)it is easy to prove that: c W T ( z + z ′ ) = c W T ( z ) c W T ( z ′ ) exp {− iω T ( z, z ′ ) / ~ } (5.73)Therefore, c W T defines a Weyl system, but for ( S , ω T ) an not for ( S , ω D ). Mim-icking the analysis that has been done previously, we conclude that: c W T ( λz ) = exp { iλ b G ( z ) } (5.74)and that: h b G ( z ) , b G ( z ′ ) i = − i ~ ω T ( z, z ′ ) (5.75)Now we are in a position to consider Weyl systems for a vector space withan arbitrary and translationally invariant symplectic structure ω . By Darboux’theorem [1, 4], there exists always an invertible linear transformation T suchthat: ω = e T ω T (5.76)Then, the sequence of transformations:( S , ω ) T → ( S , ω ) W → U ( H ) (5.77)defines a Weyl system W ◦ T = W T for ( S , ω ) such that: c W T ( z ) =: c W ( T z ) (5.78)
Remark 47
The matrix T in Eq.(5.76) is clearly ambiguous by left multipli-cation by any matrix T ′ such that f T ′ ω T = ω . However, as: ω ( T ′ T z, T ′ T z ′ ) = ω ( T z, T z ′ ) = ω ( z, z ′ ) (5.79) the Weyl systems associated with T and T ′ T are unitarily equivalent. As is well known [4], a conspicuous example of a one-parameter group of sym-plectic transformations is provided by the time evolution of a Hamiltonian sys-tem. So, let’s study some simple examples.116 .4.1 The free particle
In this case, the one-parameter group is given by: ( q, p ) → ( q + tp/m, p ) and isrepresented by the matrix: (cid:12)(cid:12)(cid:12)(cid:12) q ( t ) p ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = F ( t ) (cid:12)(cid:12)(cid:12)(cid:12) qp (cid:12)(cid:12)(cid:12)(cid:12) ; F ( t ) = (cid:12)(cid:12)(cid:12)(cid:12) t/m (cid:12)(cid:12)(cid:12)(cid:12) ; F ( t ) F ( t ′ ) = F ( t + t ′ ) (5.80)Then: c W t ( q, p ) = c W ( q ( t ) , p ( t )) = exp n ( i/ ~ ) h q ( t ) b P + p ( t ) b Q io =: exp n ( i/ ~ ) h q b P t + p b Q t io (5.81)where: b P t = b P , b Q t = b Q + t b P /m (5.82)There exists therefore a one-parameter family n b F ( t ) o t ∈ R of unitary operatorssuch that: exp n ip b Q t / ~ o = b F † ( t ) exp n ip b Q/ ~ o b F ( t ) (5.83)and: exp n iq b P t / ~ o = b F † ( t ) exp n iq b P / ~ o b F ( t ) (5.84)Setting then: b F ( t ) = exp n − i b Ht/ ~ o (5.85)using Eq.(5.82) and expanding for small q, p and t , one finds the commutationrelations: h b P , b H i = 0 , h b Q, b H i = i ~ m b P (5.86) Remark 48
Note that the previous equation does not specify what are the ba-sic commutation relations between b Q and b P . Stated otherwise, we are not yetspecifying what should be the symplectic structure that appears on the r.h.s. ofEq.(5.19), and this is very much in the spirit [229] of Wigner’s approach. Inwhat follows, however, and as we are dealing with this and the following Exam-ples only to exhibit simple instances of Weyl systems, we shall assume that q and p are Darboux coordinates, and hence that the basic commutation relationsare of the standard form of Eq.(5.24). The only unknown quantity in Eq.(5.86)will be then the Hamiltonian b H . As the generators of linear and homogeneous canonical transformations arequadratic functions, it is natural to look for a quantum operator b H that is alsoa quadratic function: b H = a b P + b b Q + c (cid:16) b P b Q + b Q b P (cid:17) (5.87)117hen the solution of the previous commutation relations is precisely: b H = b P m + λ b I (5.88)where b I is the identity operator and λ and arbitrary (real) constant. Apart fromthis, the quantum operator associated with the time evolution is the standardquantum Hamiltonian. The classical Hamiltonian is: H = p m + 12 mω q (5.89)and the solution of the equations of motion is: q ( t ) = q cos ωt + p sin ωtmω (5.90) p ( t ) = p cos ωt − qmω sin ωt The matrix F ( t ) is then: F ( t ) = (cid:12)(cid:12)(cid:12)(cid:12) cos ωt sin ωtmω − mω sin ωt cos ωt (cid:12)(cid:12)(cid:12)(cid:12) (5.91)Proceeding just as in the previous case we find again: c W t ( q, p ) = exp n ( i/ ~ ) h q b P t + p b Q t io (5.92)with, now: b Q t = b Q cos ωt + b P sin ωtmω (5.93)and: b P t = b P cos ωt − b Qmω sin ωt (5.94)Defining again: b F ( t ) = exp n − i b Ht/ ~ o and working out the commutationrelations of b H with b Q and b P , that read now: h b Q, b H i = i ~ m b P (5.95)just as before, and: h b P , b H i = − i ~ mω b Q (5.96)one finds : b H = b P m + 12 mω b Q + λ b I (5.97)i.e., again ”modulo” an additive multiple of the identity, the standard quantumHamiltonian. 118 .4.3 A Charged Particle in a Constant Magnetic Field The equations of motion for a particle of mass m and charge q in a constantmagnetic field B are [7, 184] (in units c = 1): d x dt = v (5.98) m d v dt = q v × B (5.99)The vector field is therefore:Γ = v · ∂∂ x + qm v × B · ∂∂ v (5.100)The equations of motion can be derived either from the Lagrangian: L = 12 m v + q v · A (5.101)where A is the vector potential: ∇ × A = B , or from the Hamiltonian: H = p m (5.102)with the symplectic form: ω B = − qε ijk B k dx i ∧ dx j + dx i ∧ dp i (5.103)where: p = π − q A (5.104) π is the canonical momentum: π = ∂ L ∂ v (5.105)and p = m v is the kinetic momentum.We will consider here a field: B = (0 , , B ). As the motion along x is trivialand decouples, we will ignore it and concentrate on the dynamics in the (cid:0) x , x (cid:1) plane. Among the various gauges that one can employ the most popular are the Landau gauges : A = B (cid:0) x , , (cid:1) ; A = B (cid:0) , − x , (cid:1) (5.106)or the symmetric gauge : A s = B (cid:0) x , − x , (cid:1) = 12 B × r = A + A For an analysis at the quantum level, see [28, 62, 49, 102, 126] z = (cid:0) z , ..., z (cid:1) with: (cid:0) z , z (cid:1) = (cid:0) x , x (cid:1) , (cid:0) z , z (cid:1) = ( p , p ) and setting q = m = 1, the symplectic form can be writtenas: ω B = 12 Ω ij dz i ∧ dz j (5.108)where Ω is the matrix: Ω = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − B B − − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.109)Explicitly: ω B = − Bdx ∧ dx + dx ∧ dp + dx ∧ dp (5.110)The inverse of Ω : Λ = − Ω − = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − B − − B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.111)defines the Poisson tensor: Λ = 12 Λ ij ∂∂z i ∧ ∂∂z j (5.112)or, explicitly: Λ = ∂∂x ∧ ∂∂p + ∂∂x ∧ ∂∂p + B ∂∂p ∧ ∂∂p (5.113)A transformation that reduces ω B to the standard Darboux form, definedby the matrix: Ω = (cid:12)(cid:12)(cid:12)(cid:12) × I × − I × × (cid:12)(cid:12)(cid:12)(cid:12) (5.114)is: z → e z = T z with: T = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.115)i.e., explicitly: p → e p = p − Bx (5.116)with all the other coordinates unchanged and: ω B ( z, z ′ ) = ω D ( T z, T z ′ ) = ω ( e z, e z ′ ) (5.117)which implies, as can also be checked by direct calculation on the representativematrices: 120 T ω T = Ω (5.118)Notice that this amounts to the transformation: r → r , p → p − A (5.119)One could have used instead, e.g., the transformation: r → r , p → p − A s (5.120)that is defined by the matrix: T ′ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) B/ − B/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.121)and here too: e T ′ ω T ′ = Ω (5.122)Notice that, defining: V =: T ′ T − , Eqns. (5.118) and (5.122) imply: e V ω V = ω (5.123)i.e.: V ∈ Sp (cid:0) R (cid:1) , and this too can be checked by direct calculation.Concentrating now on the transformation defined by Eq.(5.115) and follow-ing the procedure outlined in Sect.4 .
3, we define the Weyl system: c W T ( z ) = c W ( T z ) = (5.124)= exp n i he x b P + e x b P + e p b Q + e p b Q io or, explicitly: c W T ( z ) = exp i X i =1 , h x i b P ( T ) i + p i b Q ( T ) i i (5.125)where: b Q ( T ) i = b Q i , i = 1 , , b P ( T )1 = b P , b P ( T )2 = b P + B b Q (5.126)Notice that: h b Q ( T ) i , b Q ( T ) j i = 0; h b Q ( T ) i , b P ( T ) j i = iδ ij (5.127)while: h b P ( T )1 , b P ( T )2 i = − iB (5.128)Time evolution is given by: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ( t ) x ( t ) p ( t ) p ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = F ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x x p p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.129)121here: F ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin( Bt ) B − cos( Bt ) B − − cos( Bt ) B sin( Bt ) B Bt ) sin ( Bt )0 0 − sin ( Bt ) cos ( Bt ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (5.130)is a linear symplectic map. Explicitly: x ( t ) = x + p sin ( Bt ) B + p − cos ( Bt ) B (5.131) x ( t ) = x − p − cos ( Bt ) B + p sin ( Bt ) B and: p ( t ) = p cos ( Bt ) + p sin ( Bt ) (5.132) p ( t ) = − p sin ( Bt ) + p cos ( Bt )Following the procedure outlined in the previous examples, we define thenthe Weyl system: c W ( t ) T ( z ) = c W ( T z ( t )) = exp i X i =1 , h x i ( t ) b P ( T ) i + p i ( t ) b Q ( T ) i i (5.133)or: c W ( t ) T ( z ) = exp i X i =1 , h x i b P ( T ) i ( t ) + p i b Q ( T ) i ( t ) i (5.134)where the b P ( T ) i ( t )’s and b Q ( T ) i ( t )’s are defined by: X i =1 , h x i b P ( T ) i ( t ) + p i b Q ( T ) i ( t ) i =: X i =1 , h x i ( t ) b P ( T ) i + p i ( t ) b Q ( T ) i i (5.135)and Eqns.(5.131) and (5.132) have to be used on the r.h.s. Here too we concludethat there exists a unitary operator b F ( t ) = exp {− it H / ~ } such that:exp n ip b Q ( T ) i ( t ) / ~ o = b F † ( t ) exp n ip b Q ( T ) i / ~ o b F ( t ) (5.136)exp n iq b P ( T ) i ( t ) / ~ o = b F † ( t ) exp n iq b P ( T ) i / ~ o b F ( t ) ; i = 1 , q, p, t and using Eq.(5.126) we find the commu-tation relations: h b P , H i = 0 , h b P , H i = i ~ B (cid:16) b P − B b Q (cid:17) (5.137)and: h b Q , H i = i ~ (cid:16) b P − B b Q (cid:17) , h b Q , H i = i ~ b P (5.138)122nd it is easy to conclude that the Hamiltonian operator is now: H = 12 (cid:20)(cid:16) b P − B b Q (cid:17) + b P (cid:21) (5.139)which corresponds to the ”minimal coupling” prescription: b P → b P − A (5.140)with: A = A . We will exhibit in this final Subsection another example [184] of a Weyl system,which is provided by the implementation at the quantum level of the groupof translations in a two-dimensional electron gas in a constant (perpendicular)magnetic field that has been studied in the previous Subsection.Reinstating the constants ( c, m, q ) in the proper places, the Hamiltonian isgiven by (cfr.Eq.(5.102)): H = 12 m [ π − qc A ] (5.141)Introducing complex coordinates: ζ = x + iy , the equations of motion be-come: ddt { · ζ + i Ω ζ } = 0; Ω = qBmc (5.142)and they have the solution: ζ ( t ) = X + A exp {− i Ω t } (5.143)where: X = ζ − i Ω · ζ = const. (5.144)The associated total energy is: E = p m ≡ m | · ζ | ≡ m Ω | A | (5.145)and the orbits are circles of radius | A | and center: X = x + iy ; x = x + p y m Ω , y = y − p x m Ω (5.146)The Poisson brackets for the components of the kinetic momentum are: { p i , p j } = qc ( ∂ i A j − ∂ j A i ) ≡ qc ε ijk B k (5.147)123.e.: { p x , p y } = m Ω (5.148)and hence: { x , y | = − m Ω = − cqB (5.149)The Cartan form: θ L = ∂ L ∂ v · d q ≡ { p i + qc A i } dx i (5.150)leads to: ω L =: − dθ L = dx i ∧ dp i + q c ( ∂ i A j − ∂ j A i ) dx i ∧ dx j (5.151)i.e.: ω L = dx i ∧ dp i + q c ε ijk B i dx j ∧ dx k (5.152)The dynamical vector field is given by (cfr. Eq.(5.100)):Γ = p i m ∂∂q i + qmc ε ijk p j B k ∂∂v i (5.153)N¨other’s theorem [186] states that, if X c is a tangent lift , and: L X c L = L Γ h, h = π ∗ b G, i.e. : h = h ( r ) (5.154)then the associated constant of the motion is: χ = i X c θ L − h (5.155)For translations in the plane: X i ≡ X ci = ∂∂x i (5.156)Hence: L X i L = qmc ( ∂ i A j ) p j ≡ qc ( ∂ i A j − ∂ j A i ) + qc ∂ j A i (5.157)i.e.: L X i L = qc ddt ( A + r × B ) i (5.158)( h = qc ( A + r × B )), and the associated N¨other’s constants of the motion are: χ i = i X i θ L − h i = ( p + qc B × r ) i (5.159) We recall [186] that,if: X = X i ∂/∂q i , X i = X i ( q ) is a vector field on some manifold M ,its tangent lift X c is the vector field on T M defined by: X c = X i ∂/∂q i + L Γ ( X i ) ∂/∂v i , withΓ any second-order vector field. x = cqB ( p y + qBc x ) ≡ cqB ( p + qc B × r ) y (5.160) y = − cqB ( mv x − qBc y ) ≡ − ( p + qc B × r ) x Hence: χ x = − qBc y , χ y = qBc x (5.161)It follows then that the P.B. ’s among the N¨other’s constants of the motionare: { χ i , χ j } = − qc ε ijk B k (5.162)Following then the standard rules for the implementation of symmetries atthe quantum level, we associate with a (finite) translation by a vector a in theplane the magnetic translation operator b T ( a ) [76, 184, 236, 237] defined by: b T ( a ) = exp (cid:26) i ~ b χ op · a (cid:27) (5.163)where ( p = π − q A /c ): b χ op = b π − qc ( B × r − A ) ; b π = ~ i ▽ (5.164)with the commutation relations:[ b χ i , b χ j ] = i ~ qc ε ijk B k (5.165)Of course, b χ op commutes with the Hamiltonian, and so does b T ( a ).Using then the identities:exp { A + B } = exp { A } exp { B } exp (cid:18) −
12 [
A, B ] (cid:19) (5.166)exp { A } exp { B } = exp { B } exp { A } exp ([ A, B ]) (5.167)valid whenever: [ A, [ A, B ]] = [ B, [ A, B ]] = 0, and noting that: h π i , ( B × r − A ) j i = i ~ ∂A j ∂x i (5.168)one finds for the action of b T ( a ) on wavefunctions: (cid:16) b T ( a ) ψ (cid:17) ( r ) = exp n − i q ~ c a · A o ψ ( r + a ) (5.169)in the symmetric gauge, and, e.g.: (cid:16) b T ( a ) ψ (cid:17) ( r ) = exp n − i q ~ c Ba ( y − a / o ψ ( r + a ) (5.170)125n the Landau gauge: A = B (0 , x, b T ( a ) b T ( b ) = exp (cid:26) iq ~ c B · a × b (cid:27) b T ( a + b ) (5.171)and: b T ( a ) b T ( b ) = b T ( b ) b T ( a ) exp (cid:26) iq ~ c B · a × b (cid:27) (5.172)But: qc B · a × b = ω L ( a , b ) (5.173)and hence: b T ( a ) b T ( b ) = exp (cid:26) i ~ ω L ( a , b ) (cid:27) b T ( a + b ) (5.174)The magnetic translation operators are therefore an instance [238] of a Weylsystem on the configuration space. 126 Quantum Mechanics in Phase Space
We will work in S ≈ R for simplicity. Generalizations to higher dimensions areeasy to work out.As a preliminary remark, let’s observe that we have the identity ( f ∈L (cid:0) R (cid:1) ): Z Z Z Z dξdηdq ′ dp ′ (2 π ~ ) f ( q ′ , p ′ ) e − iω (( q ′ ,p ′ ) , ( ξ,η ) ) / ~ e i ( ξp + ηq ) / ~ ≡ f ( q, − p ) (6.1)This can also be rewritten as: Z Z dξdη π ~ (cid:20) ~ F s ( f ) (cid:18) η ~ , ξ ~ (cid:19)(cid:21) e i ( ξp + ηq ) / ~ = f ( q, − p ) (6.2)where F s ( f ) is the symplectic Fourier transform [77, 240]: F s ( f ) ( η, ξ ) = Z Z dqdp π f ( q, p ) e − iω (( q,p ) , ( ξ,η )) (6.3)and, as usual: ω (( q, p ) , ( ξ, η )) = qη − pξ . Digression.
