Conformal Spinning Quantum Particles in Complex Minkowski Space as Constrained Nonlinear Sigma Models in U(2,2) and Born's Reciprocity
aa r X i v : . [ h e p - t h ] J un Conformal Spinning Quantum Particles inComplex Minkowski Space as ConstrainedNonlinear Sigma Models in U (2 , andBorn’s Reciprocity M. Calixto , ∗ and E. P´erez-Romero Departamento de Matem´atica Aplicada y Estad´ıstica, Universidad Polit´ecnica deCartagena, Paseo Alfonso XIII 56, 30203 Cartagena, Spain Instituto de Astrof´ısica de Andaluc´ıa (IAA-CSIC), Apartado Postal 3004, 18080Granada, Spain
Abstract
We revise the use of 8-dimensional conformal, complex (Cartan) domains as abase for the construction of conformally invariant quantum (field) theory, eitheras phase or configuration spaces. We follow a gauge-invariant Lagrangian ap-proach (of nonlinear sigma-model type) and use a generalized Dirac method forthe quantization of constrained systems, which resembles in some aspects thestandard approach to quantizing coadjoint orbits of a group G . Physical wavefunctions, Haar measures, orthonormal basis and reproducing (Bergman) kernelsare explicitly calculated in and holomorphic picture in these Cartan domains forboth scalar and spinning quantum particles. Similarities and differences withother results in the literature are also discussed and an extension of Schwinger’sMaster Theorem is commented in connection with closure relations. An adap-tation of the Born’s Reciprocity Principle (BRP) to the conformal relativity,the replacement of space-time by the 8-dimensional conformal domain at shortdistances and the existence of a maximal acceleration are also put forward. PACS:
MSC:
Keywords:
Coherent States, Reproducing Kernels, Cartan Domain, Conformal Relativ-ity, Nonlinear Sigma Models, Constrained Quantization, Born Reciprocity. ∗ Corresponding author: [email protected] Introduction
Complex manifolds and, in particular, Cartan classical domains have been studied formany years by mathematicians and theoretical physicists (see e.g. [1] and referencestherein for a review). In this article we are interested in the Lie ball D = SO (4 , / ( SO (4) × SO (2)) = SU (2 , /S ( U (2) × U (2)) , which can be mapped one-to-one onto the 8-dimensional forward/future tube domain T = { x µ + iy µ ∈ C , , y > k ~y k} of the complex Minkowski space C , through a Cayley transformation (see next Section formore details). Both manifolds can be considered as the phase space of massive conformalparticles and there is a renewed interest in its quantization (see e.g. [2] and referencestherein for a survey). The presentation followed in the literature is of geometric (twistor[3, 4] and Konstant-Kirillov-Souriau [5, 6] descriptions) and representation-theoretic [7, 8]nature. Here we shall adopt a (sigma-model-type) Lagrangian approach to the subjectand we shall use a generalized Dirac method for the quantization of constrained systemswhich resembles in some aspects the particular approach to quantizing coadjoint orbits ofa group G developed many years ago in [9] (see also [10] and [11] for interesting examplesin G = SU (3)).We share with many authors (namely, [1, 2, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22])the believe that the use of complex Minkowski 8-dimensional space as a base for theconstruction of quantum (field) theory is not only useful from the technical point of viewbut can be of great physical importance. Actually, as suggested in [14], the conformaldomain D could be considered as the replacement of the space-time at short distances (atthe “microscale”). This interpretation is based on Born’s Reciprocity Principle (BRP) [15,16], originally intended to merging quantum theory and general relativity. The reciprocitysymmetry between coordinates x µ and momenta p µ states that the laws of nature are (orshould be) invariant under the transformations( x µ , p µ ) → ( ± p µ , ∓ x µ ) . (1)The word “reciprocity” is used in analogy with the lattice theory of crystals, where somephysical phenomena (like the theory of diffraction) are sometimes better described inthe p -space by means of the reciprocal (Bravais) lattice. The argument here is thatBorn’s reciprocity implies that there must be a reciprocally conjugate relativity principleaccording to which the rate of change of momentum (force) should be bounded by auniversal constant b , much in the same way the usual relativity principle implies a boundof the rate of change of position (velocity) by the speed of light c . As a consequence ofthe BRP, there must exist a minimum (namely, Planck) length ℓ min = p ~ c/b .This symmetry led Born to conjecture that the basic underlying physical space is the8-dimensional { x µ , p µ } and to replace the Poincar´e invariant line element dτ = dx µ dx µ
2y the Finslerian-like metric (see [17, 18] for an extension to Born-Clifford phase spaces) d ˜ τ = dx µ dx µ + ℓ ~ dp µ dp µ . (2)From the BRP point of view, local (versus extended) field theories like Klein-Gordon’srepresent the “point-particle limit” ℓ min →
0, for which the reciprocal symmetry is broken.Also, the Minkowski spacetime is interpreted either as a local ( ℓ min →
0) version or as ahigh-energy-momentum-transfer limit ( b → ∞ ) of this 8-dimensional phase-space domain.Moreover, putting dp µ /dτ = md x µ /dτ = ma µ , with m = bℓ min /c a (namely, Planck)mass and a µ the proper acceleration (with a ≤
