Clifford Algebras in Symplectic Geometry and Quantum Mechanics
aa r X i v : . [ m a t h - ph ] M a r Clifford Algebras in Symplectic Geometry andQuantum Mechanics
Ernst Binz , Maurice A. de Gosson , Basil J. Hiley . Fakult¨at f. Mathematik und Informatik A5/6, D-68131 Mannheim, Germany. Universit¨at Wien, NuHAG, Fakult¨at f¨ur Matematik, A-1090 Wien. TPRU, Birkbeck, University of London, London, WC1E 7HX.Accepted. To appear in Fond. Phys.arXiv:1112.2378 (math-ph)DOI:1007/s10701-012-9634-z
Abstract
The necessary appearance of Clifford algebras in the quantum descrip-tion of fermions has prompted us to re-examine the fundamental roleplayed by the quaternion Clifford algebra, C , . This algebra is essen-tially the geometric algebra describing the rotational properties of space.Hidden within this algebra are symplectic structures with Heisenberg alge-bras at their core. This algebra also enables us to define a Poisson algebraof all homogeneous quadratic polynomials on a two-dimensional sub-space, F a of the Euclidean three-space. This enables us to construct a PoissonClifford algebra, H F , of a finite dimensional phase space which will carrythe dynamics. The quantum dynamics appears as a realisation of H F interms of a Clifford algebra consisting of Hermitian operators. In a series of lectures at the annual Askloster Seminars, hosted by Georg Wik-man, one of us (BJH) developed a new approach to quantum mechanics basedon the notion of ‘structure process’, which provides a very different way of look-ing at quantum phenomena [9]. There it was shown how orthogonal Cliffordalgebras arise from a collection of what we call ‘elementary processes’. It shouldbe no surprise that Clifford algebras play an essential role in quantum mechan-ics since these algebras are necessary for a description of both the Pauli andDirac particles. The fact that the Schr¨odinger particle can also be described interms of a Clifford algebra may come as a surprise.In this paper we will not discuss these elementary processes in their fullgenerality. Rather we will take these ideas to motivate a discussion of the roleof quaternions in bringing quantum mechanics and geometry closer together.Another motivation for our present discussion comes directly from the workof Binz and Pods [2]. This shows that a natural symplectic structure can beconstructed using the quaternions. This, in turn, gives rise to the Heisenberg1lgebra and it is the appearance of this algebra as a geometric feature thatenables us to make contact with quantum mechanics.We start by constructing the skew-field of quaternions, a Clifford algebra.We call this algebra, together with the Clifford algebra of an Euclidean plane,the elementary Clifford algebra. Combining this structure with the complexnumbers (which drop out of the real quaternions), the general theory of realClifford algebras as presented in Greub [7], then allows us to construct the(finite-dimensional) Clifford algebras of bilinear forms of arbitrary index outof the elementary ones, thus enabling us to discuss the Schr¨odinger, Pauli andDirac particles in one geometric structure.In quantum mechanics the symplectic geometry plays a fundamental rolethrough the Heisenberg group as has been shown by de Gosson [4]. How is thisstructure hidden in Clifford algebras? To answer this question we will focus, inparticular, on the geometric aspects of the quaternion Clifford algebra whichencodes a two-dimensional symplectic space F a , characterised by selecting aparticular pure quaternion a . F a is the tangent plane to the two-sphere at a .The choice of a is arbitrary. We could have chosen another pure quaternion,say b , and this will give rise to a different two-dimensional symplectic space F b . However these pure quaternions are related by the Clifford group SU (2), sothat we have an infinity of Heisenberg algebras within the quaternion Clifford, H = R · e ⊕ E , with E being the Euclidean three-space.Each symplectic space gives rise to a two-dimensional classical phase space ,while the whole structure gives rise to a 2 n -dimensional phase space, which inturn, gives rise to a Poisson algebra. We then find the Poisson algebra of allhomogenous quadratic polynomials over a two-dimensional phase space deter-mines a natural Clifford algebra as well, which we will call the Poisson Cliffordalgebra of the phase space. This is isomorphic to the geometric Clifford alge-bra. The isomorphism is based on the construction of Hamiltonian vector fields.The Poisson algebra of homogenous quadratic polynomials of any phase spaceof dimension higher than two is the linear image of a natural subspace of thePoisson Clifford algebra. Finally we introduce a realization of the characteris-tic Clifford algebra of a phase space generated by Hermitian operators. Thisrealization is based on the quantization map of the above mentioned Poissonalgebra. This final step is where we make contact with quantum mechanics.This elementary presentation follows essentially the book “The Geometry ofHeisenberg Groups” [2].