Allowing also for distribution-valued transforms, we have, in particular: F s ( q ) ( η, ξ ) = 2 πiδ ′ ( η ) δ ( ξ ) (6.4)and: F s ( p ) ( η, ξ ) = − πiδ ( η ) δ ′ ( ξ ) (6.5)The Weyl map [225] amounts to the replacement, in Eq.(6.2):exp { i ( ξp + ηq ) / ~ } → exp n i (cid:16) ξ b P + η b Q (cid:17) / ~ o ≡ c W ( ξ, η ) (6.6)whereby one obtains the map:Ω : F (cid:0) R (cid:1) → O p ( H ) (6.7)defined by: Ω ( f ) = Z Z dξdη π ~ (cid:20) ~ F s ( f ) (cid:18) η ~ , ξ ~ (cid:19)(cid:21) c W ( ξ, η ) = (6.8) ≡ Z Z dξdη π F s ( f ) ( η, ξ ) c W ( ~ ξ, ~ η ) See Appendix C . The fact that we get a change in sign in the second variable is preciselya byproduct of the use of the symplectic Fourier transform. Had we used instead the ordinaryFourier transform we would have obtained of course f ( q, p ) instead of f ( q, − p ) on the r.h.s.of (6.2). f is real, then: F s ( f ) ( η, ξ ) = F s ( f ) ( − η, − ξ ) (6.9)and this proves that Ω ( f ) is (at least) a symmetric [201] operator (more on thislater on). Using then: (cid:16)c W ( ξ, η ) ψ (cid:17) ( x ) = exp { iη [ x + ξ/ / ~ } ψ ( x + ξ ) (6.10)we obtain:(Ω ( f ) ψ ) ( x ) = Z Z dξdη π F s ( f ) ( η, ξ ) exp [ iη ( x + ~ ξ/ ψ ( x + ~ ξ ) (6.11)In particular, using (6.4) and (6.5):(Ω ( q ) ψ ) ( x ) = xψ ( x ) , (Ω ( p ) ψ ) ( x ) = i ~ dψdx (6.12)In other words: Ω ( q ) = b Q (6.13)while (cfr. the discussion in the previous footnote):Ω ( p ) = − b P (6.14)More generally, for arbitrary integers n and m : F s ( q n p m ) ( η, ξ ) = 2 π ( − ) m i n + m δ ( n ) ( η ) δ ( m ) ( ξ ) (6.15)which implies:(Ω ( q n p m ) ψ ) ( x ) = (cid:18) i ddξ (cid:19) m [( x + ~ ξ/ n ψ ( x + ~ ξ )] | ξ =0 (6.16)which can be rearranged [240] in the form:(Ω ( q n p m ) ψ ) ( x ) = 12 n n X k =0 (cid:18) nk (cid:19) x k (cid:18) i ~ ddx (cid:19) m (cid:2) x n − k ψ ( x ) (cid:3) (6.17)Hence: Ω ( q n p m ) = 12 n n X k =0 (cid:18) nk (cid:19) [Ω ( q )] k · [Ω ( p )] m · [Ω ( q )] n − k (6.18)In particular, for n = m = 1:Ω ( qp ) = 12 (Ω ( q ) · Ω ( p ) + Ω ( p ) · Ω ( q )) (6.19)Notice that: Ω ( qp ) = Ω ( pq ) (6.20)128ut: Ω ( qp ) = Ω ( q ) · Ω ( p ) (6.21)Also, as can be shown on examples, in general:Ω ( f g ) = 12 (Ω ( f ) · Ω ( g ) + Ω ( g ) · Ω ( f )) (6.22)as can be seen already from Eq.(6.18) when m and/or n = 1, i.e. the ”Weylsymmetrization procedure” (6.22) [225] holds only in very special cases.Using Eq.(6.11) we obtain, for the matrix elements of the Weyl operatorΩ ( f ): h φ | Ω ( f ) | ψ i = Z dxdξdη π F s ( f ) ( η, ξ ) e iη ( x + ~ ξ/ φ ( x ) ψ ( x + ~ ξ ) (6.23)In particular, in a plane-wave basis ( ψ ( x ) = (cid:0) / √ π (cid:1) exp { ikx } etc.): h k ′ | Ω ( f ) | k i = Z dξ π F s ( f ) ( k ′ − k, ξ ) exp { i ~ ξ ( k + k ′ ) / } (6.24)or: h K + k/ | Ω ( f ) | K − k/ i = Z dξ π F s ( f ) ( k, ξ ) exp { i ~ ξK/ } (6.25)Inserting then the explicit form of the symplectic Fourier transform we findeventually: h K + k/ | Ω ( f ) | K − k/ i = Z dq π f ( q, − ~ K ) exp {− ikq } (6.26)For example, f ( q, p ) = p yields: h K + k/ | Ω ( p ) | K − k/ i = − ~ Kδ ( k ) (6.27)which (cfr. Eq.(6.14)) is the correct result.The Weyl map can be inverted, i.e there exists a map, called the Wignermap : Ω − : O p ( H ) → F (cid:0) R (cid:1) (6.28)such that: Ω − (Ω ( f )) = f (6.29)In general, given any operator b O such that T r h b O c W ( x, k ) i exists , theWigner map is defined as:Ω − (cid:16) b O (cid:17) ( q, p ) =: Z Z dxdk π ~ exp {− iω (( x, k ) , ( q, p )) / ~ } T r h b O c W † ( x, k ) i (6.30) As W is a bounded operator, this will be granted, e.g., if A is trace-class. T r [ c W ( x, k ) c W † ( ξ, η )] = Z dhdh ′ D h (cid:12)(cid:12)(cid:12)c W ( x, k ) (cid:12)(cid:12)(cid:12) h ′ E D h ′ (cid:12)(cid:12)(cid:12)c W † ( ξ, η ) (cid:12)(cid:12)(cid:12) h E (6.31)Using Eq. (5.35) we obtain: T r hc W ( x, k ) c W † ( ξ, η ) i = 2 π ~ δ ( x − ξ ) δ ( k − η ) (6.32)Inserting then (6.32) into (6.30) and using this result, we obtain:Ω − (Ω ( f )) ( q, p ) = Z dξdη π F s ( η, ξ ) exp {− iω (( ξ, η ) , ( q, p )) } = f ( q, p ) (6.33) (cid:4) Going back to Eq.(6.8), which reads:Ω ( f ) = Z Z dξdη (2 π ~ ) c W ( ξ, η ) Z Z dqdpe − i ( qη − pξ ) / ~ f ( q, p ) (6.34)and, to the extent that it is legitimate to apply Fubini’s theorem, we obtain:Ω ( f ) = Z Z dqdp π ~ f ( q, p ) b A ( q, p ) (6.35)where the symplectic Fourier transform of c W ( ξ, η ) , i.e.: b A ( q, p ) =: Z Z dξdη π ~ e − i ( qη − pξ ) / ~ c W ( ξ, η ) (6.36)defines the so-called ”phase-point operators” [38, 39, 68, 125, 233, 234]. It isnot hard to prove, using: c W † ( ξ, η ) = c W ( − ξ, − η ) and Eqns.(5.37) and (6.32)that: • The phase-point operators are of unit trace:
T r b A ( q, p ) = 1 (6.37) • They are Hermitian: b A † ( q, p ) = b A ( q, p ) (6.38)and: • They are trace-orthogonal, i.e.:
T r (cid:16) b A ( q, p ) b A ( q ′ , p ′ ) (cid:17) = 2 π ~ δ ( q − q ′ ) δ ( p − p ′ ) (6.39)130 Moreover, a simple calculation shows that:
Z Z dqdp π ~ b A ( q, p ) = c W (0 ,
0) = b I (6.40)with b I the identity operator.All this proves that the phase-point operators are a complete (trace) or-thonormal set of Hermitian operators. In particular, substituting the Wignerfunction Ω − (cid:16) b O (cid:17) for the function f in Eq.(6.35) and as: Ω (cid:16) Ω − (cid:16) b O (cid:17)(cid:17) = b O ,we obtain at once the reconstruction: b O = Z Z dqdp π ~ Ω − (cid:16) b O (cid:17) ( q, p ) b A ( q, p ) (6.41)in terms of the Wigner function and the phase-point operators, as well as, usingEq.(6.39): Ω − (cid:16) b O (cid:17) ( q, p ) = T r n b O b A ( q, p ) o (6.42)An explicit representation of phase-point operators satisfying all of the aboveproperties is: b A ( q, p ) = Z dq ′ | q + q ′ / i exp ( iq ′ p/ ~ ) h q − q ′ / | (6.43)with matrix elements: D x (cid:12)(cid:12)(cid:12) b A ( q, p ) (cid:12)(cid:12)(cid:12) x ′ E = 2 δ ( x + x ′ − q ) exp { ip ( x − x ′ ) / ~ } (6.44) It is useful to have an expression for the Wigner map directly in terms of thematrix elements of the operators. Working again for simplicity in R and intro-ducing resolutions of the identity in terms of plane waves: { | m i} :Ω − (cid:16) b O (cid:17) ( q, p ) = Z dxdπdldm π ~ exp {− i ( xp − πq ) / ~ } D l | b O | m E D m | c W † ( x, π ) | l E (6.45)or ( π = ~ k ):Ω − (cid:16) b O (cid:17) ( q, p ) = Z dxdkdldm π exp {− i ( xp/ ~ − kq ) } D l | b O | m E D m | c W † ( x, ~ k ) | l E (6.46)and using: D m | c W † ( x, ~ k ) | l E = exp {− ix ( m + l ) / } δ ( l − m − k ) (6.47) h x | m i = √ π exp { imx } , and: Z dm | m i h m | = I . − (cid:16) b O (cid:17) ( q, p ) = Z dke iqk D − p/ ~ + k/ | b O | − p/ ~ − k/ E (6.48)with obvious generalizations to higher dimensions. As an example, if: b A = − b P ,then, as: b P | m i = ~ m | m i : D − p/ ~ + k/ | ( − b P ) | − p/ ~ − k/ E = ( p ~ + k/ h− p/ ~ + k/ | − p/ ~ − k/ i≡ pδ ( k ) (6.49)and we find: Ω − (cid:16) ( − b P ) (cid:17) ( q, p ) = p (6.50)as expected.Also, it is easy to prove that:Ω − (cid:16)c W ( q ′ , , p ′ ) (cid:17) ( q, p ) = exp { iω (( q, p ) , ( q ′ , p ′ )) / ~ } (6.51)Introducing now resolutions of the identity relative to the coordinates:Ω − (cid:16) b A (cid:17) ( q, p ) = Z dkdxdx ′ e iqk D − p/ ~ + k/ | x i h x | b A | x ′ i h x ′ | − p/ ~ − k/ E (6.52)the integration over k yields a delta-function, and we obtain, eventually, thecelebrated Wigner formula [125, 228], or
Wigner transform :Ω − (cid:16) b O (cid:17) ( q, p ) = Z dξe ipξ/ ~ D q + ξ/ | b O | q − ξ/ E (6.53)Here too, setting: b A = b Q , we find at once: Ω − (cid:16) b Q (cid:17) ( q, p ) = q , as expected. Asanother example, consider, e.g.: b A = | φ ih ψ | (which is a prototype of a finite-rankoperator). Then it is immediate to see that:Ω − ( | φ ih ψ | ) ( q, p ) = ∞ Z −∞ dξe ipξ/ ~ φ ( q + ξ/ ψ ( q − ξ/
2) (6.54)
Remark 49
From Eq.(6.54) we obtain: (cid:12)(cid:12) Ω − ( | φ ih ψ | ) ( q, p ) (cid:12)(cid:12) ≤ ∞ Z −∞ dη | φ ( q + η ) | | ψ ( q − η ) | (6.55) and, using Schwartz’s inequality: (cid:12)(cid:12) Ω − ( | φ ih ψ | ) ( q, p ) (cid:12)(cid:12) ≤ k φ k k ψ k (6.56)132 n particular, if | ψ i = | φ i and: h φ | φ i = 1 , i.e. for a one-dimensional projector: P φ = | φ ih φ | : (cid:12)(cid:12) Ω − ( P φ ) ( q, p ) (cid:12)(cid:12) ≤ As every density matrix can be written as a convex linear combination of one-dimensional projectors, we obtain eventually the uniform bound [68]: (cid:12)(cid:12) Ω − ( b ρ ) ( q, p ) (cid:12)(cid:12) ≤ if b ρ is a density matrix.Proceeding in a somewhat heuristic manner, let now b O be a self-adjointoperator with a completely discrete spectrum: b O | φ n i = λ n | φ n i , h φ n | φ m i = δ nm and: P n | φ n ih φ n | = I . Then: Ω − (cid:16) b O (cid:17) ( q, p ) = X n λ n Z dξe ipξ/ ~ φ n ( q + ξ/ φ n ( q + ξ/
2) (6.59) and hence, proceeding as before: (cid:12)(cid:12)(cid:12) Ω − (cid:16) b O (cid:17) ( q, p ) (cid:12)(cid:12)(cid:12) ≤ X n | λ n | = 2 T r (cid:12)(cid:12)(cid:12) b O (cid:12)(cid:12)(cid:12) (6.60) where [201]: (cid:12)(cid:12)(cid:12) b O (cid:12)(cid:12)(cid:12) =: p b O † b O . Trace-class operators are defined [201] by requir-ing finiteness of T r (cid:12)(cid:12)(cid:12) b O (cid:12)(cid:12)(cid:12) . Therefore:The Wigner function of any trace-class operator b O will be uniformly boundedby T r (cid:12)(cid:12)(cid:12) b O (cid:12)(cid:12)(cid:12) . It is easy to check that the Wigner transform inverts to: D x | b O | x ′ E = Z dp π ~ exp {− ip ( x − x ′ ) / ~ } Ω − (cid:16) b O (cid:17) (cid:18) x + x ′ , p (cid:19) (6.61)As an example, let’s consider the Wigner transform of b O = | φ ih ψ | as givenby Eq.(6.54). Then it is immediate to check that, indeed: Z dp π ~ e {− ip ( x − x ′ ) / ~ } Ω − (cid:16) b O (cid:17) (cid:18) x + x ′ , p (cid:19) = φ ( x ) ψ ( x ′ ) = h x | φ ih ψ | x ′ i (6.62) Example 50
As a less simple example as compared to the previous ones, letus consider a D harmonic oscillator of mass m and proper frequency ω . Thecorresponding Hamiltonian is: b H = b P m + 12 mω b Q (6.63) Note that we are using here a slightly different normalization than that used in Ref.[68]. A class of operators comprising, in particular, finite-rank projection operators as well asdensity states. ith eigenvalues: E n = ( n + 1 / ~ ω , n ≥ and eigenfunctions: ψ n ( x ) = r mωπ ~ √ n n ! exp (cid:0) − ζ / (cid:1) H n ( ζ ) (6.64) where ζ is the dimensionless variable: ζ = x p mω/ ~ and the H n ’s are theHermite polynomials [69]. We want to evaluate here the Wigner function (theWigner map) associated with the ”Boltzmann factor” b A = exp (cid:16) − β b H (cid:17) , with β the inverse temperature . Of course: D x (cid:12)(cid:12)(cid:12) e − β b H (cid:12)(cid:12)(cid:12) x ′ E = ∞ X n =0 e − βE n ψ n ( x ) ψ n ( x ′ ) (6.65) Inserting the explicit form (6.64) of the eigenfunctions: D x (cid:12)(cid:12)(cid:12) e − β b H (cid:12)(cid:12)(cid:12) x ′ E = r mωzπ ~ ∞ X n =0 z n n n ! e − ( ζ + ζ ′ ) / H n ( ζ ) H n ( ζ ′ ) , z = exp ( − β ~ ω )(6.66) Now, it turns out that [77, 128]: ∞ X n =0 z n n n ! H n ( ζ ) H n ( ζ ′ ) = 1 √ − z exp ( zζζ ′ − z (cid:0) ζ + ζ ′ (cid:1) − z ) , | z | < and therefore the matrix element (6.65)can be expressed in closed form as: D x (cid:12)(cid:12)(cid:12) e − β b H (cid:12)(cid:12)(cid:12) x ′ E = r mωπ ~ e − ( ζ + ζ ′ ) / r z − z exp[ 2 zζζ ′ − z (cid:0) ζ + ζ ′ (cid:1) − z ] (6.68) Setting then: x = q + ξ/ , x ′ = q − ξ/ and inserting the result into Eq.(6.53)one finds eventually the Wigner function: Ω − (cid:16) e − β b H (cid:17) ( q, p ) = 1cosh ( β ℏ ω/
2) exp (cid:26) − tanh ( β ℏ ω/ (cid:20) mω ℏ q + p m ℏ ω (cid:21)(cid:27) (6.69)Coming back now to the main object of this Section, an interesting conse-quence of Eq.(6.48) is the following. Let’s calculate the L norm of Ω − (cid:16) b A (cid:17) ( q, p ),i.e.: (cid:13)(cid:13)(cid:13) Ω − (cid:16) b A (cid:17)(cid:13)(cid:13)(cid:13) = Z dqdp π ~ (cid:12)(cid:12)(cid:12) Ω − (cid:16) b A (cid:17) ( q, p ) (cid:12)(cid:12)(cid:12) (6.70)Explicitly: (cid:13)(cid:13)(cid:13) Ω − (cid:16) b A (cid:17)(cid:13)(cid:13)(cid:13) = (6.71)= Z dqdp π ~ dkdk ′ e i ( k ′ − k ) q D p/ ~ − k/ | b A † | p/ ~ + k/ E D p/ ~ + k ′ / | b A | p/ ~ − k ′ / E This is known also as
Mehler’s formula. q , which produces a delta-function, and shiftingvariables: p → p + ~ k/ (cid:13)(cid:13)(cid:13) Ω − (cid:16) b A (cid:17)(cid:13)(cid:13)(cid:13) = Z d ( p/ ~ ) dk D p/ ~ | b A † | p/ ~ + k E D p/ ~ + k | b A | p/ ~ E (6.72)The integration over k yields a resolution of the identity, and we end up with: (cid:13)(cid:13)(cid:13) Ω − (cid:16) b A (cid:17)(cid:13)(cid:13)(cid:13) = Z d ( p/ ~ ) D p/ ~ | b A † b A | p/ ~ E (6.73)i.e., eventually: (cid:13)(cid:13)(cid:13) Ω − (cid:16) b A (cid:17)(cid:13)(cid:13)(cid:13) = T r n b A † b A o (6.74)and, if: b A = Ω ( f ): k f k = T r n Ω ( f ) † Ω ( f ) o (6.75)The condition of finiteness (positivity is obvious) of T r (cid:8) A † A (cid:9) characterizes A as a Hilbert-Schmidt [201] operator. Therefore [198]:
Theorem 51 f will be square-integrable ( f ∈ L (cid:0) R (cid:1) ) if and only if Ω ( f ) isHilbert-Schmidt . Quite similarly: Ω − (cid:16) b A (cid:17) will be square-integrable if and onlyif b A is Hilbert-Schmidt. The Weyl and Wigner maps establish therefore a bijection [77, 93] betweenHilbert-Schmidt operators and square-integrable functions on phase space. Thisis consistent with the fact that both spaces are Hilbert spaces. Moreover,Eqs.(6.74) and (6.75) prove that the bijection, being an isometry, is also (strongly)bicontinuous.The fact that: F s ( η, ξ ) = F s ( − η, − ξ ) as well as that: c W † ( ξ, η ) = c W ( − ξ, − η )allows also to prove at once that the Weyl and Wigner maps ”preserve conju-gation” , i.e. that: Ω (cid:0) f (cid:1) = Ω ( f ) † (6.76)as well as: Ω − (cid:16) b O † (cid:17) = Ω − (cid:16) b O (cid:17) (6.77)Therefore, in particular, if f is real, then, as already mentioned, Ω ( f ) will be a symmetric operator.As a final remark, we observe that Eq. (6.61) implies also: T r x (cid:16) b O (cid:17) =: Z dx h x | O | x i = Z dqdp π ~ Ω − (cid:16) b O (cid:17) ( q, p ) (6.78)135with the same result for the similarly defined T r p (cid:16) b A (cid:17) ) as well as, of course: Z dqdp π ~ f ( q, p ) = T r (Ω ( f )) (6.79)and this defines formally a ”trace” operation on phase space: T r ( f ) =: Z dqdp π ~ f ( q, p ) (6.80)Of course, all these results will make sense when all the quantities in the previousequations are finite. For example, if: b A = P ψ = | ψ ih ψ | , h ψ | ψ i = 1 is a one-dimensional projector, then:Ω − ( P ψ ) ( q, p ) = Z dξe ipξ/ ~ h q + ξ/ | ψ i h ψ | q − ξ/ i (6.81)and: Z dqdp π ~ Ω − ( P ψ ) ( q, p ) = Z dq h q | ψ i h ψ | q i = k ψ k = 1 (6.82)As a less trivial example, in the case of the harmonic oscillator we find withsome long but elementary algebra using Eq.(6.69): T r n Ω − (cid:16) e − β b H (cid:17)o = Z dqdp π ~ Ω − (cid:16) e − β b H (cid:17) = 12 sinh ( β ~ ω/