0, space-like), one can write the previousextended line element as d ˜ τ = dτ s − | a | a , (3)which naturally leads to a maximal (proper) acceleration a max = c /ℓ min . The existenceand physical consequences of a maximal acceleration was already derived by Caianiello[19]. Many papers have been published in the last years (see e.g. [20] and referencestherein), each one introducing the maximal acceleration starting from different motiva-tions and from different theoretical schemes. Among the large list of physical applicationsof Caianiello’s model we would like to point out the one in cosmology which avoids aninitial singularity while preserving inflation. Also, a maximal-acceleration relativity prin-ciple leads to a variable fine structure “constant” α [20], according to which α could havebeen extremely small (zero) in the early Universe and then all matter in the Universecould have emerged via the Fulling-Davies-Unruh-Hawking effect (vacuum radiation dueto the acceleration with respect to the vacuum frame of reference) [23, 24, 25, 26].There has been group-theoretical revisions of the BRP like [21, 22] replacing thePoincar´e by the Canonical (or Quaplectic) group of reciprocal relativity, which enjoysa richer structure than Poincar´e. In this article we pursue a different reformulation ofBRP as a natural symmetry inside the conformal group SO (4 ,
2) and the replacement ofspace-time by the 8-dimensional conformal domain D or T at short distances. We believethat new interesting physical phenomena remain to be unravelled inside this framework.Actually, in a coming paper [27] (see also [28] for a previous related work), we shall discussa group-theoretical revision of the Unruh effect [25] as a spontaneous breakdown of theconformal symmetry and the consequences of a maximal acceleration. Also, a wavelettransform on the tube domain T , based on the conformal group, could provide a way toanalyze wave packets localized in both: space and time. Important developments in thisdirection have been done in [29, 30] for electromagnetic (massless) signals and [31] forfields with continuous mass spectrum.In this article we shall study the geometrical and quantum mechanical underlyingframework. We shall follow a gauge-invariant (singular) Lagrangian approach of nonlinearsigma-model type and we shall use a generalized Dirac method for the quantization ofconstrained systems.The paper is organized as follows. In Section 2 we briefly review the conformal group SO (4 , ≃ SU (2 , D and T ; in this Section we also introducethe concept of BRP in a conformally invariant setting. Section 3 is devoted to the La-grangian formulation of conformally invariant nonlinear sigma-models on the conformaldomains (either as configuration or phase spaces) and the study of their gauge invari-ance. The quantization of these models (for the case of Lagrangians linear in velocities)is accomplished in Section 4 by using a generalized Dirac method for the quantization ofconstrained systems which resembles in some aspects the particular approach to quan-tizing coadjoint orbits of G . Physical wave functions, Haar measures, orthonormal basisand reproducing (Bergman) kernels are explicitly calculated in an holomorphic picturein the Cartan domain D , for both scalar and spinning quantum particles in subsections4.1 and 4.2, respectively. Similarities and differences with other results in the literatureare also discussed and an extension of the Schwinger Master Theorem is commented inconnection with closure relations. In Section 5 we translate (through an equivariant map)all the constructions above to the tube domain T , where we enjoy more physical intuition.We comment on K¨ahler structures and generalized Born-like line elements and the exis-tence of a maximal acceleration for conformal (quantum) particles. The last Section 6 isdevoted to comments and outlook where we point out an interesting connection betweenBRP and CPT symmetry inside the conformal group and discuss on the appearance of amaximal acceleration in this scheme. The conformal group SO (4 ,
2) is comprised of Poincar´e (spacetime translations b µ ∈ R , and Lorentz Λ µν ∈ SO (3 , ρ = e τ ∈ R + ) andrelativistic uniform accelerations (special conformal transformations, SCT, a µ ∈ R , )which, in Minkowski spacetime, have the following realization: x ′ µ = x µ + b µ , x ′ µ = Λ µν ( ω ) x ν ,x ′ µ = ρx µ , x ′ µ = x µ + a µ x ax + a x , (4)respectively. The interpretation of SCT as transitions from inertial reference frames tosystems of relativistic, uniformly accelerated observers was identified many years ago by(see e.g., [32, 33, 34]), although alternative meanings have also been proposed. Oneis related to the Weyl’s idea of different lengths in different points of space time [35]:“the rule for measuring distances changes at different positions”. Other is Kastrup’sinterpretation of SCT as geometrical gauge transformations of the Minkowski space [36](for this point see later on Eq. (40)).The generators of the transformations (4) are easily deduced: P µ = ∂∂x µ , M µν = x µ ∂∂x ν − x ν ∂∂x µ ,D = x µ ∂∂x µ , K µ = − x µ x ν ∂∂x ν + x ∂∂x µ (5)4nd they close into the conformal Lie algebra[ M µν , M ρσ ] = η νρ M µσ + η µσ M νρ − η µρ M νσ − η νσ M µρ , [ P µ , M ρσ ] = η µρ P σ − η µσ P ρ , [ P µ , P ν ] = 0 , [ K µ , M ρσ ] = η µρ K σ − η µσ K ρ , [ K µ , K ν ] = 0 , [ D, P µ ] = − P µ , [ D, K µ ] = K µ , [ D, M µν ] = 0 , [ K µ , P ν ] = 2( η µν D + M µν ) . (6)We shall argue later that P µ and K µ are conjugated variables (they can not be simultane-ously measured) and that D can be taken to be the generator of (proper) time translations(i.e., the Hamiltonian). A BRP-like symmetry manifests here in the form: P µ → K µ , K µ → P µ , D → − D, (7)which leaves the commutation relations (6) unaltered. This symmetry can also be seenin the quadratic Casimir operator: C = D − M µν M µν + 12 ( P µ K µ + K µ P µ ) = D − M µν M µν + P µ K µ + 4 D, (8)which generalizes the Poincar´e Casimir P = P µ P µ , just as d ˜ τ in (2) generalizes thePoincar´e invariant line element dτ . We shall provide a conformal invariant line elementsimilar to d ˜ τ later in Section 5.Any group element g ∈ SO (4 ,