In Hiley [9] the notion of an elementary process was introduced. A set of theseelementary processes can then be ordered by defining a simple multiplicationrule, which then produces a structure that is isomorphic to the quaternion Clif-ford algebra. A detailed discussion of the physical motivation and the underly-ing philosophical grounds for introducing the fundamental notion of a process and its history, we refer to Hiley [9], Kauffman [13] as well as to Penrose [15].2he fundamental idea behind this approach is to show how a geometricpicture of an Euclidean three-space, E , emerges from the order of the elementaryprocesses encapsulated in the algebra. We can represent each process as a directed one-simplex [ P P ], connecting the opposite distinguishable poles ofthe process P and P . As a basic assumption, we regard these process as indivisible , meaning that the poles are merely distinguishing features of theprocess that cannot be separated from the process itself. Since we are assumingour processes are directed, we assume [ P P ] = − [ P P ]. The totality of one-simplexes forms a real linear space isomorphic to the rotational symmetries of E . An inner multiplication for the one-simplexes is defined by[ P P ] • [ P P ] = [ P P ] . (1)This multiplication rule encapsulates the Leibnizian notion of the order of suc-cession.Starting with the three elementary processes [ P P ], [ P P ] and [ P P ] wefind 1 [ P P ] [ P P ] [ P P ][ P P ] − e − [ P P ] [ P P ][ P P ] [ P P ] − e − [ P P ][ P P ] − [ P P ] [ P P ] − e (2)In this structure [ P P ], [ P P ] and [ P P ] are easily shown to act as two sidedunities, which we identify and denote by a single unity e . In this way wehave generated a group from the elementary processes. We can quickly identifythis group by making the following notation change i := [ P , P ], j := [ P , P ]and k := [ P , P ]. In this way we immediately see that e , i , j , k generate thequaternion group. The resulting skew-field H of quaternions is a real Cliffordalgebra [7], which can be verified by the defining universal property presentedbelow. The quaternions are often called a hypercomplex system.In [9] it was shown how to generalise the process algebra to include thePauli- and Dirac-Clifford algebras. This has enabled Hiley and Callaghan [12][10] [11] to apply these algebras in a novel way to the quantum mechanicsof the Schr¨odinger, Pauli and Dirac particles. This has led to a systematicgeneralisation of the Bohm approach [3] to the relativistic domain.Our approach should not be confused with the attempt made by Finkelstein et al. [5] to develop a quaternion quantum mechanics. These authors wereinvestigating the possibility of using a Hilbert space over the quaternion field.Our approach is exploring a very different structure.
For convenience we repeat the notion of a Clifford algebra by means of its universal property . Let A be an associative algebra with a unit element e A . A Clifford map from the R -linear space F to A is a linear map ϕ which satisfies3 ϕ ( h )) = b ( h, h ) e A . Here b is an R -valued, symmetric bilinear map defined on F . The universal property mentioned above reads as follows:A Clifford algebra C F over F is an associative algebra C F with unit element e , together with a Clifford map i F from F to C F subject to the following twoconditions: i F ( F ) ⊂ C F F generates C F and to every Clifford map ϕ from F to C F , there is a unique algebra homomorphism Φ from C F to A satisfying ϕ = Φ ◦ i F . H is a very basic Clifford algebra as we will see in the next section. The bilin-ear map is − <, > , the negative of a scalar product <, > on a two-dimensionalsubspace F in E (cf. [7]).Let us point out at this stage that the R -algebra End F with the composition as its product is a Clifford algebra. It is determined by a scalar product <, > on F (cf. [7]). The multiplication table for J, A, B ∈ End F with respectivematrices J = (cid:18) − (cid:19) ; A = (cid:18) − (cid:19) ; B = (cid:18) (cid:19) (3)is id J