2) (6.83)which is the expected result [183] for the canonical partition function of a 1 D harmonic oscillator. Remark 52
The mere existence of the phase-space trace of Ω − (cid:16) b O (cid:17) , i.e. finite-ness of R ( dqdp/ π ~ ) Ω − (cid:16) b O (cid:17) ( q, p ) does not however guarantee that b A be trace-class, as this requires, as already recalled [201], the more stringent condition that: T r (cid:16)(cid:12)(cid:12)(cid:12) b O (cid:12)(cid:12)(cid:12)(cid:17) < ∞ , (cid:12)(cid:12)(cid:12) b O (cid:12)(cid:12)(cid:12) =: √ O † O (6.84) and (cid:12)(cid:12)(cid:12) b O (cid:12)(cid:12)(cid:12) is not connected to the Wigner function Ω − (cid:16) b O (cid:17) in any simple manner. Working again for simplicity in S ≈ R , the Wigner map allows for the defi-nition of a new algebra structure on the space of functions F (cid:0) R (cid:1) , the Moyal ” ∗ ”- product [94, 189, 228], that is defined as: f ∗ g =: Ω − (cid:16)b Ω ( f ) · b Ω ( g ) (cid:17) (6.85) We stress once again that extensions to higher dimensions are essentially straightforward. b Ω ( f ) · b Ω ( g ) = b Ω ( g ) · b Ω ( f ), it is clear that, again generically: f ∗ g = g ∗ f ).This product is associative (as the algebra of operators is), it is distributive w.r.t. the sum (as b Ω ( . ) is linear), but it is non-local and non-commutative. Indeed:( f ∗ g ) ( q, p ) = Z Z dxdk π ~ exp {− iω (( x, k ) , ( q, p )) / ~ } T r hb Ω ( f ) · b Ω ( g ) c W † ( x, k ) i (6.86)and: T r hb Ω ( f ) · b Ω ( g ) c W † ( x, k ) i == Z dξdηdξ ′ dη ′ (2 π ) F s ( f ) ( η, ξ ) F s ( g ) ( η ′ , ξ ′ ) T r hc W ( ~ ξ, ~ η ) c W ( ~ ξ ′ , ~ η ′ ) c W † ( x, k ) i (6.87)Now: T r hc W ( α, β ) c W ( σ, τ ) c W † ( x, k ) i == 2 πδ ( α + σ − x ) δ ( β + τ − k ) exp {− i [ β ( α + σ ) + k ( σ − x )] / ~ } (6.88)Hence: T r hb Ω ( f ) · b Ω ( g ) c W † ( x, k ) i = Z dξdξ ′ dηdη ′ π ~ F s ( f ) ( η, ξ ) F s ( g ) ( η ′ , ξ ′ ) e − i ( ηx − kξ ) / δ ( ξ + ξ ′ − x/ ~ ) δ ( η + η ′ − k/ ~ )(6.89)and, using the deltas to get rid of the ξ ′ , η ′ integrations and the explicit form ofthe symplectic Fourier transforms: T r hb Ω ( f ) · b Ω ( g ) c W † ( x, k ) i =4 Z dadbdsdt π ~ f ( a, b ) g ( s, t ) e − i ( sk − tx ) / ~ δ ( k − t − b )) δ ( x − s − a )) (6.90)Inserting this result into Eq.(6.86) we eventually obtain:( f ∗ g ) ( q, p ) = 4 Z dadbdsdt (2 π ~ ) f ( a, b ) g ( s, t ) exp (cid:26) − i ~ [( a − q ) ( t − p ) + ( s − q ) ( p − b )] (cid:27) (6.91)or:( f ∗ g ) ( q, p ) = 4 Z dadbdsdt (2 π ~ ) f ( a, b ) g ( s, t ) exp { iω (( q − a, p − b ) , ( q − s, p − t )) / ~ } (6.92)and this exhibits explicitly the non-locality of the Moyal product.It can be shown that: f ∗ ( g ∗ h ) = ( f ∗ g ) ∗ h f ∗ ( g + h ) = f ∗ g + f ∗ h See,e.g., Ref.[240] for details.
The Moyal product can be recast in the form:( f ∗ g ) ( q, p ) = ∞ X n,m =0 (cid:18) i ~ (cid:19) n + m ( − n n ! m ! (cid:26) ∂ m + n f ( a, b ) ∂a m ∂b n ∂ m + n g ( a, b ) ∂a n ∂b m (cid:27) | a = q,b = p (6.93)and that: • Eq.(6.93) can be rewritten in compact form as:( f ∗ g ) ( q, p ) = f ( q, p ) exp ( i ~ " ←− ∂∂q −→ ∂∂p − ←− ∂∂p −→ ∂∂q g ( q, p ) (6.94)Other equivalent forms of the Moyal product are:( f ∗ g ) ( q, p ) = f q + i ~ −→ ∂∂p , p − i ~ −→ ∂∂q ! g ( q, p ) (6.95)or: ( f ∗ g ) ( q, p ) = f ( q, p ) g q − i ~ ←− ∂∂p , p + i ~ ←− ∂∂q ! (6.96) Remark 53
All the above expressions for the Moyal product apply of course tofunctions that are regular enough for the right-hand side of the defining equa-tions to make sense. In particular, they will hold when f, g are ”Schwartzian”functions [203] in S (cid:0) R (cid:1) , i.e. they are of class C ∞ and of fast decrease atinfinity. The form (6.93) exhibits explicitly the Moyal product as a series expansionin powers of ~ . To lowest order: f ∗ g = f g + i ~ { f, g } + O (cid:0) ~ (cid:1) (6.97)where { ., . } is the Poisson bracket. The Planck constant ~ acts then as a ”de-formation parameter” of the usual associative product structure on the algebraof functions, making the product non-commutative. Indeed, it can be seen, e.g.,from the expansion of the exponential in Eq.(6.94), that terms proportionalto even powers of ~ are symmetric under the interchange f ↔ g , but termsproportional to odd powers are anti symmetric, and this makes the productnon-commutative. Example 54 f ≡ or g ≡ . Then: (1 ∗ g ) ( q, p ) = g ( q, p ) , ( f ∗
1) ( q, p ) = f ( q, p ) (6.98) • f = q . Then, at least if g ∈ S ∞ (cid:0) R (cid:1) : ( q ∗ g ) ( q, p ) = 4 Z dadbdsdt (2 π ~ ) ag ( s, t ) exp (cid:8) i ~ [( a − q ) ( t − p ) + ( s − q ) ( p − b )] (cid:9) == 4 Z dadbdsdt (2 π ~ ) g ( s, t ) (cid:0) q + i ~ ∂∂t (cid:1) exp (cid:8) i ~ [( a − q ) ( t − p ) + ( s − q ) ( p − b )] (cid:9) (6.99) and, integrating by parts in the second integral and using the previousresult: ( q ∗ g ) ( q, p ) = (cid:18) q + i ~ ∂∂p (cid:19) g ( q, p ) (6.100) Then, in view of the symmetry properties of the various terms in the ex-pansion of the Moyal product in powers of ~ : ( g ∗ q ) ( q, p ) = (cid:18) q − i ~ ∂∂p (cid:19) g ( q, p ) (6.101) • In the same way, if f = p : ( p ∗ g ) ( q, p ) = (cid:18) p − i ~ ∂∂q (cid:19) g ( q, p ) (6.102) etc. • If f = q, g = p (or viceversa), then, using, e.g., Eq. (6.95): ( q ∗ p ) ( q, p ) = qp + i ~ p ∗ q ) ( q, p ) = qp − i ~ Notice that Eq.(6.100) implies: b Ω ( q ) · b Ω ( g ) = b Ω ( qg ) + i ~ b Ω (cid:18) ∂g∂p (cid:19) (6.104) and similarly for the others. The generalization of these results, as well as of those of the following Sub-sections, to higher dimensions, i.e. to: S = R n with n >
1, are straightforward,so we will omit details here. 139 .5 The Moyal Bracket(s), ”Moyal” Quantum Mechanicsand the Quantum-Classical Transition
Using the Moyal product we can define the
Moyal Bracket { ., . } M as: { ., . } M : F (cid:0) R (cid:1) × F (cid:0) R (cid:1) → F (cid:0) R (cid:1) ; { f, g } M =: 1 i ~ ( f ∗ g − g ∗ f ) (6.105)Hence, in particular: { f, g } M = { f, g } + O (cid:0) ~ (cid:1) (6.106)where { ., . } is the standard Poisson bracket .Being defined in terms of an associative product, the Moyal bracket fulfillsall the properties of a Poisson bracket (linearity, anti-symmetry and the Jacobiidentity), and defines a new Poisson structure on the (non-commutative) algebraof functions with the Moyal product. In particular, just as for the ordinaryPoisson brackets, the Jacobi identity implies: { f, g ∗ h } M = { f, g } M ∗ h + g ∗ { f, h } M (6.107)i.e. that { f, . } is a derivation (with respect to the ∗ -product) on the algebraof functions. Writing down explicitly the second term in (6.106): { f, g } M = { f, g } + ~ { f, g } + ... , we obtain: { f, g } ( q, p ) = 124 (cid:26) ∂ f∂q ∂ g∂p − ∂ f∂p∂q ∂ g∂q∂p + 3 ∂ f∂p ∂q ∂ g∂q∂q − ∂ f∂p ∂ g∂q (cid:27) (6.108)Therefore, { f, g } M contains, besides first-order derivatives, third and higher-order derivatives, and, although it is a derivation on the algebra of functionswith the ” ∗ ” product, it is not a vector field (while { f, . } is a vector field).The reason for that is precisely that the Moyal bracket is non-local, and henceWillmore’s theorem [231] connecting (inner) derivations with vector fields doesnot apply. It is only when f is at most a quadratic polynomial that { f, . } M becomes a derivation on the usual pointwise product. Indeed, if this is the case,the Moyal and Poisson brackets of f with other functions coincide. As a check,we see that, in simple cases, we obtain: { q, p } M = 1 , { q, g } M = ∂g∂p , { p, g } M = − ∂g∂q (6.109)Using the definitions of the Weyl and Wigner maps we have, in general: { f, g } M = i Ω − (cid:16)b Ω ( f ) · b Ω ( g ) − b Ω ( g ) · b Ω ( f ) (cid:17) / ~ (6.110) The difference between the Moyal and Poisson brackets is O ( ~ ), and not O ( ~ ) as onecould expect, and that because the difference f ∗ g − g ∗ f contains only odd powers of ~ . hb Ω ( f ) , b Ω ( g ) i = − i ~ b Ω ( { f, g } M ) (6.111)In particular, using (6.109) (and: b Ω (1) = I ) : hb Ω ( q ) , b Ω ( p ) i = − i ~I , hb Ω ( q ) , b Ω ( H ) i = − i ~ b Ω ( ∂ H /∂p ) (6.112) hb Ω ( p ) , b Ω ( H ) i = i ~ b Ω ( ∂ H /∂q ) (6.113)Unless f and/or g are at most quadratic, { f, g } M = { f, g } . Therefore, thecommutator of the quantum operators associated with observables on phasesspace is not ( ”modulo” a multiplicative constant) the quantum operator asso-ciated with the Poisson bracket [56]. Generically, it becomes so only to lowestorder in ~ , and reproduces the Ehrenfest theorem [69]. First of all, it is of some interest, in view of the relevant rˆole they play inQuantum Mechanics, to see here which phase-space functions correspond toprojection operators on the Hilbert space. The latter, that we will denote as b P ,are completely characterized by: • b P = b P , idempotency (6.114) • b P † = b P , self − adjointness (6.115)As to (6.115), this requires the associated Wigner function Ω − (cid:16) b P (cid:17) to be real . As to (6.114), this implies, in terms of the Moyal product (cfr. Eq. (6.85)):Ω − (cid:16) b P (cid:17) = Ω − (cid:16) b P (cid:17) = Ω − (cid:16) b P (cid:17) ∗ Ω − (cid:16) b P (cid:17) (6.116)Moreover: T r (cid:16) Ω − (cid:16) b P (cid:17)(cid:17) = T r ( b P ) (6.117)and: T r (cid:16) Ω − (cid:16) b P (cid:17)(cid:17) will be finite iff b P is a finite-rank projection operator.Therefore: Projection operators are represented in phase space by real, uniformly-bounded(cfr. Eq.(6.58)) functions satisfying: f ∗ f = f (6.118) The minus sign in the first commutator stems from the fact that Ω ( p ) = − b P , i.e. ulti-mately from the fact that we are using the symplectic and not the ordinary Fourier transform. nd : T r ( f ) < + ∞ (6.119) iff the associated projector is of finite rank. Density states will be represented inturn by real, again uniformly-bounded, phase-space functions f ( q, p ) satisfying: T r ( f ) = 1 and: T r ( f ∗ f ) ≤ P H . Once thisis identified (via the Hermitian structure, see the discussion in Chapt.4) withthe space of rank-one projectors, it is natural to pose eigenvalue problems notfor vectors in the Hilbert space but for the associate rank-one projectors, i.e. inthe form: b O b P = λ b P ; b P † = b P , b P = b P , T r b P = 1 (6.121)with b O an observable and λ ∈ R the corresponding eigenvalue . Put in thisform, the eigenvalue problem can be easily formulated on phase space. Indeed,denoting by simplicity as f b O = Ω − (cid:16) b O (cid:17) the Wigner function associated with b O , the equivalent phase-space formulation will be: f b O ∗ f = λf ; f ∗ f = f, f ∈ L ( T ∗ Q ) (6.122)for a real (and uniformly-bounded) function f . This will qualify f as the Wignerfunction associated with a projection operator: f = Ω − (cid:16) b P (cid:17) , with: T rf = 1 ifit corresponds to a pure state.A superposition rule capturing also interference phenomena can be formu-lated in terms of Wigner functions [144, 145, 146, 147, 148] following the linesof the discussion of Sect.4.1. If we denote as f the Wigner function associatedwith a reference (pure) state (see Sect.4.1 for more details) and as f , f thoseassociated with two orthogonal (i.e.: f ∗ f = 0) pure states, then to the linearsuperposition with coefficients c and c , | c | + | c | = 1, there correspondsthe Wigner function associated with Eq.(4.15), namely: f = X i,j =1 c i c ∗ j f i ∗ f ∗ f j p T r ( f i ∗ f ∗ f j ∗ f ) (6.123)where the phase-space trace has been defined in Eq.(6.80).Coming now to quantum evolution, an observable (a self-adjoint operator) b O will evolve in time as: b O ( t ) = b U † ( t ) · b O · b U ( t ) (6.124) To avoid unnecessary technical complications, we pose here the problem in the discretespectrum. Also, the last condition in Eq.(6.121) can be relaxed in favor of P becoming then anot necessarily one-dimensional eigenprojector onto the subspace spanned by the eigenvalue λ . b U ( t ) = exp (cid:16) − it b H/ ~ (cid:17) (6.125) b H being the Hamiltonian operator. Denoting again the Wigner function asso-ciated with b O as f b O , and from the very definition of the Moyal product: f b O ( t ) = f b U † ( t ) · b O · b U ( t ) = f b U † ( t ) ∗ f b O ∗ f b U ( t ) (6.126)Using the (formal) series expansion of the evolution operator (6.125) we canalso write explicitly the evolution operator in phase space f b U ( t ) as [12]: f b U ( t ) = exp ∗ (cid:0) − itf b H / ~ (cid:1) =: ∞ X n =0 ( − it/ ~ ) n n ! (cid:0) f b H (cid:1) n ∗ (6.127)where ( . ) n ∗ stands for an n -fold star-product.Now, to lowest order in t : f b U ( t ) ≈ − ( it/ ~ ) f b H etc., and we obtain easily: ddt f b O ( t ) = n f b O ( t ) , f b H o M (6.128)or, more generally: ddt f ( t ) = (cid:8) f ( t ) , f b H (cid:9) M ; f (0) = f (6.129)with f any suitable function (e.g., a square-integrable function) on phase space,leading to: f ( t ) = exp ∗ (cid:0) itf b H / ~ (cid:1) ∗ f ∗ exp ∗ (cid:0) − itf b H / ~ (cid:1) (6.130)and this is the phase-space description of quantum dynamics. As the classical( ~ →
0) limit of the Moyal bracket is the Poisson bracket, Eqs.(6.128) and/or(6.129) reduce, in the classical limit, to the description of the dynamics in termsof Poisson brackets.