2) (near the identity element 1) could be written as theexponential map g = exp( u ) , u = τ D + b µ P µ + a µ K µ + ω µν M µν , (9)of the Lie-algebra element u (see [37, 38]). The compactified Minkowski space M = S × Z S ≃ U (2) can be obtained as the coset M = SO (4 , / W , where W denotes theWeyl subgroup generated by K µ , M µν and D (i.e., a Poincar´e subgroup P = SO (3 , s R augmented by dilations R + ). The Weyl group W is the stability subgroup (the little groupin physical usage) of x µ = 0.There is another interesting realization of the conformal Lie algebra (6) in terms ofgamma matrices in, for instance, the Weyl basis γ µ = (cid:18) σ µ ˇ σ µ (cid:19) , γ = iγ γ γ γ = (cid:18) − σ σ (cid:19) , where ˇ σ µ ≡ σ µ (we are using the convention η = diag(1 , − , − , −
1) for the Minkowskimetric) and σ µ are the Pauli matrices σ = (cid:18) (cid:19) , σ = (cid:18) (cid:19) , σ = (cid:18) − ii (cid:19) , σ = (cid:18) − (cid:19) . Indeed, the choice D = γ , M µν = [ γ µ ,γ ν ]4 = (cid:18) σ µ ˇ σ ν − σ ν ˇ σ µ
00 ˇ σ µ σ ν − ˇ σ ν σ µ (cid:19) ,P µ = γ µ γ = (cid:18) σ µ (cid:19) , K µ = γ µ − γ = (cid:18) σ µ (cid:19) (10)5ulfils the commutation relations (6). These are the Lie algebra generators of the funda-mental representation of the four cover of SO (4 , SU (2 ,
2) = (cid:26) g = (cid:18) A BC D (cid:19) ∈ Mat × ( C ) : g † Γ g = Γ , det( g ) = 1 (cid:27) , (11)with Γ a 4 × −− ). In particular, taking Γ = γ , the2 × A, B, C, D in (11) satisfy the following restrictions: g − g = I × ⇔ D † D − B † B = σ A † A − C † C = σ A † B − C † D = 0 , (12)together with those of gg − = I × . In this article we shall work with G = U (2 , SO (4 ,
2) and we shall use a set of complex coordinates to parametrize G . Thisparametrization will be adapted to the non-compact complex Grassmannian D = G/H of the maximal compact subgroup H = U (2) . It can be obtained through a block-orthonormalization process with metric Γ = γ of the matrix columns of: (cid:18) σ Z † σ (cid:19) → g = (cid:18) σ ZZ † σ (cid:19) (cid:18) ∆
00 ∆ (cid:19) , ∆ = ( σ − ZZ † ) − / ∆ = ( σ − Z † Z ) − / . Actually, we can identify Z = Z ( g ) = BD − , Z † = Z † ( g ) = CA − , ∆ = ( AA † ) / , ∆ = ( DD † ) / . (13)From (12), we obtain the positive-matrix conditions AA † > DD † >
0, which areequivalent to: σ − ZZ † > , σ − Z † Z > . (14)Moreover, from the top condition of (12), we arrive at the determinant restriction:det( ZZ † ) = det( B † B ) det( σ + B † B ) − < , (15)which, together with det( σ − ZZ † ) = 1 − tr( ZZ † )+det( ZZ † ) >
0, implies that tr( ZZ † ) <
2. Thus, we can identify the symmetric complex Cartan domain D = G/H = { Z ∈ Mat × ( C ) : σ − ZZ † > } (16)with an open subset of the eight-dimensional ball with radius √
2. Moreover, the com-pactified Minkowski space M is the Shilov boundary U (2) = { Z ∈ Mat × ( C ) : Z † Z = ZZ † = σ } of D .There is a one-to-one mapping from D onto the future tube domain T = { W = X + iY ∈ Mat × ( C ) : Y > } , (17)6f the complex Minkowski space C , , with X = x µ σ µ and Y = y µ σ µ hermitian matricesand Y > ⇔ y > k ~y k . This map is given by the Cayley transformation and its inverse: Z → W ( Z ) = i ( σ − Z )( σ + Z ) − , W → Z ( W ) = ( σ − iW ) − ( σ + iW ) . (18)This is the 3+1-dimensional analogue of the usual map form the unit disk onto theupper half-plane in two dimensions. Actually, the forward tube domain T is naturallyhomeomorphic to the quotient G/H in a new realization of G in terms of matrices f whichpreserve Γ = γ , instead of Γ = γ ; that is, f † γ f = γ . Both realizations of G are relatedby the map g → f = Υ g Υ − , Υ = 1 √ (cid:18) σ − σ σ σ (cid:19) . (19)We shall come again to this “forward tube domain” realization later on Section 5.Let us proceed by giving a complete local parametrization of G adapted to the fibration H → G → D . Any element g ∈ G (in the present patch, containing the identity element)admits the Iwasawa decomposition g = (cid:18) A BC D (cid:19) = (cid:18) ∆ Z ∆ Z † ∆ ∆ (cid:19) (cid:18) U U (cid:19) , (20)where the last factor U = ∆ − A, U = ∆ − D belongs to H ; i.e., U , U ∈ U (2). Likewise, a parametrization of any U ∈ U (2) (in apatch containing the identity), adapted to the quotient S = U (2) /U (1) , is (the Hopffibration) U = (cid:18) a bc d (cid:19) = (cid:18) δ zδ − ¯ zδ δ (cid:19) (cid:18) e iα e iβ (cid:19) , (21)where z = b/d ∈ C ≃ S (the one-point compactification of C by inverse stereographicprojection), δ = (1 + z ¯ z ) − / and e iα = a/ | a | , e iβ = d/ | d | .Sometimes it will be more convenient for us to use the following compact notation forthe sixteen coordinates of U (2 , α z Z Z − ¯ z β Z Z ¯ Z ¯ Z α z ¯ Z ¯ Z − ¯ z β = x x x x x x x x x x x x x x x x = { x αβ ( g ) } , (22)The set of coordinates { x αβ } is adapted to the new Lie algebra basis of step operatormatrices ( X βα ) νµ ≡ δ να δ βµ fulfilling the commutation relations: (cid:2) X β α , X β α (cid:3) = δ β α X β α − δ β α X β α , (23)and the usual orthogonality properties:tr( X βα X ργ ) = δ ρα δ βγ . u (1) ⊂ G is made of diagonal operators { X αα , α = 1 , . . . , } .Another realization of the conformal Lie algebra that will be useful for us is the onegiven in terms of left- and right-invariant vector fields, as generators of right- and left-translations of G , [ U Rg ψ ]( g ′ ) = ψ ( g ′ g ) , [ U Lg ψ ]( g ′ ) = ψ ( g − g ′ ) , (24)on complex functions ψ : G → C , respectively. Denoting by θ L = − ig − dg = θ αβ X βα = θ αµβν dx νµ X βα (25)the left-invariant Maurer-Cartan 1-form, the left-invariant vector fields L βα are defined byduality θ αβ ( L σρ ) = δ αρ δ σβ . The same applies to right-invariant 1-forms θ R = − idgg − inrelation with right-invariant vector fields R βα . They can also be computed through thegroup law g ′′ = g ′ g as: L βα ( g ) ≡ ∂x µν ( gg ′ ) ∂x αβ ( g ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ′ =1 ∂∂x νµ ( g ) , R βα ( g ) ≡ ∂x µν ( g ′ g ) ∂x αβ ( g ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g ′ =1 ∂∂x νµ ( g ) . (26)The quadratic Casimir operator (8) now adopts the compact form: C = L βα L αβ = R βα R αβ . Both sets of vector fields will be essential in our quantization procedure, the first ones( L ) as generators of gauge transformations and the second ones ( R ) as the symmetryoperators of our theory. G The actual Lagrangian for quantum mechanical geodesic free motion on G , as a configu-ration space, is given by: L G ( g, ˙ g ) = 12 tr( ϑ L ) = 12 ϑ αβ ϑ βα = 12 g νσµρ ( x ) ˙ x µν ˙ x ρσ , (27)where we are denoting by ϑ L = − ig − ˙ g = ϑ αβ X βα = ϑ ανβµ ˙ x µν X βα the restriction of (25) to trajectories g = g ( t ) and writing the natural metric on G ,g νσµρ = ϑ ανβµ ϑ βσαρ , in terms of vielbeins ϑ αβ . The equations of motion derived from (27) are:˙ ϑ L = 0, which can be converted into the standard form of geodesic motion¨ x a + Γ abc ( x ) ˙ x b ˙ x c = 08y introducing the Levi-Civita connection Γ abc [here we used an alternative indexation a = ( αβ ) = 1 , . . . ,
16, to simplify expressions]. The phase space of this theory is thecotangent bundle T ∗ G , which can be identified with the product of G and its Lie algebra G in a suitable way.It can be shown that the Lagrangian (27) is G -invariant under both: left- and right-rigid transformations, g ( t ) → g ′ g ( t ) and g ( t ) → g ( t ) g ′ , respectively; that is, L G is chiral.This chirality is partially broken when we reduce the dynamics from G to certain cosets G/G , with G the isotropy subgroup of a given Lie algebra element of the form X = X α =1 λ α X αα (28)(with λ α some real constants) under the adjoint action X → gX g − of G on its Liealgebra G . Actually, the new Lagrangian on G/G can be written as a “partial trace”: L G/G ( g, ˙ g ) = 12 tr G/G ( ϑ L ) ≡
12 tr([ X , ϑ L ]) = 12 N X α,β =1 ( λ α − λ β ) ϑ αβ ϑ βα . (29)For example, choosing X = λ γ = λD (the dilation) we have G = H = U (2) (the maxi-mal compact subgroup) and G/G the eight-dimensional domain D . For λ α = λ β , ∀ α, β =1 , . . . ,
4, the isotropy subgroup of X is the maximal Abelian subgroup G = U (1) and G/G = F is a twelve-dimensional “pseudo-flag” (non-compact) manifold. It is obvi-ous that L G/G is still invariant under general rigid left-transformations g ( t ) → g ′ g ( t ).However, this Lagrangian is now singular or, equivalently: Proposition 3.1.
The Lagrangian (29) is gauge invariant under local right-transformations g ( t ) → g ( t ) g ( t ) , ∀ g ( t ) ∈ G (30) Proof: we have that: ϑ L = − ig − ˙ g → ϑ ′ L = − ig − g − ( ˙ gg + g ˙ g ) = g − ϑ L g − ig − ˙ g and [ X , ϑ ′ L ] = g − [ X , ϑ L ] g , since G is the isotropy subgroup of X , which means [ X , g ] = 0 = [ X , ˙ g ]. The cyclicproperty of the trace completes the proof (cid:4) We have considered so far
G/G as a configuration space. In this article, we shall berather interested in G/G as a phase space. For example, we shall consider D [or thetube domain (17) of the complex Minkowski space C , ] as a (complex) phase space offour-position x µ and four-momenta y ν , in itself. This situation will require a new singularLagrangian of the form: L ( g, ˙ g ) = tr( X ϑ L ) = X α =1 λ α ϑ αα . (31)Again, this Lagrangian is left- G -invariant under rigid transformations. The differencenow is that it is linear in velocities ˙ x . Moreover, we shall prove that:9 roposition 3.2. The Lagrangian (31) is gauge (semi-)invariant under local right- trans-formations g ( t ) → g ( t ) g ( t ) , ∀ g ( t ) ∈ G (32) up to a total time derivative, i.e., L → L + ∆ L , ∆ L = − i tr( X g − ˙ g ) = dτdt , τ = X α =1 λ α x αα . (33) Proof:
We shall just consider the two important cases for us:1. λ α = λ β , ∀ α = β ⇒ G = U (1) , G/G = F X = λD = λ γ ⇒ G = H = U (2) , G/G = D .For the first case, any g ∈ G can be written as g = exp( ix αα X αα ) and ˙ g = ig ˙ x αα X αα because G is Abelian; therefore∆ L = − i tr( X g − ˙ g ) = X β =1 λ β ˙ x αα tr( X ββ X αα ) = X α =1 λ α ˙ x αα . For the second case, g = exp( iϕI + iτ ′ D + iω µν M µν ) = (cid:18) U U (cid:19) ∈ H . Disregardingthe trivial global phase ϕ , it is clear that for dilations g = d = e iτ ′ D we have ˙ d = i ˙ τ ′ De iτ ′ D and − i tr( X d − ˙ d ) = λ ˙ τ tr( D ) = λ ˙ τ ′ = ˙ τ , where τ ≡ λτ ′ . For Lorentz transformations g = m = exp( iω µν M µν ) we have ∆ L = 0since tr( DM µν ) = 0, which is a direct consequence of the orthogonality properties of thePauli matrices tr( σ µ σ ν ) = 2 δ µν . (cid:4) Remark . We can always fix the gauge to τ ( t ) = t . In the case X = λD , thisimplies that the dilation operator D will play the role of the Hamiltonian of the quantumtheory. The replacement of time translations by dilations as dynamical equations ofmotion has been considered in [39] and in [40] when quantizing field theories on space-likeLorentz-invariant hypersurfaces x = x µ x µ = τ =constant. In other words, if one wishesto proceed from one surface at x = τ to another at x = τ , this is done by scaletransformations; that is, D is the evolution operator in a proper time τ (cid:3) . G/G λ α label the (lowest weight) irreducible representationsof G on which the Hilbert space of our theory is constructed. There are several ways ofseing that the values of λ α are quantized. One way is through the path integral method.To examine this explicitly, consider the transition amplitude from an initial point g at10 = t to a final point g at t = t . For each path g ( t ) connecting g and g , there aremany gauge equivalent paths g ′ ( t ) = g ( t ) g ( t ) , g ( t ) ∈ G , g ( t ) = g ( t ) = 1that must contribute to the sum of the path integral with the same amplitude, that is: e i R t t dt L ( g, ˙ g ) = e i R t t dt L ( g ′ , ˙ g ′ ) = e i R t t dt L ( g, ˙ g ) e i R t t dt ∆ L ( g, ˙ g ) ⇒ e i R t t dt ∆ L ( g, ˙ g ) = 1 . Using (33), the last expression can be written as exp( i ( τ ( t ) − τ ( t )) = 1 which, togetherwith the fact that g ( t , ) = e i P α x αα ( t , ) = 1 ⇔ x αα ( t , ) = 2 πn α , , n α , ∈ Z , means that λ α must be an integer number. Considering coverings of G , one can relax theinteger to a half-integer condition, as happens with SU (2) in relation with SO (3).Other alternative way to the path-integral description of realizing the integrality of λ α is though the following operator (representation-theoretic) description. At the quantumlevel, finite-right gauge transformations like (32) induce constraints on “physical” wavefunctions ψ ( g ) as: ψ ( gg ) = U λ ( g ) ψ ( g ) , g ∈ G (34)where we are allowing ψ to transform non-trivially according to a representation U λ of G of index λ . This could be seen as a generalization of the original Dirac approach to thequantization of constrained systems (where U λ is taken to be trivial) which allows newinequivalent quantizations labelled by λ α (see e.g. [41, 42, 43, 44] for several approachesto the subject). The finite constraint condition (34) can be written in infinitesimal formas L αα ψ = λ α ψ, α = 1 , . . . , , (35)where we have used the fact that left-invariant vector fields (26) are generators of finiteright-transformations. In the parametrization { x αβ } , the left-invariant vector fields L βα fulfill the same commutation relations as the step operator matrices (23). Therefore,when acting on physical/constrained states (35), they satisfy creation and annihilationharmonic-oscillator-like commutation relations:[ L βα , L αβ ] = ( λ β − λ α ) (no sum on α, β ) . We shall work in a holomorphic picture, which means that constrained wave functions(35) will be further restricted by holomorphicity conditions: L βα ψ = 0 , ∀ α > β = 1 , , . (36)In fact, looking at (26), for g ∈ G near the identity we have L βα ( g ) ∼ ∂/∂x αβ so that L βα ψ =0 means, roughly speaking, that ψ ( g ) does not depend on the variables x αβ , α > β = 1 , , ψ is holomorphic. The complementary option L βα ψ = 0 , ∀ β > α = 1 , , polarization conditions(see also [46] for a Group Approach to Quantization scheme and [47] for the extensionof first-order polarizations to higher-order polarizations), intended to reduce the left-representation U L (24) of G , on complex wave functions ψ , to G/G . Also, the constraints(35) and (36) are exactly the defining relations of a lowest-weight representation. Firstly we shall consider the (spin-less) case λ = λ = − λ = − λ ≡ − λ/
2, that is, X = λ γ = λD , and we shall call λ the conformal, scale or mass dimension . In this casethe gauge group is the maximal compact subgroup G = H = U (2) and the phase spaceis the eight-dimensional domain D = G/G . The constraint conditions (35) can now be enlarged to D L ψ = −
12 ( L + L − L − L ) ψ = λψ, M Lµν ψ = 0 . (37)which renders translation ( P µ ) and acceleration ( K ν ) generators into conjugated variables.In fact, the last commutator of (6), on constrained (physical) wave functions (37), gives: (cid:2) K Lµ , P Lν (cid:3) ψ = 2 λη µν ψ, (38)which states that K µ and P µ can not be simultaneously measured, the conformal dimen-sion λ playing here the role of the Planck constant ~ . Note that K µ and P µ are conjugatedbut not canonically conjugated as such. We address the reader to Refs. [48, 49] for otherdefinitions of quantum observables associated with positions in space-time, namely X µ = M νµ · P ν P + D · P µ P (39)(dot means symmetrization), fulfilling canonical commutation relations [ X µ , P ν ] = η µν inside the conformal (enveloping) algebra (6).A further restriction K Lµ ψ = 0 (40)selects the holomorphic (“position”) representation. Indeed, let us prove that: Theorem 4.1.
The general solution to (37) and (40) can be factorized as: ψ λ ( g ) = W λ ( g ) φ ( Z ) , (41) where the “ground state” W λ ( g ) = det( D ) − λ = det( σ − Z † Z ) λ/ det( U ) − λ = (1 − tr( Z † Z ) + det( Z † Z )) λ/ det( U ) − λ (42)12 s a particular solution of (37,40) and φ is the general solution for the trivial representation λ = 0 of G = H (actually, an arbitrary, analytic holomorphic function of Z ), for thedecomposition (20) of an element g ∈ G . Proof:
A generic proof (also valid for other symmetry groups) that the general solutionof (37,40) admits a factorization of the form (41) can be found in the Proposition 3.3 of[50]. Here we shall just prove that (41) is a solution of (37,40). Indeed, by applying afinite right translation (24) on W λ ( g ):[ U Rg ′ W λ ]( g ) = W λ ( gg ′ ) = det( D ′′ ) − λ = det( CB ′ + DD ′ ) − λ = det( D ′ ) − λ det( CZ ′ + D ) − λ , (43)we see that W λ ( gg ′ ) is not affected by translations by Z ′† = Z † ( g ′ ) = C ′ A ′− . In-finitesimally, it means that K Lµ W λ ( g ) = 0, according to the lower-triangular choice ofthe generator K µ in (10). For Lorentz transformations we have B ′ = 0 = C ′ anddet( A ′ ) = 1 = det( D ′ ) and therefore W λ ( gg ′ ) = W λ ( g ), that is M Lµν W λ ( g ) = 0. Fordilations we have B ′ = 0 = C ′ and A ′ = e iτ/ σ = D ′† , which gives W λ ( gg ′ ) = e iλτ W λ ( g )or D L W λ ( g ) = λ W λ ( g ) for small τ . It remains to prove that φ ( Z ) is the general solutionof (37,40) for λ = 0. From (13) we have Z ′′ = Z ( gg ′ ) = B ′′ D ′′− = ( AB ′ + BD ′ )( CB ′ + DD ′ ) − , (44)which is not affected by C ′ and gives Z ′′ = Z for dilations and Lorentz transformations( B ′ = 