A BJ -id -B AA B id JB -A -J id (4)This is due to A ◦ J = B, J ◦ B = A, A ◦ B = J, and [ A, B ] = 2 · J (5)as a direct calculation shows. (See Binz and Pods [2]). Here [ A, B ] is the commutator of A and B .This multiplication table (4) differs only in the signs down the diagonal fromtable (2) and so can be regarded as arising from the structure process in F . Inthis case they are isomorphic to the Clifford algebra End F .In fact, any real finite-dimensional Clifford algebra of a symmetric bilinearform of arbitrary index is a finite graded tensor product of the Clifford algebras C , H and End F (cf. [7]). In this sense the processes introduced above determineall these Clifford algebras.We leave the algebraic treatment of processes and focus on another conceptwithin real Clifford algebras. We will show how these algebras contain symplec-tic spaces, a basic entity of Hamiltonian mechanics and quantum mechanics.Of course, these two approaches to Clifford algebras hint at a relation betweenprocesses that could give rise to symplectic geometry, to Hamiltonian mechanicsand to quantum mechanics. The latter relations are discussed in Hiley [9]. In this subsection we will show how a characteristic Clifford algebra emergesfrom each ( ) symplectic space.4o this end we start with F considered as a two-dimensional , real liner spaceequipped with a (constant) non-degenerated, skew-symmetric bilinear two-form ω , a symplectic form . Hence F is a two-dimensional symplectic space (cf. [4]). Intwo dimensions, ω serves a volume form on F . In other words F is an oriented ,two-dimensional linear space (cf. [4]).Let us analyse this assumption in a little more detail. For simplicity wechoose a scalar product <, > on F and represent ω by a skew-adjoint isomor-phism S so that ω ( v, w ) = < S ( v ) , w > ∀ u, v ∈ F . (6)There is an orthonormal basis in the Euclidean space, F , such that S has theform S = (cid:18) κ − κ (cid:19) ; κ ∈ R . Without loss of generality, we may choose κ = 1 if <, > is rescaled suitably; inthis case we write J instead of S and its matrix is J = (cid:18) − (cid:19) (7)as represented in (3). This map gives rise to an almost complex structure .Therefore, we conclude that, given a symplectic space F with skew form ω ,there is some scalar product and an isomorphism J satisfying J = − id (8)representing ω in the sense of (6).Vice versa, given ω and some J fulfilling (8), there is a unique scalar product <, > , given by − < v, w > := ω ( J ( v ) , w ) ∀ u, v ∈ F . (9)In fact, J ∈ Sp ( F ), the group of all isomorphisms of F preserving thesymplectic form. This group is called the symplectic group .To construct a phase space out of F we have to specify a configuration space ,i.e. the space of positions in F . This space is a real line R · e ⊂ F spannedby the vector e ∈ F , say. Then the line R · J ( e ) ⊂ F represents the space of momenta . F decomposed in this way is a phase space . For simplicity of notationlet e := J ( e ). The basis { e , e } ⊂ F can be assumed to be symplectic, i.e. ω ( e , e ) = 1. We will denote the coordinates by q and p , respectively. Thesymplectic form ω on the pase space F is then the canonical one (cf. [14]).Hence Lemma 1
Given a symplectic space F with symplectic form ω and some almostcomplex structure J ∈ Sp ( F ) with J = − id , there is a unique scalar product <, > defined by < v, w > = − ω ( J ( v ) , w ) ∀ u, v ∈ F . oreover, given any unit vector e ∈ F the basis { e , e } with e := J ( e ) is asymplectic and orthonormal one. This is to say it spans the phase space F = R · e ⊕ R · e . Let us remark that associated with J , the symplectic space F is turned into a complex line . To see this, we form the two-dimensional R -linear space C J := span { id, J } . Here id, J ∈ Sp ( F ). The operations on C J will be defined in a similar way tothose on a field of complex variables; J plays the role of the complex unity i in C . The action of the field C J on F is determined by the action of the linearmaps id and J on F . This turns