We can begin by recalling a theorem due to Dirac (see [56] and [89] for a moregeneral discussion) which states that, given an associative, non-Abelian andmaximally non-commutative algebra A with identity over R or C , and defin-ing a ”Poisson bracket” on A as a map: { ., . } : A × A −→ A (6.131)
That is, such that [89] the derived algebra: A ′ = Span { [ a, b ] } ; a, b ∈ A , together with theidentity, spans the whole of A . Having in mind the algebra of operators on a Hilbert space, Dirac [56] calls it a ”QuantumPoisson bracket”. { a, { b, c }} + { b, { c, a }} + { c, { a, b }} = 0 ∀ a, b, c ∈ A (6.132)and acts as a derivation on the product on the algebra, i.e.: { a, bc } = { a, b } c + b { a, c } ∀ a, b, c ∈ A (6.133)then the Poisson bracket { a, b } is necessarily proportional to the ”standard”commutator ab − ba .This theorem was actually one of the main motivations why, in Chapt.1, wediscussed alternative approaches to Quantum Mechanics involving modificationsof the Hermitian product or, equivalently, of the associative product betweenoperators.Sticking to this last approach, we consider now a ”deformed” associativeproduct between operators defined as: b A · ( b K ) b B =: b A · b K · b B (6.134)where b A, b B are linear operators and b K is a fixed, positive operator which isalso a constant of the motion. This leads to the definition of the ”deformed”commutator: h b A, b B i ( b K ) =: b A · ( b K ) b B − b B · ( b K ) b A (6.135)which satisfies again the Jacobi identity .Given then two phase-space functions f and g , Eq.(6.134) leads to the ”de-formed” Moyal product: f ∗ ( k ) g = f ∗ k ∗ g (6.136)where: k =: Ω − (cid:16) b K (cid:17) (6.137)is the Wigner function associated with the operator b K , and to the ”deformed”Moyal bracket: { f, g } M,k =: 1 i ~ ( f ∗ ( k ) g − g ∗ ( k ) f ) ≡ i ~ ( f ∗ k ∗ g − g ∗ k ∗ f ) (6.138)and, of course: { f, g } M, ≡ { f, g } M (6.139) Remark 55
Requiring the operator b K to be strictly positive is a necessary con-dition for the definition of a sensible ”deformed” Hermitian product on the See, e.g., Ref.[240] for details of the proof.
See Sect.1.2.2 for further details.
See again Sect.1.2.2. ilbert space. If this is the case, then b K is invertible and the new (asso-ciative) algebra structure defined by Eq.(6.136) will have an identity e , givennow by the ” ∗ -inverse” of k : e = k ∗− , where: k ∗− =: Ω − (cid:16) b K − (cid:17) (i.e.: k ∗ k ∗− = k ∗− ∗ k = 1 ). This is of course the counterpart of the fact that theinverse of b K , b K − , plays the rˆole of the identity for the deformed associativeproduct (6.134) on the algebra of operators. Again with reference to the discussion in Sect.1.2.2, and in particular toEq.(1.71), we see that now the dynamics will be described, in phase space, bythe equation: ddt f = (cid:8) f, f b H ′ (cid:9) M,k (6.140)in such a way that (cfr. Eq.(6.129)): (cid:8) f, f b H ′ (cid:9) M,k = (cid:8) f, f b H (cid:9) M (6.141)where the new Hamiltonian function will be given by: f b H ′ = Ω − (cid:16) b H · b K − (cid:17) = Ω − (cid:16) b K − · b H (cid:17) = f b H ∗ f b K − (6.142)Moreover (cfr. Eq.(1.74)), time evolution will act again as a derivation on thedeformed algebra of functions, i.e.: ddt (cid:18) f ∗ ( k ) g (cid:19) = dfdt ∗ ( k ) g + f ∗ ( k ) dgdt , ∀ f, g (6.143)Turning now to the classical limit and using Eq.(6.97), a simple computationshows that, for ~ →
0, Eq.(6.136) becomes: f ∗ ( k ) g ≃ f kg + i ~ { f, g } k + O (cid:0) ~ (cid:1) (6.144)with a ”deformed” bracket is given now by: { f, g } k = lim ~ → { f, g } M,k (6.145)and, explicitly: { f, g } k = k { f, g } + f { k, g } − g { k, f } (6.146)(once again: { f, g } ≡ { f, g } ). Being defined in terms of an associative product,this new bracket satisfies the Jacobi identity, but, at variance with the Poissonbracket and as it is clear from Eq.(6.146), { f, . } k fails to be (for fixed f ) aderivation on the algebra of functions (it is not even zero on constant functions). Also called [129] a
Jacobi bracket . .6.1 Alternative Moyal-like brackets In Section we go back to the GNS construction for the finite-dimensional C ∗ -algebra B ( C n ) we have discussed in 4.3. Recall that different states over B ( C n )give rise to different representations and hence to different realizations of thecorresponding Hilbert space. We have already noticed that any such state is rep-resented by a positive n × n matrix K which can be used to define an alternativescalar product on C n of the form z · K w := n X j,k =1 ¯ z j K jk w k (6.147)for any z, w ∈ C n . In turn, we can define a different multiplication rule in B ( C n )by means of: A · K B = A · K · B (6.148)for any A, B ∈ B ( C n ). This product is associative, so that ( B ( C n ) , · K ) is a C ∗ -algebra. Accordingly, we can define alternative Lie algebra and Jordan algebrastructures via: [ A, B ] K := i A · K B − B · K A ) (6.149) A ◦ K B := 12 ( A · K B + B · K A ) (6.150)Let us consider now a quantum system whose dynamics is specified by a Hamil-tonian H , yielding the standard Heisenberg equation: i ~ ˙ A = [ A, H ] (6.151)Suppose that [
H, K ] = H · K − K ˙ H = 0. By setting H K = K − · H , one caneasily verify that, for any for any A ∈ B ( C n ):[ A, H ] = A · K H K − H K · A = [ A, H K ] K (6.152)Hence we have an alternative Hesienberg-like description which makes use ofthe alternative product (6.148): i ~ ˙ A = [ A, H K ] K (6.153)These alternative structures are therefore analogue to those we have examinedin classical dynamics when we have studied bi-Hamiltonian systems.We can analyze these structures also in terms of the Wigner-Weyl formalismintroduced in the previous paragraphs. We already know (see Sect. 4.2.4) thaton the space of K¨ahler functions on the projective space, F C ( P H ), we can definea star-product, that of formula (4.98), such that: f A ⋆ f B = f AB (6.154)146e can then define an antisymmetric star-bracket according to: { f, g } ⋆ := 12 i ( f ⋆ g − g ⋆ f ) (6.155)for any f, g ∈ F C ( P H ), which yields the standard Poisson bracket in the classicallimit. Now, it is known [206] that any associative local product in F C ( P H ) isof the form: f · k g := f · k · g (6.156)for some k ∈ F C ( P H ), k >
0. With this product, we can now define analternative ⋆ k -product and ⋆ k Lie and Jordan brackets: f A ⋆ k f B = f A ⋆ k ⋆ f B (6.157) { f A , f B } ⋆k = 12 i ( f ⋆ k g − g ⋆ k f ) (6.158) f A ◦ k f B = 12 ( f ⋆ k g + g ⋆ k f ) (6.159)We are back here to the construction of ”deformed” Moyal brackets we havediscussed in the previous paragraph. We have already seen that, in the classicallimit, we get:lim ~ → ~ { f A , f B } ⋆ k = { f, g } + f { k, g } − g { k, f } := { f, g } k (6.160)obtaining the standard Poisson bracket only if k = 1. In a similar way, we seethat: lim ~ → f ◦ k g = f · k · := f · k g (6.161)This shows that the alternative quantization schemes we have introduced in theprevious paragraph depend on the associative products ⋆ k one can define on theoriginally commutative algebra k ∈ F C ( P H ). From now on we will consider the case S = R n for generic n >
1, the mainreason being that most of what will be said becomes trivial for n = 1.As discussed in previous Sections, assigning a Poisson bracket is equivalentto assigning a bi-vector field, i.e. a totally antisymmetric tensor of type (2 , Poisson tensor , given, in local collective coordinates, as:Λ = 12 Λ ij ∂∂ξ i ∧ ∂∂ξ j ; Λ ij + Λ ji = 0 (6.162)and such that: { f, g } = Λ ( df, dg ) (6.163)147n general, on can define, on multivectors, a bracket, the Schouten bracket [194, 209], that associates to every pair
X, Y of multivectors of ranks n and m respectively a multivector [ X, Y ] S of rank n + m −
1. Limiting ourselves tobi-vectors, if X and Y are monomials: X = χ ∧ χ , Y = η ∧ η (6.164)(with the χ ’s and η ’s vector fields), then:[ X, Y ] S = [ χ , η ] ∧ χ ∧ η − [ χ , η ] ∧ χ ∧ η − [ χ , η ] ∧ χ ∧ η +[ χ , η ] ∧ χ ∧ η (6.165)It follows that, if f, g are functions:[ f X, gY ] S = f g [ X, Y ] S + (6.166)+ f ( L χ g ) χ ∧ η ∧ η − f ( L χ g ) χ ∧ η ∧ η ++ g ( L η f ) χ ∧ χ ∧ η − g ( L η f ) χ ∧ χ ∧ η and then the Schouten bracket can be extended by linearity to arbitrary bi-vectors.The Jacobi identity can be expressed in terms of the Poisson tensor as:[Λ , Λ] S = 0 (6.167)and this is equivalent, whenever the Poisson tensor is not degenerate and allowsthen for the definition of a symplectic two-form ω , to the closure of the latter. Remark 56
As, in dimension two, there are no non-vanishing tri-vector fields(and all two-forms are closed), it is clear why what we are saying here becomesessentially void in dimension two. There, every pair of bi-vector fields has avanishing Schouten bracket.
The ”deformed” bracket (6.146) can be rewritten as: { f, g } k = Λ ′ ( df, dg ) + f L X k g − gL X k f (6.168)where: X k =: { k, . } is the Hamiltonian vector field associated with the function k , and: Λ ′ = k Λ (6.169)is what is called [12, 13] a conformal Poisson tensor with conformal factor k .Equivalently: { f, g } k = Λ ′ ( df, dg ) + f { k, g } − g { k, f } (6.170)Due to the presence of the conformal factor, the Schouten bracket of the con-formal Poisson tensor with itself does not vanish anymore. Instead [12]:[Λ ′ , Λ ′ ] S = − X k ∧ Λ ′ (6.171)and also, as X k is a Hamiltonian vector field: L X k Λ ′ ≡ kL X k Λ = 0 (6.172)148 emark 57
The bracket (6.146) is R -linear homogeneous in the conformal fac-tor k . So, any two such brackets with conformal factors, say, k and k , will giverise to a bracket of the same form ( a ”compatible” bracket, in this sense) withconformal factor: k = k + k . This seems to imply that, in order to obtain non-compatible classical limits, one should introduce some amount of non-linearity.This can be done by using non-linearly related Poisson structures. Remark 58
Extrapolating now the Jacobi bracket (6.168) to dimension one,one finds nonetheless an interesting consequence. In this case, and ”a fortiori”’,
Λ = Λ ′ ≡ , and hence: { f, g } k = f L X k g − gL X k f (6.173) If we consider a circle S with angular coordinate ϕ ∈ [0 , π ] and measure dϕ/ π ,consider periodic functions that can be expanded in Fourier series on the O.N. basis: f n = e inϕ , n ∈ Z (6.174) and take: X k = i ∂∂ϕ (6.175) then Eq.(6.173) yields at once: { f n , f m } k = ( n − m ) f n + m (6.176) which is nothing but the classical conformal algebra [55] (i.e. the Virasoro alge-bra without central charge). KMS
Condition in PhaseSpace
We will consider here the algebra A of functions on phase space equipped withthe ∗ -product (the Moyal product for the time being) and with the associatedbracket.Evolution in time on this algebra is an automorphism of A described byEqs.(6.129) and (6.130). In particular, the latter states that: A ∋ f → f ( t ) = exp ∗ (cid:0) itf b H / ~ (cid:1) ∗ f ∗ exp ∗ (cid:0) − itf b H / ~ (cid:1) (6.177)Let now ω be a state on the algebra. Correlation functions will be ingeneral of the form: ω ( f ( t ) ∗ g ( t ′ )), f, g ∈ A . Time-translational invariancewill be assumed [187] for equilibrium states [95]. Hence: ω ( f ( t ) ∗ g ( t ′ )) = ω ( f ( t − t ′ ) ∗ g ) = ω ( f ∗ g ( t ′ − t )) (6.178) i.e. [95] a linear functional that is real, positive and normalized , the latter condition beingequivalent [95] to: ω (1) = 1. g = 1 in Eq.(6.178), weobtain: ω ( f ( t )) ≡ ω ( f ) ∀ f, t (6.179)With any pair f, g ∈ A we can associate the correlation functions [187]: G fg ( t ) = ω ( f ( t ) ∗ g ) (6.180)and: F fg ( t ) = ω ( g ∗ f ( t )) (6.181)Making t into a complex variable, the state ω will be said to be a (Kubo, Martin,Schwinger) KMS state at (inverse) temperature β [3, 95, 96, 103, 117, 181, 187,193] if: • G fg ( t ) is bounded and continuous in the strip: − ~ β ≤ Im t ≤ • The same for F fg ( t ) but in the strip 0 ≤ Im t ≤ ~ β and: • The two are connected by: G fg ( t ) = F fg ( t + i ~ β ) , − ~ β < Im t < KMS condition: ω ( f ( t ) ∗ g ) = ω ( g ∗ f ( t + i ~ β )) (6.183) Remark 59
In the operator language, the KMS condition is usually proved(at least for bounded operators), using the cyclic invariance of the trace [110,187] for systems whose (thermodynamic) equilibrium states are described by thecanonical ensemble or (with minor modifications) by the grand-canonical ensem-ble.