0) (cid:4) Remark . In the last theorem, we are implicitly restricting ourselves to gauge trans-formations g ′ ∈ S ( U (2) ), which means det( g ′ ) = det( U U ) = 1. If we allow for transfor-mations g ′ ∈ U (2) with det( g ′ ) = 1 (like e iα I ) and we want them to leave physical wavefunctions strictly invariant ψ ( gg ′ ) = ψ ( g ) (i.e., we restrict ourselves to representationswith λ + λ + λ + λ = 0), we must choose a symmetrical form for the ground state W λ ( g ) = det( A † ) − λ/ det( D ) − λ/ = det( σ − Z † Z ) λ/ det( U † ) − λ/ det( U ) − λ/ , (45)which reduces to (42) for det( U U ) = 1.Moreover, instead of (40), we could have chosen the complementary constraint P Lµ ψ =0 which would have led us to a anti-holomorphic (“acceleration”) representation ψ λ ( g ) =ˇ W λ ( g ) φ ( Z † ) with the new ground stateˇ W λ ( g ) = det( A ) − λ/ det( D † ) − λ/ = det( σ − Z † Z ) λ/ det( U ) − λ/ det( U † ) − λ/ , (46)which, for g ∈ SU (2 , W λ ( g ) = det( A ) − λ = det( σ − ZZ † ) λ/ det( U ) − λ = W λ ( g ) . Therefore, the BRP-like symmetry K Lµ ↔ P Lµ , D L → − D L in (7) manifest here as a chargeconjugation and time reversal (CT) operations. See later on Section 6 for more details ona “BRP-CPT connection” proposal inside the conformal group. (cid:3) .1.2 Irreducible representation, Haar measure and Bergman kernel The finite left-action of G on physical wave functions (41),[ U Lg ′ ψ λ ]( g ) = ψ λ ( g ′− g ) = det( D ( g ′− g )) − λ φ ( Z ′ )= W λ ( g ) det( D ′† − B ′† Z ) − λ φ ( Z ′ ) , (47) Z ′ ≡ Z ( g ′− g ) = ( A ′† Z − C ′† )( D ′† − B ′† Z ) − , provides a unitary irreducible representation of G under the invariant scalar product h ψ λ | ψ ′ λ i = Z G dµ L ( g ) ψ λ ( g ) ψ ′ λ ( g ) (48)given trough the left-invariant Haar measure [the exterior product of left-invariant one-forms (25)] which can be decomposed as: dµ L ( g ) = c V α,β =1 ϑ αβ = c det( ϑ αµβν ) V µ,ν =1 dx νµ = c dµ ( g ) L (cid:12)(cid:12) G/H dµ L ( g ) (cid:12)(cid:12) H ,dµ L ( g ) (cid:12)(cid:12) G/H = det( σ − ZZ † ) − | dZ | ,dµ ( g ) | H = dv ( U ) dv ( U ) , (49)where we are denoting by dv ( U ) the Haar measure on U (2), which can be in turn decom-posed as: dv ( U ) ≡ dv ( U ) | U (2) /U (1) dv ( U ) | U (1) ,dv ( U ) | U (2) /U (1) = dv ( U ) | S ≡ ds ( U ) = (1 + z ¯ z ) − | dz | , (50) dv ( U ) | U (1) ≡ dαdβ. We have used the Iwasawa decomposition of an element g given in (20,21) and denotedby | dz | and | dZ | the Lebesgue measures in C and C , respectively. The normalizationconstant c = π − ( λ − λ − ( λ − (cid:18) (2 π ) (cid:19) − (51)is fixed so that the ground state (42) is normalized, i.e. hW λ | W λ i = 1 (see AppendixB of Ref. [31] for orthogonality properties), the factor (2 π ) / v ( U (2)). The scalar product (48) is finite as long as λ ≥ R αβ ( g ) in (26)and constitute the operators (observables) of our quantum theory. For example, fromthe general expression (47), we can compute the finite left-action of dilations g ′ = e iτD ( B ′ = 0 = C ′ and A ′ = e − iτ/ σ = D ′† ) on physical wave functions, ψ λ ( g ′ g ) = e iλτ W λ ( g ) φ ( e iτ Z ) ,
14r infinitesimally: D R ψ λ ( g ) = −
12 ( R + R − R − R ) ψ λ ( g ) = W λ ( g ) λ + X i,j =1 Z ij ∂∂Z ij ! φ ( Z ) ≡ W λ ( g ) D λ φ ( Z ) , (52)where we have defined the restriction of the dilation operator on holomorphic functionsas: D λ ≡ λ + X i,j =1 Z ij ∂∂Z ij , (53)for future use. As we justified in Remark 3.3, the dilation generator D R plays the role ofthe Hamiltonian operator of this theoryˆ H = − i ∂∂τ = D R . (54)The conformal or mass dimension λ can be then interpreted as the zero point (vacuum)energy and the corresponding eigenfunctions are homogeneous polynomials φ n ( Z ) of acertain degree (eigenvalue) n , according to Euler’s theorem. We shall come back to thisquestion later in Theorem 4.3.Let us introduce bracket notation and write: W λ ( g ) ≡ h g | λ, i = h λ, |U Lg − | λ, i , ψ λ ( g ) ≡ h g | ψ λ i . (55)Here we are implicitly making use of the Coherent-States machinery (see e.g. [51, 52]).Actually, we are denoting by | g i ≡ U Lg | λ, i the set of vectors in the orbit of the ground(“fiducial”) state | λ, i (the lowest-weight vector) under the left action of the group G (this set is called a family of covariant coherent states in the literature [51, 52]). We caneasily calculate the coherent state overlap: h g ′ | g i = h λ, | U Lg ′− g | λ, i = W λ ( g − g ′ ) = det( D ( g − g ′ )) − λ = det( D † D ′ − B † B ′ ) − λ = det( D † ) − λ det( σ − ( BD − ) † B ′ D ′− ) − λ det( D ′ ) − λ = W λ ( g ) det( σ − Z † Z ′ ) − λ W λ ( g ′ ) . (56)The set of coherent states {| g i , g ∈ G } constitutes a tight frame (see [31] for a proof inthe context of Conformal Wavelets) with resolution of unity:1 = Z G dµ ( g ) | g i h g | . Actually, the coherent state overlap (56) is a reproducing kernel satisfying the integralequation of a projector operator h g | g ′′ i = Z G dµ L ( g ′ ) h g | g ′ i h g ′ | g ′′ i ψ λ ( g ′ ) = Z G dµ L ( g ) h g ′ | g i ψ λ ( g ) . Since the ground state W λ is a fixed common factor of all the wave functions (41), wecould factor it out and define the restricted left-action[ U λg ′ φ ]( Z ) ≡ W − λ ( g )[ U Lg ′ ψ λ ]( g ) = det( D ′† − B ′† Z ) − λ φ ( Z ′ ) ≡ φ ′ ( Z ) (57)of G on the arbitrary (holomorphic) part φ of ψ λ , instead of (47). In standard (in-duced) representation theory, the factor det( D ′† − B ′† Z ) − λ is called a “multiplier” (Radon-Nicodym derivative) and fulfils cocycle properties. For the representation (57) of G onholomorphic functions φ ( Z ) to be unitary, the left- G -invariant Haar measure (49) has tobe accordingly modified as: dµ λ ( Z, Z † ) ≡ c λ |W λ ( g ) | dµ L ( g ) (cid:12)(cid:12) G/H = c λ det( σ − ZZ † ) λ − | dZ | , (58)where dµ L ( g ) (cid:12)(cid:12) G/H in (49) is the projection of the left- G -invariant Haar measure dµ L ( g )onto G/H . Roughly speaking, we are integrating out the coordinates of H and redefiningthe normalization constant c in (51) as c λ = c/v ( U (2)) = π − ( λ − λ − ( λ −