F into a complex line.The symplectic space F can naturally be enlarged to a three-dimensional,oriented, Euclidean, real linear space E := F ⊕ R · J. (10)We extend <, > orthogonally to a scalar product <, > E on all of E for which < J, J > E = 1 and fix a metric volume form µ E on E . If its restriction to F is ω , then µ E ( e , e , J ) = 1 . (11)The oriented, Euclidean linear space E then admits a cross product . With theseingredients we turn the real linear space H := F ⊕ C J = E ⊕ R · id (12)into a skew-field by setting( λ e + u ) · ( λ e + u ) := ( λ λ − < u , u > ) · e + λ u + λ u + u × u (13)for all λ , λ ∈ R and any u , u ∈ E . Here e replaces id . This skew-field iscalled the quaternions H F (cf. [6]).From the matrix structure of J , we deduce J ( v ) = j × v = j · v ∀ v ∈ F . (14)for a unique unit vector j ∈ E , perpendicular to F ; hence F is the tangent planeto the unit sphere S in E at j . As a consequence j · j = − e . Because of (14),we identify the vector j ∈ E with the map J on F . Hence, in dealing with H we will replace J by j in what follows.By the universal property, H F is the Clifford algebra of the plane F endowedwith − <, > (cf. [7]) and hence we call it the geometric Clifford algebra of thesymplectic space F . Occasionally, we express this by the symbol F j . Vice versa,by (13) the algebra H F determines the bilinear form − <, > on F , because h = − < h, h > holds for any h ∈ F . Moreover, C j is a commutative subfield6f H F and complements the linear space F within H F and thus the Cliffordalgebra H F is Z graded.The space F is embedded as a phase space if an orthonormal basis h, k ⊂ F is chosen. In this case we have the orthonormal basis h, k, j, e in H F .The unit sphere S ⊂ H F is a group, namely SU (2) (cf. [1], of which thetwo-sphere S in E is the equator. S constitutes all j ∈ E with j = − e .Alternately formulated, it constitutes all the almost complex structures J onall two-dimensional linear subspaces in E .Clearly ω is algebraically encoded in the Clifford algebra H F by (14) and(6), or more sophisticatedly expressed, the Lie bracket of the Heisenberg algebra on E = F ⊕ R · j given by ω is determined by the product in H F (cf. [2]). Themetric − <, > E , being part of the product in H F , plays a central role in specialrelativity (cf. [2]). More generally, H F encodes the Euclidean geometry of theline, plane, three- and four-space, as well as the Minkowski geometry of theplane and four-space.Finally, given a Clifford algebra in terms of a skew-field of quaternions, H ,then H = H F (15)for some symplectic plane F . In fact, the tangent plane F , say, at any j ∈ S ⊂ H is symplectic with symplectic structure ω given by ω ( v, w ) = < j · v, w > = < j, v · w > ∀ v, w ∈ F . (16)As easily seen, (14) holds true. Thus the symplectic plane F can be recon-structed from H ; in this sense H is characteristic for F .In fact the Clifford algebra H in (15), the skew-field of quaternions encodesall symplectic planes in the three-dimensional real linear space E , as is easilyseen.As mentioned in section 2.1 the Clifford algebra H emanates from processes;since the symplectic space F can be reconstructed from its geometric Cliffordalgebras by (15), similarly the symplectic space F can also be reconstructedfrom process as well.The construction of the geometric Clifford algebra of a sym-plectic space F is based on the following lemma (cf. [6]): Lemma 2