Remark 60
Although the KMS condition is usually stated for equilibriumstates at non-zero temperature, there is a similar condition [103] characterizingthe ground state(s) at zero temperature, namely that G fg ( t ) be, for real times, theboundary value on the real axis of an entire function that is uniformly boundedfor Im t ≤ . Noticing that: f ( t + i ~ β ) = exp ∗ (cid:0) i ( t + i ~ β ) f b H / ~ (cid:1) ∗ f ∗ exp ∗ (cid:0) − i ( t + i ~ β ) f b H / ~ (cid:1) == exp ∗ (cid:0) − βf b H (cid:1) ∗ f ( t ) ∗ exp ∗ (cid:0) βf b H (cid:1) (6.184)and expanding the exponentials in the last expression:: f ( t + i ~ β ) ≃ f ( t ) + i ~ β (cid:8) f ( t ) , f b H (cid:9) M + O (cid:0) ~ (cid:1) (6.185)150nd, as: { ., . } M = { ., . } (the classical Poisson bracket) to lowest order in ~ , weobtain the (correct) expansion: f ( t + i ~ β ) ≃ f ( t ) + i ~ β (cid:8) f ( t ) , f b H (cid:9) + O (cid:0) ~ (cid:1) (6.186)and hence the classical KMS condition [3, 12, 13]: ω ( { f ( t ) , g } ) = βω (cid:0) g (cid:8) f ( t ) , f b H (cid:9)(cid:1) (6.187)Interchanging the rˆoles of f and g and taking differences, we obtain also: ω ( { f ( t ) , g } ) = 12 βω (cid:0) f ( t ) (cid:8) f b H , g (cid:9) − g (cid:8) f b H , f ( t ) (cid:9)(cid:1) (6.188) Remark 61
Setting g = 1 in Eq.(6.188) we obtain: ω (cid:0)(cid:8) f b H , f ( t ) (cid:9)(cid:1) = 0 ∀ f ∈A . Adding then ( − / βω (cid:0)(cid:8) f b H , f ( t ) g (cid:9)(cid:1) = 0 to the r.h.s. of Eq.(6.188) were-obtain Eq.(6.187), and the two are therefore equivalent. Noticing further that:12 β (cid:8) f b H , . (cid:9) ≡ − e (1 / βf c H n e − (1 / βf c H , . o (6.189)we can rewrite Eq.(6.188) in the form: ω (cid:16) e (1 / βf c H h e − (1 / βf c H { f ( t ) , g } + f ( t ) n e − (1 / βf c H , g o (6.190) − g n e − (1 / βf c H , f ( t ) oi(cid:17) = 0Comparison with Eq.(6.170) shows then that: The classical
KM S condition (6.187) is equivalent to the condition ω (cid:16) e (1 / βf c H { f ( t ) , g } k (cid:17) = 0 ∀ f, g ∈ A (6.191) where the bracket on the l.h.s. of Eq.(6.191) is the conformal bracket (6.170)with conformal factor k = exp (cid:0) − (1 / βf b H (cid:1) (6.192)We turn now to the full quantum case (i.e. away from the limit ~ → k β =: exp ∗ (cid:0) − (1 / βf b H (cid:1) (6.193)where: exp ∗ f =: 1 + ∞ X n =1 n ! f ∗ f ∗ ... ∗ f | {z } n times (6.194)151hose ∗ -inverse is k − β . This defines the automorphism: σ : f → σ ( f ) = f ( i ~ β/
2) = k β ∗ f ∗ k − β (6.195)(notice that: σ ( f ∗ g ) = σ ( f ) ∗ σ ( g ) ∀ f, g ) and the KM S condition (6.183) canbe written as: ω ( f ( t ) ∗ g ) = ω (cid:0) g ∗ σ ( f ( t )) (cid:1) (6.196)Substituting now σ ( g ) for g in Eq.(6.196) we find: ω (cid:0) σ ( g ) ∗ σ ( f ( t )) (cid:1) = ω ( σ ( g ∗ σ ( f ( t )))) = ω ( g ∗ σ ( f ( t ))) (6.197)the last passage following from time-translational invariance (Eq.(6.179))and, eventually: ω ( f ( t ) ∗ σ ( g )) = ω ( g ∗ σ ( f ( t ))) (6.198)or, in terms of the deformed Moyal bracket (6.138) with deformation factor k = k β : ω (cid:16) { f ( t ) , g } M,k β ∗ k − β (cid:17) = 0 (6.199)But: { f ( t ) , g } M,k β ∗ k − β = σ h k − β ∗ { f ( t ) , g } M,k β i (6.200)and, using again Eq.(6.179) , we obtain eventually [12, 13]: ω (cid:16) k − β ∗ { f ( t ) , g } M,k β (cid:17) ≡ ω (cid:16) exp ∗ (cid:0) (1 / βf b H (cid:1) ∗ { f ( t ) , g } M,k β (cid:17) = 0 (6.201)which is the quantum version of the classical KM S condition, with exponen-tials replaced by ” ∗ -exponentials” and (deformed) Poisson brackets replaced by(deformed) Moyal brackets. If time-translational invariance is not assumed, then Eq.(6.196) leads, setting g = 1, to: ω (( σ − f ) = 0. As what is needed to complete the argument is instead the condition (seebelow): ω (( σ − f ) = 0, one has then to assume [12] the mapping σ + 1 to be invertible. Additional Topics and Concluding Remarks
Weyl systems, the way we have presented them, have been built with the use ofa specific prescription (whose basic ingredients (see Chapt.5) are a vector spaces E and a symplectic structure over E ) to deal with a specific prescription for theordering problem that arises in the quantization procedure, one that is knownas the ”Weyl ordering” prescription (see Sect.6.1).To deal with other ordering prescriptions that are available in the literature(say, normal, antinormal or other ” s -ordering” prescriptions (see, e.g. Ref.[121])one has to enlarge slightly the setting of Weyl systems.Consider then a symplectic vector space with symplectic form ω ( ., . ), equippedhowever with an additional complex structure and therefore (see Chapt.4) withan Hermitian structure h . | . i . In this way, having the Hermitian structure athand, one can replace the (conventional) Weyl map, i.e.: c W ( v ) c W ( v ) c W − ( v ) c W − ( v ) = e − iω ( v , v ) I (7.1)with the following one: c W ( v ) c W ( v ) c W − ( v ) c W − ( v ) = e −h v | v i I (7.2)Here the r.h.s. is no more a unitary transformation, i.e. an element of U (1),but it is instead an element of C ≡ U (1) × R + .More generally, by splitting, as we have done repeatedly, the Hermitianstructure into its real and imaginary parts: h . | . i = g ( ., . ) + iω ( ., . ), it is possibleto consider a further generalization by setting: c W ( v ) c W ( v ) c W − ( v ) c W − ( v ) = e − sg ( v , v ) − iω ( v , v ) I (7.3)with the ”deformation parameter” s taking values in [ − , Remark 62
Notice that, the metric tensor g having been replaced by sg , thelink between the real and imaginary parts of the Hermitian structure and thecomplex structure gets lost here for all s = ± . From our point of view, this kind of generalization raises a new problemconcerning the Moyal product. Namely, besides the bi-differential operatorexp " i ←−− ∂∂x µ ∧ −−→ ∂∂p µ ! (7.4)we will be forced to consider in addition also the bi-differential operator153xp " s δ µν ←−− ∂∂x µ ⊗ −−→ ∂∂x ν + δ µν ←−− ∂∂p µ ⊗ −−→ ∂∂p ν ! (7.5)In the framework of our ”deformation” construction, and with reference to thediscussion of Nijenhuis operators and of the Hochschild cohomology that issummarized in App. A , it is possible however to show that these additionalterms do not change the cohomology class of the algebra we obtain by usingonly the Poisson tensor, i.e. the bi-differential operator (7.4), as the followingexample shows. Example 63
To illustrate the situation, it will be enough to consider the newproduct on functions defined on R along with the deformation of the usual point-wise product. We can consider then the bilinear map: ( f, g ) → ” f ∗ g ” := f exp ( − s ←− ∂∂x ⊗ −→ ∂∂x ) g (7.6) Now, it is possible to show that the linear map T defined by: T = exp (cid:26) − s ∂ ∂x (cid:27) (7.7) is such that: ” f ∗ g ” = T ( f · g ) − T ( f ) · g − f · T ( g ) (7.8) (with the dot denoting the usual pointwise product), thus proving (see againApp. A ) that the bilinear map (7.6) is indeed a coboundary in the Hochschildcohomology of the algebra of functions with the pointwise product. It is appropriate at this point of our exposition to mention that many aspectsof our mathematical considerations have also appeared in a setting that hasa completely different origin, namely the field of pseudo-Hermitian QuantumMechanics. Pseudo-Hermitian Quantum Mechanics (PHQM) is an attempt togeneralize Quantum Mechanics due mainly to C.M.Bender and collaborators(see, e.g., [15] and references therein). One starts with a Hilbert space equippedwith an Hermitian product h· , , ·i and a Hamiltonian H which is diagonalizablebut is not Hermitian, i.e., in general: h ψ, Hφ i 6 = h Hψ, φ i (7.9)We shall assume for simplicity the spectrum of H to be entirely discrete, thismeaning that the eigenvalue equation H | ψ n i = λ n | ψ n i (7.10)154dmits of a complete set {| ψ n i} n of eigenfunctions which cannot, in general, bechosen to be orthonormal. Suppose in addition that {| ψ n i} n admits of a bi-orthonormal extension {| ψ n i} n , | φ n i} n , i.e. that there exists another completeset {| φ n i} n such that : h φ m | ψ n i = δ mn (7.11)Notice that this implies: h φ m | Hψ n i = λ n δ mn (7.12)which implies in turn: ( λ m h φ m | − h φ m | H ) | ψ n i = 0 ∀ n (7.13)and hence: h φ m | H = λ m h φ m | (7.14)i.e. that the h φ m | ’s are (a complete set of) left eigenvectors of H .Then one has a resolution of identity: I = X n | ψ n ih φ n | = X n | φ n ih ψ n | (7.15)Now one defines a new operator η : η = X | φ n ih φ n | (7.16)which can be easily shown [188] to be invertible, with inverse η − = X | ψ n ih ψ n | (7.17)and positive. Thus one can define a new new Hermitian product that will berelated to the original one by: h ( · , · ) = h· , , η ·i (7.18)In other words, η is a positive operator that behaves as a (1, 1)-type tensorconnecting the new and the old metrics. The latter is then used only to identifythe topology of the vector space of states, which turns out to be equivalent [188]to the one defined by the new scalar product.It is immediate to see that: ( i ) the complete set of eigenfunctions {| ψ n i} n becomes orthonormal w.r.t. h ( · , · ), h ( ψ n , ψ m ) = δ nm and: ( ii ) the Hamiltonian H becomes Hermitian, i.e.: h ( ψ, Hφ ) = h ( Hψ, φ ) (7.19)
Such a set always exists provided {| ψ n i} n is a Riesz basis, i.e. provided one can find abounded invertible operator A and an orthonormal basis {| χ n i} n such that | ψ n i = A | χ n i .Indeed in this case one has: | ψ n i = P m A mn | χ n i with A mn = h χ m | Aχ n i and can set: | φ m i = P j ( A − ) ∗ jm | χ j i . H † = ηHη − (7.20)which is true iff the spectrum of H is real, as one can easily find after checkingthat: H = P m λ m | ψ m ih φ m | , while: ηHη − = P m λ m | φ m ih ψ m | and: H † = P m λ ∗ m | φ m ih ψ m | . Hermiticity of H w.r.t. to the new Hermitian product impliesof course that h ( · , · ) is preserved by the dynamical evolution (while h· , , ·i is not).It is clear that, from our point of view, the problem appears as a sort of inverseproblem, i.e. the problem of determining all Hermitian products which arepreserved by the flow defined by the Hamiltonian H . Clearly, once a solutionhas been found, there exist many others that can be found by using appropriateoperators in the commutant of H . Indeed, if A is such that [ A, H ] = 0, then:( ηA ) H ( ηA ) − ≡ ηHη − = H † (7.21)and this defines the new Hermitian product: h A ( · , · ) = h ( · , A · ) = h· , , ηA ·i (7.22)The appropriate conditions on A will be that it be invertible and that ηA be stilla positive operator, and the conditions on H , namely that it be diagonalizablewith a real and discrete spectrum, appear simply as conditions for the inverseproblem to have a solution (and hence in general many others). Thus, while inthe usual approach one fixes a Hilbert space (and hence an Hermitian product)and looks for observables and unitary evolution, in PHQM it is the dynamicalevolution that is given, and one looks for the Hermitian products that are pre-served by the evolution. Recalling our discussion of Sect. 1.2, we notice alsothat the new scalar product h ( · , · ) induces a new associative product betweenoperators: A · η B = AηB (7.23)It is clear that even if [
A, B ] = 0 then [
A, B ] η = AηB − BηA = 0 in general.For example, if both A and B admits the following decomposition in term ofthe bi-orthonormal system: A = X n a n | ψ n ih φ n | , B = X n b n | ψ n ih φ n | (7.24)so that [ A, B ] = 0, one has:[
A, B ] η = X mn ( a m b n − a n b m ) h φ m | φ n i| ψ m ih φ n | (7.25)which is not zero since not all h φ m | φ n i are necessarily zero. When operatorswith continuous spectra are involved, it may be the case that the Hermitianproducts rendering the Hamiltonian Hermitian need not induce commutationrelations for which the operator is localizable. By this we mean that the positionoperators need not commute w.r.t. the new associative product that has beeninduced on the operators. 156et us end this section by giving a simple example of a pseudo-hermitianoperator [116] . We consider the Hilbert space L ([0 , d ]) with the standardscalar product h· , ·i and an operator H α defined on twice (weakly) differentiablefunctions in L ([0 , d ]) given by the quadratic form: h α ( φ, ψ ) = h φ ′ , ψ ′ i + iαφ ( d ) ∗ ψ ( d ) − iαφ (0) ∗ ψ (0) (7.26)where α is any real number. Some straightforward algebra shows that theeigenvalue problem admits the following solutions: ψ ( x ) = A exp( − iαx ) λ = α (7.27) ψ j ( x ) = A j h cos( k j x ) − i αk j sin( k j x ) i λ j = k j (7.28) λ j = k j , k j = j πd , j = 1 , , · · · provided that αd/π / ∈ Z − { } . It also easy to see that H † α = H − α and itseigenfunctions and eigenvalues are given by: φ ( x ) = B exp( iαx ) λ = α (7.29) φ j ( x ) = B j h cos( k j x ) + i αk j sin( k j x ) i λ j = k j (7.30) λ j = k j , k j = j πd , j = 1 , , · · · Both the sets {| ψ n i} ∞ n =0 and {| φ n i} ∞ n =0 are complete [116] and the coefficients A n , B n can be chosen so that h φ j | ψ k i = δ jk (7.31)which shows that {| ψ n i , | φ n i} ∞ n =0 is a bi-orthonormal basis. Thus the invertiblepositive operator η α , that can now be used to define a new scalar product w.r.t.which H α becomes hermitian, assumes the form: η α = ∞ X j =0 h φ j , ·i φ j (7.32)In ref. [116] it is shown that it can be recast in the following form: η α = I + h φ , ·i φ + θ + iαθ + α θ (7.33)where, for any ψ ( x ) ∈ L ([0 , d ]):( θ ψ )( x ) := − d ( Jψ )( d )( θ ψ )( x ) := 2( Jψ )( x ) − xd ( Jψ )( d ) − d ( J ψ )( d )( θ ψ )( x ) := − ( J ψ )( x ) + xd ( J ψ )( d )with ( Jψ )( x ) := Z x dxψ ( x ) (7.34)which allows to prove explicitly that indeed η α is bounded, invertible and posi-tive. 157 .3 The Rˆole of Linear Structures in Statistical and Quan-tum Mechanics In Sects.3.3.2 and 3.3.3 we have examined the situation in which it is possibleto define alternative linear structures at the classical level. We will examinenow the quantum case.In general, if two non-linearly related linear structures (and associated sym-plectic forms) are available for a classical system, then one can set up twodifferent Weyl systems realized on two different Hilbert space structures madeof functions defined on the same Lagrangian subspace (see the example below)but anyhow with different Lebesgue measures. These two Lebesgue measures,call them dµ and dµ ′ , will be associated with different actions of the Abelianvector group of translations that are not linearly related. When compared bywriting both in the same coordinate system they will not be simply proportionalwith a constant proportionality factor. Functions that are square-integrable inone setting need not be such in the other. Moreover, a necessary ingredientin the Weyl quantization program is the use of the (standard or symplectic)Fourier transform. For the same reasons as outlined above, it is clear that thetwo different linear structures will define genuinely different Fourier transforms.In this way one can ”evade” the uniqueness part of von Neumann’s theo-rem. What the present discussion is actually meant at showing is that there areassumptions, namely that the linear structure (and symplectic form) are givenonce and for all and are unique, that are implicitly assumed but not explicitlystated in the usual formulations of the theorem, and that, whenever more struc-tures are available, the situation can be much richer and lead to genuinely andnon-equivalent (in the unitary sense) formulations of Quantum Mechanics.Let us illustrate these considerations by going back to the example of the1 D harmonic oscillator that has been discussed in Sect.3.3.2. To quantize thissystem according to the Weyl scheme we have first of all to select a Lagrangiansubspace L of R and a Lebesgue measure dµ on it defining then L ( L , dµ ).When we endow R with the standard linear structure ∆ = q∂/∂q + p∂/∂p ,we can choose L = { ( q, } and dµ = dq . Consider now, e.g., the change ofcoordinates: φ : ( q, p ) ↔ ( Q, P ) defined by [66]: q = Q (cid:0) λR (cid:1) , p = P (cid:0) λR (cid:1) (7.35)parametrized by: λ ≥ R = Q + P . Eqs.