3) sothat the unit constant function φ ( Z ) = 1 (the ground state) is normalized (see [31] fororthogonality properties). As before, we could also introduce a modified bracket notation φ ( Z ) ≡ ( Z | φ ) and a new set {| Z ) , Z ∈ D } of coherent states in the Hilbert space H λ ( D ) = L ( D , dµ λ ) of analytic square-integrable holomorphic functions φ on D . Thenew coherent state overlap ( Z | Z ′ ) is nothing but the so called reproducing Bergman’skernel K λ ( Z, Z ′ ). It is related to (56) by: K λ ( Z ′ , Z ) = ( Z ′ | Z ) = h g ′ | g iW λ ( g ′ ) W λ ( g ) = det( σ − Z † Z ′ ) − λ . (59)We notice that, unlike | g i , the coherent state | Z ) is not normalized. In fact, K λ ( Z, Z † ) ≡ ln ( Z | Z ) (60)is nothing but the K¨ahler potential, which defines D as a K¨ahler manifold with localcomplex coordinates Z = z µ σ µ , an Hermitian Riemannian metric g and a correspondingclosed two-form ωds = g µν dz µ ⊙ d ¯ z ν , ω = − i g µν dz µ ∧ d ¯ z ν , g µν ≡ ∂ K λ ∂z µ ∂ ¯ z ν , (61)where ⊙ denotes symmetrization. We shall come back to the Riemannian structure of D and T and the connection with the BRP later on Section 5.16 .1.3 Schwinger’s theorem, orthonormal basis and closure relations As already commented after Eq. (52), we are interested in calculating an orthonormalbasis of H λ ( D ) made of Hamiltonian eigenfunctions ϕ J ( Z ) ≡ ( Z | λ, J ), where J denotesa set of indices. This orthonormal basis would provide us with a new resolution of theidentity 1 = X J | λ, J ) ( λ, J | . Actually, we shall identify ϕ J ( Z ) by looking at the expansion of the Bergman’s kernel K λ ( Z ′ , Z ) = ( Z ′ | Z ) = X J ( Z ′ | λ, J ) ( λ, J | Z ) = X J ϕ J ( Z ′ ) ϕ J ( Z ) . Thus, the Bergman’s kernel plays here the role of a generating function. To be moreprecise:
Theorem 4.3.
The infinite set of polynomials ϕ j,mq ,q ( Z ) = s j + 1 λ − (cid:18) m + λ − λ − (cid:19)(cid:18) m + 2 j + λ − λ − (cid:19) det( Z ) m D jq ,q ( Z ) , (62) with D jq ,q ( Z ) = s ( j + q )!( j − q )!( j + q )!( j − q )! min( j + q ,j + q ) X p =max(0 ,q + q ) (cid:18) j + q p (cid:19)(cid:18) j − q p − q − q (cid:19) × z p z j + q − p z j + q − p z p − q − q (63) the standard Wigner’s D -matrices ( j is a non-negative half-integer), verifies the followingclosure relation (the reproducing Bergman kernel): X j ∈ N / ∞ X m =0 j X q ,q = − j ϕ j,mq ,q ( Z ) ϕ j,mq ,q ( Z ′ ) = 1det( σ − Z † Z ′ ) λ (64) and constitute an orthonormal basis of H λ ( D ) . This theorem has been proven in [31]. It turns out to be rooted in a extension of theSchwinger’s formula:
Theorem 4.4. (Schwinger’s Master Theorem)
The identity X j ∈ N / t j j X q = − j D jqq ( X ) = 1det( σ − tX ) (65) holds for any × matrix X , with t an arbitrary parameter. The abovementioned extension of the Theorem 4.4 can be stated as:17 heorem 4.5. ( λ -Extended Schwinger’s Master Theorem) For every λ ∈ N , λ ≥ andevery × complex matrix X the following identity holds: X j ∈ N / j + 1 λ − ∞ X m =0 t j +2 m (cid:18) m + λ − λ − (cid:19)(cid:18) m + 2 j + λ − λ − (cid:19) det( X ) m j X q = − j D jqq ( X )= det( σ − tX ) − λ . (66)We address the interested reader to Ref. [31] for a complete proof. Scketch of proof of Theorem 4.3:
Assuming the validity of (66) and replacing tX = Z † Z ′ in it, we have: X j ∈ N / j + 1 λ − ∞ X m =0 (cid:18) m + λ − λ − (cid:19)(cid:18) m + 2 j + λ − λ − (cid:19) det( Z † Z ′ ) m j X q = − j D jqq ( Z † Z ′ )= 1det( σ − Z † Z ′ ) λ . (67)Using determinant and Wigner’s D -matrix rulesdet( Z † Z ′ ) n j X q = − j D jqq ( Z † Z ′ ) = det( Z † ) n det( Z ′ ) n j X q ,q = − j D jq q ( Z ) D jq q ( Z ′ )and the definition of the functions (62), we see that (67) reproduces (64). On the otherhand, the number of linearly independent polynomials Q i,j =1 z n ij ij of fixed degree of ho-mogeneity n = P i,j =1 n ij is ( n + 1)( n + 2)( n + 3) /
6, which coincides with the numberof linearly independent polynomials (62) with degree of homogeneity n = 2 m + 2 j . Thisproves that the set of polynomials (62) is a basis for analytic functions φ ∈ H λ ( D ).Moreover, this basis turns out to be orthonormal under the projected integration measure(58). We address the interested reader to the Appendix B of Ref. [31] for a proof. (cid:4) Remark . The set (62) constitutes a basis of Hamiltonian eigenfunctions with energyeigenvalues E λn (the homogeneity degree) given by:ˆ H λ ϕ j,mq ,q = E λn ϕ j,mq ,q , E λn = λ + n, n = 2 j + 2 m, (68)with ˆ H λ = D λ defined in (53). Each energy level E λn is then ( n + 1)( n + 2)( n + 3) / E λ = λ playing the role of a zero-point energy. At this stage it is interesting to compare ourHamiltonian choice with others in the literature like [53] studying a SU (2 , D . In this case the quantum Hamiltonian is chosen to bethe Toeplitz operator corresponding to the square of the distance with respect to the SU (2 , D . (cid:3) .2 Conformal spinning quantum particles Let us use the following notation for X = X α =1 λ α X αα = λD + s Σ (3)1 + s Σ (3)2 + κI, (69)where Σ (3)1 = X − X = (cid:18) σ
00 0 (cid:19) , Σ (3)2 = X − X = (cid:18) σ (cid:19) stand for the third spin components and I the 4 × SU (2) are s ≡ ( λ − λ ) / s ≡ ( λ − λ ) /
2. The conformal dimension is λ = ( λ + λ − λ − λ ) / κ = ( λ + λ + λ + λ ) / U (1) quantum number. We shall choose,without lost of generality, κ = 0, which means that λ remains integer (as in the spin-lesscase) and that we are restricting ourselves to representations of SU (2 , ⊂ U (2 , Theorem 4.7.