Let F be a symplectic space with ω as (constant)symplectic form. The linear space can be decomposed into the direct sum F = ⊕ s F s (17) (where s runs from 1 to n ) which admits a basis { e , ..., e n } such that theintersection { e , ..., e n } T F s is a symplectic basis of ω s , the restriction of ω to F s for each s . In addition there is a unique scalar product <, > s for each s,such that ω s is represented uniquely by J s having (7) as its matrix and whichsatisfies (8). <, > on F = ⊕ s F s yielding <, > s inlemma 2 if restricted to F s for each s. From lemma 1 each F s in (17) is a phasespace. If H F s denotes the Clifford algebra of − <, > s for all s, the Cliffordalgebra H F of − <, > on F is given by the 2 n -dimensional algebra H F = ⊗ s H F s . (18)Here s runs from 1 to n and ⊗ denotes the graded tensor product (cf. [7]. Thistensor product is called the anticommutative tensor product) of the Z gradedalgebras H F s . Again here we will call it the geometric Clifford algebra of thesymplectic space F ; obviously H F is characteristic for F . Clearly each H F s isa subalgebra of H F .As an example, consider m particles moving in R endowed with the canon-ical scalar product and its natural orthonormal basis. The configuration spaceis the m-fold direct sum R ⊕ .... ⊕ R which is isomorphic to the m-foldCartesian product of R with itself, a 3 m -dimensional linear space. Let the q -coordinates be enumerated from q to q m . The phase space is isomorphicto ( R ⊕ .... ⊕ R ) × ( R ⊕ .... ⊕ R ), a 2 · m -dimensional linear space. Thefirst factor of this cartesian product constitutes all possible spatial positions ofthe m particles, while the second factor contains all the momenta. This phasespace together with the canonical symplectic form ω , say, is a 2 n dimensionalsymplectic space. Here n = 3 m . Now we regroup the coordinates in this phasespace. We take the plane F generated by the first spatial coordinate q and itsmoment p together with the canonical symplectic form. Passing to the secondspatial coordinates and its momenta yields F etc. In this way we obtain F := ⊕ s F s ∼ = ( R ⊕ .... ⊕ R ) × ( R ⊕ .... ⊕ R ) , (19)and endow it with the natural scalar product. Here s varies from 1 to 3 m . Thesymplectic bases of F in which the canonical symplectic form ω decomposesaccording to lemma 2 is identical with the canonical orthonormal basis of F in(19). sp ( F ) Another realization of the geometric Clifford algebra of the two-dimensionalsymplectic space F is obtained by means of sp( F ), the Lie algebra of the sym-plectic group Sp ( F ). At first we will construct a natural splitting of sp( F )and a symplectic structure on a two-dimensional subspace. The basis of thisconstruction is End F , the four-dimensional R -linear space of all ( R -linear) en-domorphisms of F . This space is split intoEnd F = R · id F ⊕ sp ( F ) (20)where sp( F ) is the three-dimensional R -linear space of all traceless endomor-phisms of F . In fact, this splitting is orthogonal with respect to the naturalscalar product <, > EndF , say, defined by < A, B >
EndF := 12 trA ◦ e B ∀ A, B ∈ EndF. (21)8ere e B is the adjoint of B . In fact the splitting (20) carries a natural Heisenbergalgebra structure (cf. [2]).By means of this scalar product we observe that any almost complex struc-ture on a two-dimensional linear subspace of sp( F ) is a unit vector and viceversa any vector in the unit sphere S ⊂ sp( F ) is an almost complex structureon the tangent plane of S at that vector (cf. 16).Moreover, let µ sp ( F ) be a constant metric volume form, i.e. a determinantfunction (cf. [6]) on sp ( F ), of the scalar product < A, B > sp ( F ) in (21).For a given J with matrix (cid:18) − (cid:19) , the linear space sp( F ) decomposesinto sp ( F ) = R · J ⊕ Σwhere the matrix of an element in Σ has the form (cid:18) a bb − a (cid:19) , obviously a sym-metric matrix with vanishing trace. (These matrices are formed with respectto an orthonormal basis in F ). This yieldsEnd F = R · id + R · J + Σand hence Σ = span (cid:26) a · (cid:18) − (cid:19) , c · (cid:18) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) a, c ∈ R (cid:27) . (Thus sp ( F ) is not closed under composition, because of (5)). Also becauseof (5), the linear two-dimensional space Σ inherits a symplectic structure ω Σ defined by ω Σ ( A, B ) := < · [ A, B ] , J > EndF ∀ A, B ∈ sp ( F ) . Since [