(7.35) invert to (cid:16) r = p q + p (cid:17) : Q = qK ( r ) , P = pK ( r ) (7.36)158here K is a positive function, the (unique) real solution of the equation : λr K + K − ′ = Q∂/∂Q + P ∂/∂P and take: L ′ = { ( Q, } and: dµ ′ = dQ .Notice that L and L ′ are the same subset of R , defined by the conditions P = p = 0 and with the coordinates related by the relation Q = qK ( r = | q | ).Nevertheless the two Hilbert spaces L ( L , dµ ) and L ( L ′ , dµ ′ ) are not relatedvia a unitary map since the Jacobian of the coordinate transformations is notconstant As a second step in the Weyl scheme, we construct in L ( L , dµ ) the operatorˆ U ( α ): (cid:16) ˆ U ( α ) ψ (cid:17) ( q ) = e iαq/ ~ ψ ( q ) , ψ ( q ) ∈ L ( L , dµ ) , (7.38)whose generator is ˆ x = q , and the operator ˆ V ( h ): (cid:16) ˆ V ( h ) ψ (cid:17) ( q ) = ψ ( q + h ) ψ ( q ) ∈ L ( L , dµ ) , (7.39)which is generated by ˆ π = − i ~ ∂/∂q . The quantum Hamiltonian can be writtenas H = ~ (cid:0) a † a + (cid:1) where a = (ˆ x + i ˆ π ) / √ ~ (here the adjoint is taken withrespect to the complex structure compatible with the Lebesgue measure dµ ).Similar expressions hold in L ( L ′ , dµ ′ ), and we will obtain unitary operatorsˆ U ′ ( α ), ˆ V ′ ( h ) with infinitesimal generators: b X = Q and: b Π = − i ~ ∂/∂Q . Noticethat, when seen as an operator in the previous Hilbert space, ˆ V ′ ( h ) implements[66] translations with respect to the linear structure defined, in the notation ofSect.3.3.2 by: ( ˆ V ′ ( h ) ψ )( q ) = ψ ( q + ( φ ) h ) . (7.40)Denoting as usual with a dagger but also with an additional prime the ad-joints taken with respect to the complex structure compatible with the Lebesguemeasure dµ ′ , the quantum Hamiltonian will be now: H ′ = ~ (cid:0) A †′ A + (cid:1) with A = ( b X + i b Π) / √ ~ .It is interesting to notice that, in the respective Hilbert spaces: [ a, a † ] = I as well as: [ A, A †′ ] = I , so that we obtain two different and not linearly relatedrealizations of the Heisenberg algebra.In terms of the ”uppercase” variables, we obtain [66] with some algebra: b x = (1 + λ b X ) b X (7.41)and: b π = (1 + 3 λ b X ) − b Π (7.42)
Eq.(7.37) below shows that, actually: K = K (cid:0) λr (cid:1) . K is monotonically decreasing for λ ≥ λ = 0 ↔ K ≡
1, while: K ≈ λ →∞ (cid:0) λr (cid:1) − / . In fact: dµ = (1 + 3 λQ ) dµ ′ . b x will be self-adjoint with respect to bothmeasures, the conjugate momentum operator will be not, and indeed, while: b x † = b x †′ = b x and: b π † = b π , we obtain instead [66]: b π †′ = b π − iλ b X (1 + 3 λ b X ) − (7.43)Thus, the C ∗ -algebra generated by ˆ x, ˆ π, I seen as operators acting on L ( L , dµ )is closed, whereas the one generated by ˆ x, ˆ π, I and their adjoints ˆ x †′ , ˆ π †′ , I †′ act-ing on L ( L ′ , dµ ′ ) does not close because we generate new operators wheneverwe consider the commutator between ˆ π and ˆ π †′ . As a consequence, the operatorsˆ x, ˆ π and ˆ x ′ , ˆ π ′ close on the Heisenberg algebra only if we let them act on twodifferent Hilbert spaces generated, respectively, by the sets of the Fock states | n i = 1 √ n ! ( a † ) n | i , (7.44) | N i = 1 √ N ! ( A †′ ) N | i . (7.45) By further considering the example of the 1 D harmonic oscillator, we would liketo examine whether alternative Hamiltonian descriptions do lead to the samethermodynamical description of a given system.Let us start from the classical case, when the symplectic form can be rewrit-ten on R − { } as: ω = dp ∧ dq = dH ∧ ξ (7.46)with: ξ = dt = pdq − qdp H (7.47)and the ”time function” t will be given by: t = (1 /ω ) tan − { mωq/p } , whichemphasizes its local character. Thus R − { } can be identified with S × R + parametrized by dH and dt . The associated canonical partition function iseasily evaluated, and the well-known result [187] is: Z = h − Z R exp {− βH } ω = h − ∞ Z dE exp {− βE ) Z Σ( E ) dt = 1 β ~ ω (7.48)Here Σ( E ) denotes the one-dimensional ”surface” of constant energy E , β =1 /k B T with T the (absolute) temperature and k B the Boltzmann constant, We will restrict here to the canonical ensemble of (both classical and quantum) StatisticalMechanics. h (and: ~ = h/ π ) is a numerically undetermined constant with thedimension of an action .In order to keep track of the correct dimensions of the various physicalquantities involved, let’s consider a new Hamiltonian of the form: H f = β − f ( β H ) (7.49)where β is a ”fiducial” quantity, fixed once and for all and having dimension[ energy ] − , and f ( . ) is a real function . It is easy to prove that if Γ is Hamil-tonian w.r.t. ( H, ω ), then it will be Hamiltonian as well w.r.t. ( H f , ω f ), where ω f is defined as: ω f = dH f ∧ dt (7.50)Having redefined (through the new symplectic form) the volume element inphase space, it is natural to redefine the partition function as: Z f = h − Z R exp {− βH f } ω f (7.51)But then: Z f = h − Z dE f exp {− βE f } Z Σ( E f ) dt (7.52)We notice that the nonlinear change of coordinates (3.93) defines such a trans-formation on the Hamiltonian if we set: f ( β H ) ≡ φ ( H ).We come now to the analogous problem in the context of Quantum Me-chanics. In terms of the creation and annihilation operators a and a † , with thestandard commutation relations: [ a, a † ] = 1 (7.53)one constructs a basis in the Fock space as: | n i = ( a † ) n √ n ! | i (7.54)with | i the Fock vacuum and the standard scalar product, that we will denoteas h . | . i : h n | m i = δ nm (7.55) It is well known that one is forced [187] to introduce it in the context of classical StatisticalMechanics in order to obtain a dimensionless expression for the partition function, so as tomake sense of expressions such as : F = − β − ln Z for the (Helmoltz) free energy. The valueof h is fixed unambiguously at that of Planck’s constant at the quantum level of StatisticalMechanics. We will assume f ′ > i ) to give a sensible meaning tointegrals (see below) over phase space and: ii ) not to change the number of critical points.The original Hamiltonian will correspond of course to f ( x ) = x . T r ˆ O = ∞ X n =0 D n | ˆ O | n E (7.56)in order to be able to calculate the partition function at the quantum level as: Z ≡ T r exp {− βH } = X n h n | exp {− βH }| n i (7.57)Now, we perform a ”nonlinear change of variables” by defining [150, 67] newoperators as: A = f ( b n ) a (7.58)with f ( b n ) a positive, monotonically increasing and nowhere vanishing functionof the number operator b n = a † a .At this point, a little care is required when defining the adjoint of any oper-ator: with the scalar product h . | . i , with which a † is the adjoint of a , the adjointof A is of course: A † = a † f (ˆ n ).It is pretty clear that, b n being a constant of the motion, the equations ofmotion for A and A † will be the same as before. We can however reconstruct adifferent Fock space by assuming the same vacuum and defining new states as: | n i = ( A † ) n √ n ! | i (7.59)with a new scalar product defined as: h n | m i = δ nm (7.60)The nonlinearity of the transformation reflects itself in the fact that, despitethe fact that | n i and | n i are proportional, the linear structure in the Fockspace labeled by ”1” does not carry over to the linear structure of space ”2”.This has to do with the fact that the proportionality factors between the | n i ’sand the | n i ’s depend on n . In other words, if we try to induce on space ”2” alinear structure modeled on that of space ”1” , the latter will not be compatiblewith the bilinearity of the scalar product h . | . i that we have just defined.Now, A † is no more the adjoint of A w.r.t. the new Hermitian structure wehave introduced. If we denote by ( . ) † the adjoint of any operator w.r.t. thesecond Hermitian structure, then we find:( A † ) † = 1 f (ˆ n ) a (7.61)which is quite different from A . The pair { ( A † ) † , A † } will yield a new (”non-linear”) realization of the Heisenberg algebra, and indeed it is immediate to seethat: [( A † ) † , A † ] = 1 (7.62) Note that, with this definition: | n i = { Q N − k =0 f ( k ) }| n i A † ) † and A † will obey the same equations of motion as a and a † , that canbe derived from the previous commutation relations and from the Hamiltonian: e H = A † ( A † ) † + 1 / O as: T r ˆ O = ∞ X n =0 D n | ˆ O | n E (7.63)will lead to the same partition function. We recall here, mainly to fix the notation, what are the main ingredients forthe construction of a Weyl system that were discussed at the beginning of thisChapter. What we need is: • A real, symplectic vector space S whose symplectic form (skew-symmetricand non-degenerate) will be denotes as ω ( ., . ). If S is finite-dimensional,then: dim S = 2 n for some integer n . S will be required (see Sect.3.5.1for more details) to possess also a complex structure J , i.e. a (1 , J = − I n × n and compatible with ω , which means: ω ( z, Jz ′ ) + ω ( Jz, z ′ ) = 0 ∀ z, z ′ ∈ S (7.64)and implies that: g ( ., . ) =: ω ( ., J ( . )) (7.65)( g ( z, z ′ ) = ω ( z, Jz ′ )) will be symmetric and nondegenerate, hence a met-ric and a positive one iff: ω ( z, J.z ) > , ∀ z = 0 (7.66)It is always possible to decompose S into the direct sum of two La-grangian subspaces S and S , S = S ⊕ S , in such a way that, writing(in an unique way): z = ( z , z ) = ( z ,
0) + ( z , , z ∈ S , z ∈ S , ω can be written ”in Darboux form”, being represented by the matrix: k ω ij k = (cid:12)(cid:12)(cid:12)(cid:12) n × n I n × n − I n × n n × n (cid:12)(cid:12)(cid:12)(cid:12) (7.67)i.e.: ω ( z, z ′ ) = z · z ′ − z · z ′ (7.68)163he dot denoting the standard Euclidean scalar product. The (compatible)complex structure J will act as : J : ( z , z ) ( − z , z ) (7.69)The vector space S can be viewed either as the cotangent space of either S or S or, alternatively, as the realification [5] of a complex vector spaceof complex dimension n , in which case, writing, e.g.: z = z + iz , thecomplex structure will act as multiplication by the imaginary unit i . AWeyl system will consist then of: • A map: W : S → U ( H ) ; S ∋ z c W ( z ) ∈ U ( H ) into the set U ( H ) ofthe unitary operators over a Hilbert space H which is strongly continuousand satisfies: c W ( z ) c W ( z ′ ) = c W ( z + z ′ ) exp { iω ( z, z ′ ) / } , ∀ z, z ′ ∈ S (7.70)where (here and in the following) we have set for simplicity ~ = 1. Wehave already discusses how, using Stone’s theorem [201], one can represent c W ( z ) as: c W ( z ) = exp n i b G ( z ) o (7.71)with b G ( z ) (essentially) self-adjoint, b G ( tz ) = t b G ( z ) and: h b G ( z ) , b G ( z ′ ) i = − iω ( z, z ′ ) (7.72) Remark 64
Using the truncated Baker-Campbell-Hausdorff formula one canalso write: exp n it b G ( z ) o · exp n i c tG ( z ′ ) o = exp n it h b G ( z ) + b G ( z ′ ) io · (7.73) · exp (cid:26) − t h b G ( z ) , b G ( z ′ ) i(cid:27) whence, comparing with Eqs.(7.70) and (7.72) and expanding in t : b G ( z ) + b G ( z ′ ) = b G ( z + z ′ ) , ∀ z, z ′ (7.74) Remark 65
To be more precise, the l.h.s.’s of both Eqs.(7.72) and (7.74) shouldbe properly understood [24] as the closures of the commutator and of the sumrespectively.
We know also from Sect.5.2 that, via the von Neumann theorem [223], onecan realize concretely H as the Hilbert space of square-integrable functions overa Lagrangian submanifold Q ⊂ S , and how different realizations of H aremutually unitarily related. Notice that J is not unique. For example [24], if J is a complex structure, then also: J ′ = S − JS will be such if S is any symplectic transformation. e A e B = e A + B e [ A,B ] whenever: [ A, [ A, B ]] = [ B [ A, B ]] = 0.
As long as we do not alter (see Sect.7.3.1) the linear structure in a non-linear way. .4.2 Weyl Systems over a Hilbert Space. Second Quantization
Following the scheme set up in Sect.5.2, assume that we have realized the Hilbertspace H as the (complete) Hilbert space L ( Q ), with Q a Lagrangian submani-fold of the original (real) vector space S . To fix the ideas, and in the notation ofthe previous Subsection, we can take, e.g.: Q = S and, writing now: z = ( q , p )and: c W ( z ) = c W ( q , p ) , we have then, with: ψ ∈ L ( S ) and: x ∈ S : (cid:16)c W ( q , ψ (cid:17) ( x ) =: (cid:16) b U ( q ) ψ (cid:17) ( x ) = ψ ( x + q ) (7.75)and: (cid:16)c W (0 , p ) ψ (cid:17) ( x ) =: (cid:16) b V ( p ) ψ (cid:17) ( x ) = exp { i p · x } ψ ( x ) (7.76)(here too we are setting: ~ = 1).We will consider here H ≃ L ( Q ) as a ”single-particle Hilbert space”, andwe will proceed to setting up a description of an assembly of identical particles,fixing our attention, for the sake of illustration, on the case of particles obeyingBose statistics.We turn now explicitly to the Hilbert space L ( Q ), which is endowed withthe Hermitian (linear in the second factor) scalar product ( d x standing for theLebesgue measure): h ( ψ, ψ ′ ) =: Z d x ψ ( x ) ψ ′ ( x ) (7.77)Writing: ψ = u + iv for every ψ ∈ H , the complex Hilbert space H can berealified [5] into the real linear vector space of pairs ( u, v ), equipped with botha (positive) metric: g (( u, v ) , ( u ′ , v ′ )) = Z d x [ uu ′ + vv ′ ] = Re h ( ψ, ψ ′ ) (7.78)and a symplectic form: ω (( u, v ) , ( u ′ , v ′ )) = Z d x [ uv ′ − vu ′ ] = Im h ( ψ, ψ ′ ) (7.79)i.e.: h ( ., . ) = g ( ., . ) + iω ( ., . ) (7.80)with the complex structure (see the previous Subsection) acting as: J : ( u, v ) ( − v, u ) (7.81)(and hence: g (( u, v ) , ( u ′ , v ′ )) ≡ ω (( u, v ) , J ( u ′ , v ′ ))).One can set up now a Weyl system in the form:( u, v ) c W ( u, v ) (7.82) c W ( u, v ) c W ( u ′ , v ′ ) = c W ( u + u ′ , v + v ′ ) exp { iω (( u, v ) , ( u ′ , v ′ )) / } c W ( u, v ) as: c W ( u, v ) = exp n i b G ( u, v ) o (7.83)with a self-adjoint generator b G , we have (cfr. Eqs.(7.72) and (7.74)): h b G ( u, v ) , b G ( u ′ , v ′ ) i = − iω (( u, v ) , ( u ′ , v ′ )) (7.84)as well as: b G ( u, v ) = b Π ( u ) + b Ψ ( v ) ; b Π ( u ) =: b G ( u, , b Ψ ( v ) =: b G (0 , v ) (7.85)with the commutation relations : h b Ψ ( v ) , b Π ( u ) i = i Z d x u ( x ) v ( x ) (7.86)as well as: h b Ψ ( v ) , b Ψ ( v ′ ) i = hb Π ( u ) , b Π ( u ′ ) i = 0 (7.87)Being R -linear in their arguments, it is customary to represent both opera-tors b Ψ and b Π in the form [24, 213]: b Ψ ( v ) = Z d x b Ψ ( x ) v ( x ) ; b Π ( u ) = Z d x b Π ( x ) u ( x ) (7.88)i.e. in terms of the distribution-valued (Hermitian) field operator b Ψ ( x ) andof its conjugate momentum b Π ( x ) obeying, as a consequence of Eqs.(7.86) and(7.87), the (equal-time) commutation relations: h b Ψ ( x ) , b Π ( x ′ ) i = iδ ( x − x ′ ) (7.89)as well as: h b Ψ ( x ) , b Ψ ( x ′ ) i = hb Π ( x ) , b Π ( x ′ ) i = 0 , ∀ x,x’ (7.90)These operators are easily recognized to be appropriate for the description[213] of a bosonic field. Having constructed (admittedly in a partly heuristicway) the algebra of field operators, one should then proceed to construct thephysical vacuum and of the associated Hilbert space on which this algebra ofoperators acts via, e.g., the GN S construction [8, 95] or defining [24], in termsof the c W ’s, a generating functional for the Wightman functions [95], using the”reconstruction theorem” of Axiomatic Field Theory [216]. We shall outlinehere however a slightly different route that leads more directly to the usualFock space description of (bosonic) quantum fields. See however the Remark following Eq.(7.74).
We do not discuss here problems of uniqueness of the vacuum state. ψ = ( u, v ) for the (real) pair ( u, v ),we can use the generators: b G ( ψ ) =: b G ( u, v ) to define annihilation and creationoperators b a ( ψ ) and b a † ( ψ ) associated with the state ψ as: b a ( ψ ) = 1 √ h b G ( ψ ) + i b G ( Jψ ) i ; b a † ( ψ ) = 1 √ h b G ( ψ ) − i b G ( Jψ ) i (7.91)A little algebra shows then that:[ b a ( ψ ) , b a ( ψ ′ )] = (cid:2)b a † ( ψ ) , b a † ( ψ ′ ) (cid:3) = 0 ∀ ψ, ψ ′ (7.92)while: (cid:2)b a ( ψ ) , b a † ( ψ ′ ) (cid:3) = h ( ψ, ψ ′ ) (7.93)If we consider in particular an ON basis { ψ n } ∞ in the ”single-particle”Hilbert space H ( h ( ψ n , ψ m ) = δ nm ) and define: b a n =: b a ( ψ n ) (7.94)then: (cid:2)b a n , b a † m (cid:3) = δ nm (7.95)and all the other commutators vanish. With these operators at hand, one canthen proceed to the construction of the Fock space following, e.g., the approachdiscussed by J.M.Cook [45] already in the early Fifties.Of course, one can also work directly with the exponential form (7.70) of aWeyl system, as we will see now. The possibility to do so relies on the followingobservations that can be easily verified if we work on a finite n -dimensionalHilbert space H . We will denote with K the space of (complex) functions f ( z ) = f ( z , z , · · · , z n ), z j ∈ C , on H which are square-integrable according to the(Gaussian) measure: k f k =: Z n Y j =1 d Re z j d Im z j π e −h z,z i | f ( z ) | < ∞ (7.96)On such space, for any z ∈ H let us consider the operator: W ( z ) : f ( w ) f z ( w ) =: f ( w − z ) exp (cid:18) h w, z i − h z, z i (cid:19) (7.97)which: i ) conserves the norm: k f k = k f z k and ii ) satisfies the relation W ( z ) W ( z ′ ) = W ( z + z ′ ) exp (cid:18) i Im h z, z ′ i (cid:19) (7.98)and hence allow for the definition of a Weyl system which is irreducible on thesubspace of K of antiholomorphic functions, F K , which can be seen [8] as the For example, if H = L ( R ), we could choose [8] the basis of the eigenfunctions of the 1 D harmonic oscillator (the Hermite functions [77]). n variables z , z , · · · , z n . A straightforward calculation shows thatsetting u ( j ) = (0 , · · · , , u j , , · · · ,
0) and v ( j ) = (0 , · · · , , iv j , , · · · , u j , v j ∈ R , one has: iG ( u ( j ) ) = − ∂∂ Re w j + ¯ w j iG ( v ( j ) ) = − ∂∂ Im w j − i ¯ w j a j =: G ( u ( j ) ) − iG ( v ( j ) ) √ √ i∂ w j − i √ w j = − i √ w j on F K ˆ a † j =: G ( u ( j ) ) + iG ( v ( j ) ) √ √ i∂ ¯ w j and clearly satisfy bosonic-like commutation relations. Then the vacuum (orcyclic vector) is given by the constant unit monomial P ( z ) = 1. Notice alsothat for any unitary operator U ∈ U ( H ) we may construct a unitary operatorΓ( U ) ∈ U ( F K ) via the map:Γ( U ) : W ( z ) W ( U − z ) (7.99)A generalization of such results to an infinite dimensional Hilbert space H requires of course caution in the definition of domains of operators as well inthe definition of the spaces K and F K . This can be done by introducing theso called isonormal [8] distribution g , which determines a measure dg on theHilbert space which, when restricted to finite dimensional subspaces looks like aGaussian measure with variance σ , and defining the space K as the completionof the space of polynomials on H w.r.t. the inner product Z L P ′ ( ψ ) P ( ψ ) dg ( ψ ) (7.100) L being any finite-dimensional subspace of H on which the polynomials P, P ′ have support. The space F K is now the subspace of those functions F on H such that their restrictions F | L on any finite-dimensional subspace L areantiholomorphic in the usual sense. Thus one gets a complex representation forthe bosonic field in which the Weyl system is given by the operators [8]: W ( ψ ) : F ( φ ) F ( φ − ψ ) exp (cid:18) h ψ, φ i σ − h φ, φ i (cid:19) , ∀ ψ ∈ H (7.101)For such representation, the cyclic vector is the function on H identically equalto one. Also, for any U ∈ U ( H ) we have a unitary operator Γ( U ) ∈ U ( F K ) suchthat W ( ψ ) W ( U − ψ ) 168his completes the discussion of how Weyl’s approach can lead, in a rathernatural and elegant way, to the formalism of second quantization and hence ofField Theory. We have done that for bosons, and we refer to the literature (seein particular Refs. [8],[24] and [45]) for the parent construction for the case offermions. Alternative Hilbert space structures will give rise also to additionalambiguities in the commutation relations for the fields. By using the geometrical formulation of Quantum Mechanics we have bee ableto ”export” from the classical to the quantum framework many problems thatarise in the classical setting, and we have constructed a more direct ”bridge”which realizes Dirac’s demand [56]that problems arising in Classical Mechanicsmust be a suitable limit of analogous problems arising in Quantum Mechanics.In particular, we have addressed the problem of the quantum interpreta-tion of the bi-Hamiltonian description of completely integrable systems in theclassical setting.Alternative quantum Hamiltonian descriptions have been provided in variouspictures of Quantum Mechanics, the Schr¨odinger, Heisenberg and Weyl-Wigner-Moyal pictures.We have also shown that it is possible to deal with nonlinear transformationsin Quantum Mechanics without giving up the superposition principle which isassociated with quantum interference phenomena.The rˆole of dynamically determined structures versus pre-assigned mathe-matical structures in the formalization of Quantum Mechanics has been furtherelucidated.One may wonder if, in analogy with what happens in General Relativity,where the metric is determined by solving the Einstein equations, one can con-ceive of some field equations whose solutions would provide the Hermitian tensorto be used in the description of quantum systems.By mentioning how to deal with Second Quantization and Quantum FieldTheories in this framework we have hinted at the idea that this approach mayprovide suggestions for the introduction of interactions in a pure quantum field-theoretic setting.At the end of this journey, we believe it to be rewarding to know that manysophisticated methods of Classical Physics may find their way into the formalismof Quantum Physics. 169
Nijenhuis torsions and Nijenhuis Tensors
Nijenhuis Torsions and Tensors on Smooth Manifolds
Let us consider, to begin with, the set X ( M ) of vector fields over some (smooth)manifold M . X ( M ) has, as is well known, the structure of a (actually an infinite-dimensional) Lie algebra defined by the Lie bracket:[ ., . ] : X ( M ) → X ( M ) ; ( X, Y ) [ X, Y ] =: L X Y = −L Y X ; X, Y ∈ X ( M )(A.1)with L · the Lie derivative. Let then T be a (1 −
1) tensor viewed as a map: T : X ( M ) → X ( M ). One can associate with T an antiderivation d T ofdegree one whose actions on zero- and one-forms is given by: d T f ( X ) = df ( T X ) (A.2)on functions, and: d T θ ( X, Y ) = ( L T X θ ) ( Y ) − ( L T Y θ ) ( X ) + θ ( T [ X, Y ]) (A.3)on one-forms (recall that a (anti)derivation is entirely defined [41] by its actionon zero- and one-forms). One proves that d T is a derivation (of degree two)commuting with d : d ◦ d T = d T ◦ d . As such, its action is entirely defined [41]by that on zero-forms (functions), and one finds: (cid:0) d T f (cid:1) ( X, Y ) = − df ( N T ( X, Y )) (A.4)where [78, 152, 186, 194] the
Nijenhuis torsion N T of T is the (1 − : N T ( X, Y ) = { ( T ◦ L X ( T )) − ( L T X ( T )) } ( Y ) (A.5)or, more explicitly: N T ( X, Y ) = T [ T X, Y ] + T [ X, T Y ] − T [ X, Y ] − [ T X, T Y ] (A.6) T will be said to be a Nijenhuis tensor if its Nijenhuis torsion vanishes, i.e. if: N T = 0 (A.7) Remark 66
In local coordinates x i , if: T = T i j ∂∂x i ⊗ dx j (A.8) See Ref. [186] for more details
Note that what we call here, following the literature, the ”Nijenhuis torsion” was calledthe ”Nijenhuis tensor” in Ref. [186]. hen: N T = 12 ( N T ) i km ∂∂x i ⊗ dx k ∧ dx m (A.9) where: ( N T ) i km = ∂T i k ∂x j T j m + T i j ∂T j m ∂xk − ( k ←→ m ) (A.10) and, obviously: N T = 0 whenever the representative matrix of T is a matrixwith constant entries. Nijenhuis Torsions and Tensors on Associative Algebras
Eqn.(A.5) defines the Nijenhuis torsion on a Lie algebra. Nijenhuis-type tensorsand torsions can be given however a more general setting [33, 34]in the frame-work of associative algebras. We recall that an associative algebra ( A , ∗ )becomes also a Lie algebra under commutation, i.e. with a bracket defined as:[ A, B ] =: A ∗ B − B ∗ A ; A, B ∈ A (A.11)and associativity of the algebra guarantees that the bracket does satisfy theJacobi identity, so it is indeed a Lie bracket.Let then ( A , ∗ ) be an associative algebra over a field K ( K = R or K = C for our purposes), an let: T : A → A be a linear map. T will be a derivation ofthe algebra ( A , ∗ ) if (and only if): T ( A ∗ B ) = T ( A ) ∗ B + A ∗ T ( B ) ∀ A, B ∈ A (A.12)Be it as it may, given T one can define in general the bilinear map: ∗ T : ( A, B ) → A ∗ T B = T ( A ) ∗ B + A ∗ T ( B ) − T ( A ∗ B ) (A.13)and ∗ T will be trivial if (and only if) T is a derivation. In general (with T not a derivation), ∗ T will define a (non-trivial) new algebra structure ( A , ∗ T ).As a simple example, let’s take T ∈ A , an hence: T ( A ) = T ∗ A . Then, asimple calculation shows that: A ∗ T B = A ∗ T ∗ B (A.14)Products of this sort will be employed in the text in the discussion of alternativecommutation relations in Quantum Mechanics. It goes without saying that the ”star-product” ∗ we are talking about here has nothingto do with the Moyal product. Digression on: Hochschild Cohomologies
Given an associative algebra ( A , ∗ ) and an A -bimodule V (what we will have inmind will be the case in which V is the additive group of A and the bimodulestructure is given by left and right multiplication), an n-cochain will be an n -linear mapping: α : A × A× ... A | {z } n times → V (A.15)The space C n ( A , V ) of n -cochains has a group structure under addition. Then,for every n , the Hochschild coboundary operator : δ ∗ : C n ( A , V ) → C n +1 ( A , V )is defined ( α ∈ C n ( A , V ) , a , ..., a n +1 ∈ A ) via [101]:( δ ∗ α ) ( a , ..., a n +1 ) = a α ( a , ..., a n +1 ) ++ n X n =1 ( − i α ( a , .., a i ∗ a i +1 , .., a n +1 ) ++ ( − n +1 α ( a , ..., a n ) a n +1 (A.16)where aα ( .. ) and α ( .. ) a denote the left and right actions of A on V respectively.One can check directly that: δ ∗ ◦ δ ∗ = 0 (A.17)As an example, for n = 1:( δ ∗ α ) ( a , a ) = a α ( a ) + α ( a ) a − α ( a ∗ a ) (A.18)An n -cochain α is called an n − cocycle if δ ∗ α = 0, an n − coboundary if α = δ ∗ β for some ( n − − cochain β . n − cocycles form an additive groupusually denoted as Z n ( A , V ), and (in view of (A.17)) n − coboundaries forman subgroup B n ( A , V ) of Z n ( A , V ). The n − (Hochschild) cohomology group H n ( A , V ) is defined then as the quotient: H n ( A , V ) = Z n ( A , V ) / B n ( A , V ) (A.19)The linear mapping T can be considered as a one-cochain and, looking then atEqn.(A.13)we can conclude that: A ∗ T B = δ ∗ T ( A, B ) (A.20)and hence we can rephrase what has been said previously by saying that T willbe a derivation if and only if it is a one-cocycle in the Hochschild cohomologyassociated with the ”star-product”.The ∗− Nijenhuis torsion of T is defined as: N T ( A, B ) = T ( A ∗ T B ) − T ( A ) ∗ T ( B ) (A.21) The suffix serves here to stress that the operators and the ensuing properties are allrelative to the binary product (”star-product”) in the algebra. N T ( A, B ) = T ( T ( A ) ∗ B ) + T ( A ∗ T ( B )) − T ( A ∗ B ) − T ( A ) ∗ T ( B ) (A.22)It is clear from Eqn.(A.21) that the Nijenhuis torsion of T measures the ob-struction for the linear map T to be a homomorphism of the two products.Here too it will be said that T is a ∗− Nijenhuis tensor if its Nijenhuis torsionvanishes. For example, it is easy to see that N T = 0 if T ∈ A and the associatedproduct is given by Eqn.(A.14). Hence, T is a Nijenhuis tensor. Making Contacts
To make contact with the initial definition of the Nijenhuis torsion, we recallwhat has already been said, i.e. that an associative algebra can be made into aLie algebra using the commutator (A.11). If we substitute the ”star-product”with the commutator, then Eqn.(A.22) becomes: N T ( A, B ) = T [ T ( A ) , B ] + T [ A, T ( B )] − T [ A, B ] − [ T ( A ) , T ( B )] (A.23)which coincides with Eqn.(A.6) if we substitute for A, B, .. vector fields on amanifold and the commutator with the Lie bracket. This establishes the linkbetween the two definitions of the Nijenhuis torsion that have been given here.The Nijenhuis torsion defined on an associative algebra will play a rˆole in thediscussion, in the text, of alternative associative products on the algebra of(bounded) operators on a Hilbert space. Completeness would require discussingalso how the (Lie) algebra of vector fields can be embedded into a larger asso-ciative algebra (the enveloping algebra), but we will not insist on this point nottoo lengthen too much the discussion.173
Recursion Operators
Some Preliminaries
Let T be a (1 , T ∈ F ( M ). As is already known, the actionof T on vector fields (denoted with the same symbol) and one-forms (definedas e T ) is defined uniquely by: h T X | α i =: D X | e T α E , X ∈ X ( M ) , α ∈ X ∗ ( M ) (B.1)where h . | . i denotes the usual pairing. In coordinates, if: T = T ij dx j ⊗ ∂∂x i (B.2)is represented by the matrix : T = (cid:13)(cid:13) T i j (cid:13)(cid:13) then e T will be represented by thematrix: e T =: (cid:13)(cid:13)(cid:13) e T j i (cid:13)(cid:13)(cid:13) and Eqn.(B.1) implies: e T j i = T i j (B.3)i.e. that e T be the transpose of T : e T = T t (B.4)All this is well known and is repeated here only for completeness.One can consider extending the action of the e T on forms oh higher rank, aswell as that of T on multivectors. We will concentrate here only on the former,recollecting some results that can be found in the literature ([186]).The extension under consideration is not unique. Let, e.g., ω be a two-form.In particular, ω will be considered as the map: ω : X ( M ) → X ∗ ( M ) ; ω : Y → ω ( ., Y ) = − i Y ω h ω ( ., Y ) | X i = − i X i Y ω = ω ( X, Y ) (B.5)(( ω ( ., Y )) = ω ij Y j dx i ). Hence we can compose e T with ω to obtain the (0 , e T ◦ ω : ( X, Y ) → D e T ◦ ω ( ., Y ) | X E = h ω ( ., Y ) | T X i (B.6)i.e.: (cid:16) e T ◦ ω (cid:17) ( X, Y ) = ω ( T X, Y ) (B.7)This is a linear extension. In terms of representative matrices e T ◦ ω is representedby the matrix T t ω , i.e. (cfr. Eqn.(B.3)): e T ◦ ω = (cid:0) T t ω (cid:1) ij dx i ⊗ dx j = T k i ω kj dx i ⊗ dx j (B.8) With some abuse of notation, we will denote here with the same symbol (1 ,