A, B ] ∈ R · J for A, B in Σ with k A k = k B k = 1 µ sp ( F ) ( A, B, J ) = ω Σ ( A, B ) , and hence µ sp ( F ) ( A, B, J ) = < · [ A, B ] , J > EndF . This identifies · [ A, B ] as the cross product of A, B ∈ Σ in sp ( F ).Therefore, by (13) the four-dimensional linear space End F is turned intoa skew-field H sp ( F ) isomorphic to the quaternions , i.e. isomorphic to the geo-metric Clifford algebra H F of the symplectic space F . (Note End F is a Cliffordalgebra with the composition as multiplication, as mentioned in section 2.2. Itis not isomorphic to H sp ( F ) ).We point out that Σ and ω Σ can be naturally identified with F and ω respectively as shown in (cf. [2]). Hence H F ≡ H sp ( F ) . (22)In the following we will use either picture.9 .3 Poisson Algebras of the Phase Space In order to derive a characteristic Clifford algebra from the Poisson algebra ofall homogenous quadratic polynomials on a two-dimensional phase space F , wewill need to study the Poisson algebra of F .Our first goal is to establish a natural isomorphism between the Poissonalgebra Q of all homogeneous quadratic polynomials on the two-dimensionalphase space F (cf. [14]) and sp ( F ) (cf. [16]). The key to this isomorphismwill be the construction of Hamiltonian vector fields of homogenous quadraticpolynomials (cf. [8]).The calculation in what follows can easily be visualized if we use the quater-nions H F of the phase space F . This skew-field contains F as well as the line R · j which is perpendicular to F .The coordinate system in F determined by the orthonormal unit vectors e q and e p := j · e q respectively, of the (linear) coordinate functions f q and f p on F .This allows us to introduce the concept of a polynomial on F in two variables q and p , say. The collection Q of all homogenous quadratic polynomials on F obviously forms an R -vector space.Given a polynomial pol on F , let f pol : F → R be its polynomial function inthe variables q and p . In the following we will identify the collection P Q := { f pol | pol ∈ Q} with Q by identifying any pol with f pol , if no confusion arises.We begin our investigations of Q by the geometric study of the notion ofHamiltonian vector fields on the two-dimensional phase space. Let C ∞ ( F, R )be the collection of all smooth R -valued functions of F . It is an R -algebra underpointwisely defined operations. Given f ∈ C ∞ ( F , R ) , its Hamiltonian vectorfield X f is defined by ω ( X f , X ) = df ( X ) (23)for any smooth vector field X on F (cf. [14]). Clearly, X f is smooth as well.Throughout this paper, we will replace the vector field X f = ( a f , id ) by itsprincipal part a f , mapping F into itself. Thus (23) turns into ω ( a f ( h ) , k ) = df ( h )( k ) ∀ h, k ∈ F . (24)Here df ( h )( k ) is the derivative of f at h evaluated at k ∈ F .It follows from (9) and (16) that equation (24) can be rewritten in terms ofthe scalar product, yielding ω ( a f ( h ) , k ) = < j × a f ( h ) , k > = df ( h )( k ) ∀ h, k ∈ F which by (14) yields in turn a link to ∇ f ∈ C ∞ ( F, R ), namely that the principalpart a f of the Hamiltonian vector field which is a f ( h ) = − j × ∇ f ( h ) = − j · ∇ f ( h ) ∀ h ∈ F (25)where · denotes the multiplication in H F .10 direct calculation in the orthonormal basis given by the unit vectors e q and e p shows: a f ( h ) = ∂f ( h ) ∂p · e q − ∂f ( h ) ∂q · e p ∀ h ∈ F . (26)As an example, by (25) the Hamiltonian vector fields of the coordinatefunctions f q and f p obviously are a f q = − j · e q = − e p and a f p = − j · e p = e q . (27)Clearly, j = e q · e p holds true.We study the above notions by turning to Q ⊂ C ∞ ( F, R ). Of special interestwith respect to our goal is the linear dependence of a f ( h ) on the position h ∈ F for any f ∈ Q . By (24) this means that a f ∈ End F ; (28)hence both sides of (24) depend bilinearly on h and k for any f ∈ Q .The quality of a f will become clear if we notice that Q is generated by q , p and q · p .By (26) the Hamiltonian vector fields of f q , f p and f q · p are the followingfor all h ∈ F of the form h = q · e q + p · e p : a f q ( h ) = − q · e p a f p ( h ) = p · e q a f q · p ( h ) = q · e q − p · e p . (29)Their respective matrices M ( a f q ) , M ( a f p ) and M ( a f q · p ) formed with re-spect to e q and e p (cf. 28) are M ( a f q ) = (cid:18) − (cid:19) , M ( a f p ) = (cid:18) (cid:19) , M ( a f q · p ) = (cid:18) − (cid:19) . (30)These matrices are all tracefree which means they belong to the Lie algebrasp ( F ).Since the endomorphisms a f q , a f p and a f q · p of F generate sp ( F ), we havea surjective linear map ham : Q → sp ( F )ham ( f ) := a f ∀ f ∈ Q . (31)ham is a linear isomorphism since both the domain and the range are three-dimensional.Next we will verify that Q forms a Poisson sub-algebra of C ∞ ( F, C ). Tothis end, we define a Poisson bracket on C ∞ ( F, R ) and show that C ∞ ( F, R ) isa Poisson algebra (cf. [4]), which means that the R -linear space, C ∞ ( F, R ), hasa natural Lie algebra structure. Therefore, let f, g ∈ C ∞ ( F, R ) with a f and a g being the principal parts of their Hamiltonian vector fields. Obviously, ω ( a f ( h ) , a g ( h )) = df ( h )( a g ( h )) . df ( a g ) = < j, a f × a g > . To evaluate the cross product we use equation (26) to obtain a f × a g = − ∂f∂p ∂g∂q · e q × e p − ∂f∂q · ∂g∂p · e p × e q and, therefore, a f × a g = − (cid:18) ∂f∂q · ∂g∂p − ∂f∂p · ∂g∂q (cid:19) · j. (32)Defining the Poisson bracket of f and g in C ∞ ( F, R ) by { f, g } := ∂f∂q · ∂g∂p − ∂f∂p · ∂g∂q , (33)yields { f, g } · j = − a f × a g ; (34)hence this bracket can be expressed as { f, g } · e = j · ( a f × a g ) ∀ f, g ∈ C ∞ ( F, R ) . They satisfy a Jacobi identity, i.e. {{ f , f } , f } + {{ f , f } , f } + {{ f , f } f } = 0 . Therefore, C ∞ ( F, R ) is a Lie algebra under the Poisson bracket (33) (cf. [16]).Since { f, g } ∈ C ∞ ( F, R ), we may investigate a { f,g } . Let [ a f , a g ] denote theprincipal part of the Lie bracket [ X f , X g ] defined by[ a f , a g ] := d a g ( a f ) − d a f ( a g ) . As is well known d { f, g } ( h ) = − ω a ([ a f , a g ] , h ) ∀ h ∈ F showing a { f,g } = − [ a f , a g ] ∀ f, g ∈ C ∞ ( F, R ) . (35)Denoting by Ham F the collection of all Hamiltonian vector fields on F , thelinear isomorphism in (31) extends to the linear mapham : C ∞ ( F, R ) → Ham F (36)determined by ham f := a f ∀ f ∈ C ∞ ( F, R )which satisfies (35).Now focussing on Q again, the Poisson brackets { f q , f p } , { f q · p , f q } and { f q · p , f p } are { f q , f p } = f q · p { f q · p , f q } = f q { f q · p , f p } = f p (37)which are easily calculated either from (33) or (34). Hence we have shown(cf. [8]) 12 roposition 3 Q closes under the Poisson brackets and is a Poisson sub-algebra of C ∞ ( F , R ) . The formalism we have used here to introduce Hamiltonian vector fields andPoisson brackets is adapted to our setting for two-dimensional phase spaces.For a general formalism we refer to [14] or [8]. Q . Next, we extend the Poisson algebra of all homogenous quadratic polynomialson a two-dimensional phase space to a skew-field of quaternions, a Cliffordalgebra and generalize this to symplectic space.Because of (22) we will now relate the Lie algebra sl( F ) of SL( F ), the speciallinear group of F , to Q (cf. subsection 3.2). From (28) we havesp( F ) = sl( F ) . (38)According to (26), for any f ∈ Q the trace free matrix M ( a f ) of a f is M ( a f ) = ∂f∂p ( e q ) ∂f∂p ( e p ) − ∂f∂q ( e q ) − ∂f∂q ( e p ) ! . (39)Moreover, a direct calculation provides us with the key result on ham in (31)for the construction of the Clifford algebra containing Q : Proposition 4 ham : Q → sp ( F ) (40) is a Lie algebra isomorphism satisfying ham { f, g } = [ a f , a g ] sl ( F a ) ∀ f, g ∈ Q . (41) Here [ a f , a g ] sl ( F a ) := a f ◦ a g − a g ◦ a f ∀ f, g ∈ Q is the commutator in sl ( F a ) . Now we enlarge Q to H Q := Q ⊕ R · e and extend ham in proposition 4 toham : H Q → End F (42)by sending e to the identity id . This is a linear isomorphism. As shown insection 2.2 the linear space End F carries a natural Clifford algebra structurearising from the Lie algebra sp( F ). This structure can be pulled back to H Q by ham . The resulting skew-field is called here the Poisson Clifford algebra