1) tensors andtheir representative matrices. (cid:16) e T ◦ ω (cid:17) ( X, Y ) = ω ( T X, Y ) + ω ( X, T Y ) (B.9)Also, a nonlinear extension such as: (cid:16) e T ◦ ω (cid:17) ( X, Y ) = ω ( T X, T Y ) (B.10)may be envisaged, with even more possibilities for forms of higher rank.Notice that, while the extensions (B.9) and (B.10) map two-forms into two-forms, this is not true in general for the extension (B.7) which will yield ingeneral a (0 , T of an antideriva-tion of degree one usually denote as d T that acts on zero- and one-forms as: d T f = e T df ; d T f ( X ) =: df ( T X ) (B.11)and: ( d T θ ) ( X, Y ) = ( L T X θ ) ( Y ) − ( L T Y θ ) ( X ) + θ ( T [ X, Y ]) (B.12) d T can be shown to be nilpotent (cid:0) d T ◦ d T =: d T = 0 (cid:1) like the ordinary exteriordifferential d if and only if T has a vanishing Nijenhuis torsion, but we will notinsist on that.Returning instead to the extension (B.7), one can prove the following: The extension of the action of T on two-forms defined by: (cid:16) e T ◦ ω (cid:17) ( X, Y ) =: ω ( T X, Y ) (B.13) will be a two-form (i.e. it will be skew-symmetric) if and only if: ω ( T X, Y ) = ω ( X, T Y ) ∀ X, Y (B.14)Indeed, if the condition (B.14) holds, then: (cid:16) e T ◦ ω (cid:17) ( X, Y ) =: ω ( T X, Y ) = − ω ( Y, T X ) == − ω ( T Y, X ) = − (cid:16) e T ◦ ω (cid:17) ( Y, X ) (B.15)and e T ◦ ω is skew-symmetric. Viceversa, if ω =: e T ◦ ω is skew-symmetric, then: ω ( X, T Y ) = − ω ( T Y, X ) = − ω ( Y, X ) == ω ( X, Y ) = ω ( T X, Y ) (B.16)and (B.14) holds. (cid:4)
Notice that, in this case: ω ( T X, Y ) = 12 { ω ( T X, Y ) + ω ( X, T Y ) } (B.17)and there is no real difference between the two linear extensions.175 − weak and ω -weak Recursion Operators. Strong Recur-sion Operators Let Γ be a Hamiltonian vector field with Hamiltonian H w.r.t. a givensymplectic form ω , i.e.: i Γ ω = d H (B.18)Then [53, 120, 179, 239], a (1 , T compatible with thedynamics, i.e. such that: L Γ T = 0 (B.19)is called: • A H - weak recursion operator if it ”generates new Hamiltonians” in thesense that: d (cid:16) e T k d H (cid:17) = 0 , k = 1 , , , ... (B.20)i.e., locally at least: e T k d H = d H k , k ≥ H k ∈ F ( M ). It is called instead: • A ω - weak recursion operator if it ”generates new symplectic forms” in thesense that: ω k =: e T ◦ e T ◦ .... ◦ e T | {z } k times ω =: e T k ◦ ω, k = 1 , , ... (B.22)is closed and skew-symmetric (and hence a symplectic form if T is invert-ible). Finally, T is called: • A strong recursion operator if it is both H -weak and ω -weak.Before discussing the conditions under which a (1 ,
1) tensor is H -weak and/or ω -weak, let us examine some consequences of these definitions.First of all, if T is H -weak, it may well happen that: d H k ∧ d H = 0 forsome k (even for k = 1 ) , and the process of generating new Hamiltonianfunctions will stop at this stage. Barring this case, one can generate then a setof ω -Hamiltonian vector fields Γ k via: i Γ k ω = d H k , k ≥ T one finds at once: d ( L Γ H k ) = 0 (B.24) This seems to be the case for the Kepler problem [179] L Γ H k = const. and not that H k is a constant of the motionfor Γ. This will require some additional assumptions that will be discussedshortly below.If instead T is ω -weak, taking again the Lie derivative w.r.t. Γ of Eqn.(B.22),invariance of T leads at once to: L Γ ω k = 0 , k ≥ ω k -Hamiltonian. Then, locally at least: i Γ ω k = d e H k (B.26)for some e H k ∈ F ( M ), and this will provide alternative Hamiltonian descriptions for the same dynamics. Notice that the e H k ’s are not related (at least not in asimple way) to the H k ’s of Eqn.(B.21). Alternatively, one can define a new setof vector fields e Γ k via: i e Γ k ω k = d H (B.27)and these will be all Hamiltonian vector fields associated with different sym-plectic structure but with the same Hamiltonian function.Some relevant results concerning H -weak and/or ω -weak recursion operatorshave been proved in the literature. The main results that we will summarizehere (referring to the literature for details of the proof) are:1. If T satisfies the condition (B.20) for k = 1, i.e.: d (cid:16) e T d H (cid:17) = 0 (B.28)and has vanishing Nijenhuis torsion: N T = 0 (B.29)then it is a H -weak recursion operator (i.e. Eqn.(B.20) will hold for every k ). (cid:4)
2. If, moreover, e T ◦ ω is skew-symmetric, which means, in terms of the rep-resentative matrices, ω being already skew-symmetric: T t ω = ωT (B.30)then the H k ’s defined by Eqn.(B.21) are all constants of the motion for Γpairwise in involution: {H k , H l } =: ω (Γ l , Γ k ) = 0 ∀ k, l ≥ { ., . } denotes the Poisson bracket associated with the symplecticform ω . (cid:4) Remark 67
This last result has the following implications: As ω is non-degenerate, there can be at most a set of k ≤ n = (1 /
2) dim ( M ) (functionally) independent constants of the motion pairwise in involution,and: • If the set is maximal (i.e. k = n ), the dynamics is completely integrablein the Liouville sense. Concerning ω -weak recursion operators, it has also been proved in the lit-erature that, if T has a vanishing Nijenhuis torsion and, moreover, e T ◦ ω isclosed: d (cid:16) e T ◦ ω (cid:17) = 0 (B.32)and is skew-symmetric (Eqn.(B.30)), then T is a ω -weak recursion operator. (cid:4) All this has the consequence that: • If T has a vanishing Nijenhuis torsion: N T = 0 (B.33) If : • e T ◦ ω is skew-symmetric, i.e., in terms of the representative matrices: T t ω = ωT (B.34) and if: • both e T ◦ ω and e T d H are closed: d (cid:16) e T ◦ ω (cid:17) = d (cid:16) e T d H (cid:17) = 0 (B.35) then T is a strong recursion operator. (cid:4) In the next Section we shall discuss a relevant class of recursion operatorsthat happen to satisfy almost all of the above conditions.
Factorizable Recursion Operators
We will consider here dynamical systems that are bi-Hamiltonian . A dy-namical vector field Γ is bi-Hamiltonian if there exist two pairs ( ω , H ) and( ω , H ) such that : i Γ ω = d H (B.36) Or, for that matter, bi-Lagrangian . In the Lagrangian case the same rˆole will be played by the Lagrangian two-forms and theassociated energy functions. i Γ ω = d H (B.37)At least one of the two closed two- forms, say ω , will be assumed to be non-degenerate, hence a symplectic form. As such, it will have an inverse ω − whichwill be the bivector (actually a (2 , ω − = 12 ( ω ) ij ∂∂x i ∧ ∂∂x j ; ( ω ) ik ( ω ) kj = δ i j (B.38)Out of the two symplectic forms we can then build up the (1 ,
1) tensor T defined via: (cid:16) e T ◦ ω (cid:17) ( X, Y ) =: ω ( T X, Y ) = ω ( X, Y ) (B.39)or, for short: T = ω − ◦ ω (B.40)Explicitly: T = T i j dx j ⊗ ∂∂x i ; T i j = ( ω ) ik ( ω ) kj (B.41)(1 ,
1) tensors that can constructed via the composition of a (2 ,
0) and of a (0 , factorizable .From now on, ω and H will play the rˆole of the ω, H of the previous Section. Remark 68
It is pretty obvious from the definition (B.39) that:
Ker ( T ) ≡ Ker ( ω ) (B.42) As the kernel of a closed two-form is is a Lie subalgebra of X ( M ) , i.e. it isinvolutive, if: dim Ker ( T ) has constant dimension, it is also a distribution.Moreover, T will be invertible (cid:0) det (cid:13)(cid:13) T i j (cid:13)(cid:13) = 0 (cid:1) iff, besides ω , ω is also non-degenerate, and hence symplectic as well. The (1 ,
1) tensor T is a natural candidate for a recursion operator. Indeed,let us prove first that the closure condition for e T d H is satisfied. We have: T d H = ∂ H ∂x i T i j dx j ≡ ∂ H ∂x i ( ω ) ik ( ω ) kj dx j (B.43)But: i Γ ω = d H implies: ∂ H ∂x i ( ω ) ik = Γ k (B.44)and hence: T d H = d H (B.45)which proves that T d H is not only closed, but also exact. (cid:4) ω ( T X, Y ) = ω ( X, Y ) = − ω ( Y, X ) (B.46)= − ω ( T Y, X ) = ω ( X, T Y )which proves (cfr. Eqn. (B.14)) that e T ◦ ω is skew-symmetric.This result could have been inferred more directly from Eqn.(B.39) whichstates that: e T ◦ ω = ω (B.47)which allows us also to conclude that e T ◦ ω is a closed two-form.Therefore we obtain the following result: If the (1 , tensor field ( B.40 ) satisfies the Nijenhuis condition, i.e. if: N T = 0 (B.48) then T is a strong recursion operator. (cid:4) Symplectic Fourier Transform
Introduction
Let us consider, for simplicity [77, 240], R ≈ T ∗ R with coordinates ( q, p ). Thestandard Fourier transform (e.g. in L (cid:0) R (cid:1) ) of a function f = f ( q, p ) is definedas: F ( f ) ( η, ξ ) = Z Z dqdp π exp {− i ( qη + pξ ) } f ( q, p ) (C.1)with the known inversion formula (again in the sense of L (cid:0) R (cid:1) ) : f ( q, p ) = Z Z dηdξ π exp { i ( qη + pξ ) }F ( f ) ( η, ξ ) (C.2)Notice that, with the standard Euclidean metric in R , g = diag (1 , qη + pξ = g (( q, p ) , ( η, ξ ). Introducing the canonical symplectic form ω D = dq ∧ dp , with representative matrix:Ω D = (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) (C.3)the symplectic Fourier transform F s ( f ) is defined as: F s ( f ) ( η, ξ ) =: Z Z dqdp π exp {− iω D (( q, p ) , ( ξ, η ) } f ( q, p ) (C.4)where, explicitly: ω D (( q, p ) , ξ, η ) = (cid:12)(cid:12) q p (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) ξη (cid:12)(cid:12)(cid:12)(cid:12) = qη − pξ (C.5)Therefore: F s ( f ) ( η, ξ ) = F ( f ) ( η, − ξ ) (C.6)and the transform can be inverted into: f ( q, p ) = Z Z dηdξ π exp { iω D (( q.p ) , ( ξ, η )) } F s ( f ) ( η, ξ ) (C.7)or: f ( q, p ) = Z Z dηdξ π exp {− iω D (( ξ, η ) , ( q, p )) } F s ( f ) ( η, ξ ) (C.8)where, explicitly: ω D (( q, p ) , ( ξ, η )) = qη − pξ .A generic constant symplectic structure ω in R is of course associated witha (real) skew-symmetric matrix of the form:Ω = (cid:12)(cid:12)(cid:12)(cid:12) a − a (cid:12)(cid:12)(cid:12)(cid:12) , a = 0 (C.9)181nd there exists a nonsingular matrix T ∈ A ut (cid:0) R (cid:1) = GL (2 , R ) (a (1 ,
1) tensor)such that: Ω = e T ω D T (C.10)i.e. (always remember that, by definition: (cid:16) e T (cid:17) i j = T j i ): ω ( x, y ) = ω D ( T x, T y ) , x, y ∈ R (C.11)Indeed, if: T = (cid:12)(cid:12)(cid:12)(cid:12) λ µν ρ (cid:12)(cid:12)(cid:12)(cid:12) (C.12)then the previous condition only requires:det T = λρ − µν = a (C.13)and T will be actually defined ”modulo” left multiplication by any matrix U with det U = 1, i.e.: U ∈ Sp (2 , R ) ≈ SL (2 , R ): e U ω D U = ω D . In this slightlymore general setting, the symplectic Fourier transform is defined as: F sT ( f ) ( η, ξ ) = J π Z Z dqdp exp {− iω (( q, p ) , ( ξ, η ) } f ( q, p ) (C.14)where: J =: det T . Now, if: T ( q, p ) =: ( x, k ), then: ∂ ( q, p ) ∂ ( x, k ) = J − (C.15)Moreover, with: X =: ( q, p ) , Y =: ( x, k ), T X = Y and: Z = ( ξ, η ), we have: ω (( q, p ) , ( ξ, η ) = ω (cid:0) T − Y, Z (cid:1) = ω D ( Y, T Z ). Hence, changing variables: F sT ( f ) ( η, ξ ) = Z Z dxdk π (cid:0) f ◦ T − (cid:1) ( x, k ) exp {− iω D (( x, k ) , T ( ξ, η )) } (C.16)i.e., setting: ( ξ T , η T ) =: T ( ξ, η ): F sT ( f ) ( η, ξ ) = F s (cid:0) f ◦ T − (cid:1) ( η T , ξ T ) (C.17)Noticing that: f ( q, p ) ≡ (cid:0) f ◦ T − (cid:1) ( T ( q, p )) (C.18)we can write, using the inversion formula for the ”canonical” symplectic trans-form: f ( q, p ) = Z Z dξ T dη T π F s (cid:0) f ◦ T − (cid:1) ( η T , ξ T ) exp {− iω D (( ξ T , η T ) , T ( q, p )) } (C.19)or: f ( q, p ) = Z Z dξ T dη T π F sT ( f ) ( η, ξ ) exp {− iω D ( T ( ξ, η ) , T ( q, p )) } (C.20)and eventually ( ∂ ( ξ T , η T ) /∂ ( ξ, η ) = J ) we obtain the inversion formula: f ( q, p ) = J π Z Z dξdη F sT ( f ) ( η, ξ ) exp {− iω (( ξ, η ) , ( q, p )) } (C.21)182 quivariance What remains to be discussed is the role of the ambiguity in the definitionof T ( T and U T , U ∈ Sp (2 , R ) playing the same role). The question iswhether or not F sT ( f ) ( η, ξ ) and F sUT ( f ) ( η, ξ ), i.e. F s (cid:0) f ◦ T − (cid:1) ( η T , ξ T ) and F s (cid:0) f ◦ ( U T ) − (cid:1) ( η UT , ξ UT ) define the same symplectic Fourier transform. Fromthe definition: F s (cid:0) f ◦ ( U T ) − (cid:1) ( η UT , ξ UT ) == Z Z dqdp π (cid:0) f ◦ T − ◦ U − (cid:1) ( q, p ) exp {− iω D (( q, p ) , U ◦ T ( ξ, η )) } (C.22)Setting: U − ( q, p ) = ( x, k ) (det U = 1): F s (cid:0) f ◦ ( U T ) − (cid:1) ( η UT , ξ UT ) == Z Z dqdp π (cid:0) f ◦ T − (cid:1) ( x, k ) exp {− iω D ( U ( x, k ) , U ◦ T ( ξ, η )) } (C.23)But: ω D ( U ( . ) , U ( . )) = ω D (( . ) , ( . )), and hence: F s (cid:0) f ◦ ( U T ) − (cid:1) ( η UT , ξ UT ) == Z Z dqdp π (cid:0) f ◦ T − (cid:1) ( x, k ) exp {− iω D (( x, k ) , T ( ξ, η )) } (C.24)i.e.: F s (cid:0) f ◦ ( U T ) − (cid:1) ( η UT , ξ UT ) = F s (cid:0) f ◦ T − (cid:1) ( η T , ξ T ) (C.25) F sT depends then only on the right coset of T in GL (2 , R ) relative to thesubgroup Sp (2 , R ) of the symplectic linear maps. This result can be summarizedby writing (for T = I , otherwise we substitute f with f ◦ T − ): F s (cid:0) f ◦ U − (cid:1) ◦ U = F s ( f ) (C.26)or, according to the standard definition of ”pull-back” of a map: φ ∗ F s ( f ) = F s ( φ ∗ f ) (C.27)where: φ = U − ∈ Sp (2 , R ), which can then be rephrased by saying that thesymplectic Fourier transform is equivariant, or that it is ”natural”, w.r.t. thesymplectic group. 183 eferences [1] R. Abraham, J. E. Marsden, Foundations of Mechanics , 2nd Edition, Ben-jamin/Cummings, Reading, 1978.[2] L. V. Ahlfors,
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