of the phase space and is also denoted by H Q . Compared with the definingequation (13) for the product in H F , the Poisson bracket on Q serves as thecross product. By construction we have H F = H sp ( F ) ∼ = H Q (43)13s Clifford algebras.In the case where the phase space F is , we proceed as insection 3.1. We decompose F into F = ⊕ s F s and form for each s the Cliffordalgebra H Q s .Hence ⊗ s H Q s is a Clifford algebra, called here the Poisson Clifford algebra of the phase space F . Obviously H F = ⊗ s H s ∼ = ⊗ s H Q s (44)holds true. Q , the collection of all functions of homogenous quadratic polynomials of F , is the image of a natural subspace of the Poisson Clifford algebra ⊗ s H s , asseen as follows: Based on (29) we restrict ham in (36) to Q and calculate in ⊗ s H sp ( F s ) . The tensor q s · p r e ⊗ ... ⊗ e s ⊗ ... ⊗ e r ... ⊗ e in ⊗ s H sp ( F s ) appliedto an element of F = ⊕ s F s in (17) (via the canonical scalar product) yields q s · p r for all s, r ≤ n . Thus both q s · p r e ⊗ ... ⊗ e s ⊗ ... ⊗ e r ... ⊗ e as well as q s · p r e ⊗ ... ⊗ e r ⊗ ... ⊗ e s ... ⊗ e map linearly to q s · p r in Q for all s, r ≤ n . Inthis way we obtain all of Q . H F consisting of Operators We close our studies of characteristic Clifford algebras of symplectic spacesby introducing for any finite-dimensional phase space F a realization of H F interms of a Clifford algebra consisting of Hermitian operators . This will allow usto associate eigen-values to any element of H F . (For an alternative approach,consult [9]).This realization is constructed first for a two-dimensional phase space F andthen extended to 2 n -dimensions F = ⊕ s F s by means of (18).Given a two-dimensional phase space F , this construction is based on the quantization map Q : Q →
End( L ( R , C ))(cf. [2]). It is defined to be the composition of Q ham → sp ( F ) − i · dU Mp → End( L ( R , C )) , (45)i.e. Q := − i · dU Mp ◦ ham . (46)The right hand composite is made up by the (injective) infinitesimal meta-plectic representation dU Mp (cf. [2] or [4]) multiplied by the factor − i and theLie algebra isomorphism ham in (31). The values of dU Mp are skew-symmetricoperators acting on the Hilbert space L ( R , C ), hence ∓ i · dU Mp maps into acollection of Hermitian operators. The choice of the factor − i in Q (cf. (46))was made to guarantee that Q ( { f, g } ) = [ Q ( f ) , Q ( g )] (47)14olds true for any pair of polynomial function f, g ∈ Q . Thus Q quantizes anyhomogenous quadratic polynomial defined on F . (The linear isomorphism ham in (31) can be defined for any finite-dimensional phase space F (cf. [8])).The isomorphism (46) obviously extends to a Clifford algebra isomorphismalso denoted by ham . This is to say we haveham : H Q → H sp ( F ) , (48)mapping e ∈ H Q to id .We extend the monomorphism dU Mp to all End F = R · id F ⊕ sp ( F ) as alinear injective map (still called dU Mp ) dU Mp : R · id F ⊕ sp ( F ) → End( L ( R , C ))by setting dU Mp ( id F ) = id End ( L ( R , C ) . Together with (46), this means we have a linear map Q : H F → im Q. Here im Q is the (four-dimensional) image of Q .Pushing forward by Q , the skew-field structure of H F to im Q yields thestructure of a skew-field of quaternions on im Q , a Clifford algebra as mentionedabove. It is denoted by H Q .To extend the quantization map Q to H F = ⊗ s H s for an 2 n -dimensional phase space F as in (18), we form the quantization map for each s yielding Q s of which the image is im Q s and set Q := ⊗ s Q s . (49)The image of Q is the graded tensor product ⊗ s im Q s , a finite dimensional Clifford algebra, referd to as the realization of H F by meansof Hermitian operators . We are deeply indebted to the generosity of Georg Wikman in sponsoring theAskloster Seminars where different aspects of this work were presented anddiscussed. Without those meetings the work presented in this paper would nothave been possible. 15 eferences [1] Ian M. Benn and Robin W. Tucker.
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