The general solution to (35) and (36) can be factorized as: ψ s ,s λ ( g ) = W s ,s λ ( g ) φ ( Z, z , z ) , (70) where the ground state W s ,s λ ( g ) = det( A † ) − λ s / det( D ) − λ s / D s s ,s ( U † ) D s − s , − s ( U )= det( A † ) − λ s / det( D ) − λ s / ¯ a s d s = det(∆ U † ) − λ s / det(∆ U ) − λ s / ( δ e − iα ) s ( δ e iβ ) s = det( σ − Z † Z ) λs (1 + ¯ z z ) − s (1 + ¯ z z ) − s × e − iα (2 s − λ s / e iβ λ s / e − iα λ s / e iβ (2 s − λ s / , (71) with λ s ≡ λ − s − s , is a particular solution of (35,36) and φ is the general solution for thetrivial representation λ α = 0 of G = U (1) (actually, an arbitrary, analytic holomorphicfunction of Z, z , z ), for the decomposition (20,21) of an element g ∈ G . Proof:
On the one hand, from (43) we conclude that the factors det( D ) − λ and D s − s , − s ( U ),with U = ( DD † ) − / D fulfill the holomorphicity conditions (36) for ( β, α ) = (1 , , , , U † = A † ( AA † ) − / and we have that A ′′† = A ( gg ′ ) † = A ′† A † + C ′† B † = A ′† ( A † + ( C ′ A ′− ) † B † ) = A ′† ( A † + Z ′ B † )is not affected by Z ′† = C ′ A ′− either, according to the definition (13). On the otherhand, for g ′ ∈ H we have that¯ a ′′ = ¯ a ( gg ′ ) = ¯ a ¯ a ′ + ¯ b ¯ c ′ = ¯ a ′ (¯ a − z ′ ¯ b ) d ′′ = d ( gg ′ ) = cb ′ + dd ′ = d ′ ( d + z ′ c )are not affected by ¯ z ′ = − c ′ /a ′ = ¯ b ′ / ¯ d ′ , according to the definition (21). This proves thatthe ground state (71) fulfills the holomorphicity conditions (36) for ( β, α ) = (1 , , g ∈ G = U (1) . Finite right (gauge) dilations g = e iτD leave W s ,s λ ( gg ) = e iλτ W s ,s λ ( g ) invariantup to the phase U λ ( g ) = e iλτ (a character of G ), where we have used that det( · ) and D s ( · ) are homogeneous of degree 2 and 2 s , respectively. Infinitesimally, it means that D L ψ s ,s λ = λψ s ,s λ . For g (1 , = e iαΣ , the ground state transforms as expected: W s ,s λ ( gg (1 , ) = U λ ( g (1 , ) W s ,s λ ( g ) , U λ ( g (1 , ) = e is , α . Infinitesimally, it means that Σ L (3)1 , W s ,s λ = 2 s , W s ,s λ , ( Σ L (3)1 ≡ L − L ,Σ L (3)2 ≡ L − L . (72)Moreover, one can easily check that W s ,s λ ( gg ) = W s ,s λ ( g ) for diagonal U (1) transfor-mations g = e iθ I , that is, κ = 0. Finally, using similar arguments to those employedin (44), we can assert that z ′ , = z , ( gg ) = z , , ∀ g ∈ G , which ends up proving thegauge conditions (34) (cid:4) Remark . Instead of (36), we could have chosen the complementary constraint L βα ψ =0 , ∀ α < β which would have led us to a anti-holomorphic representation. (cid:3) As in Eq. (47), we can compute the finite left-action of G on physical wave functions(70). In particular, for the case of dilations g ′ = e iτ ′ D (i.e., B ′ = 0 = C ′ and A ′ = e − iτ ′ / σ = D ′† ) we have: ψ s ,s λ ( g ′− g ) = e iλτ ′ W s ,s λ ( g ) φ ( e iτ ′ Z, z , z ) , or infinitesimally: D R ψ s ,s λ ( g ) = W s ,s λ ( g ) λ + X i,j =1 Z ij ∂∂Z ij ! φ ( Z, z , z ) . (73)Comparing this expression with (52), we realize that the spin coordinates z , z do notcontribute to the degree of homogeneity of φ under dilations, as they correspond to “in-ternal” (versus space-time-momentum) degrees of freedom.As in the previous subsection, we can introduce a modified bracket notation φ ( Z, z , z ) ≡ ( Z, z , z | φ ) and a set {| Z, z , z ) , Z ∈ D , z , z ∈ C } of coherent states in the Hilbertspace H s ,s λ ( F ) of analytic measurable holomorphic functions φ on the twelve-dimensionalpseudo-flag manifold F = U (2 , /U (1) , locally D × C , with integration measure dµ s ,s λ ( Z, z , z ; Z † , ¯ z , ¯ z ) ≡ dµ λ s ( Z, Z † ) 2 s + 1 π ds ( U ) 2 s + 1 π ds ( U ) , (74)where dµ λ s ( Z, Z † ) and ds ( U ) are defined in (58) and (50), respectively. Note that thesquare-integrability condition λ ≥ H λ ( D ) becomes λ s ≥ H s ,s λ ( F ). The constantfactor (2 s + 1) /π is introduced so that the following set of functions is normalized.20 heorem 4.9. The infinite set of polynomials ˇ ϕ m,m ,m j,q ,q ( Z, z , z ) ≡ ( − m + s ϕ j,mq ,q ( Z ) D s s , − m ( U † ) D s m , − s ( U ) D s s ,s ( U † ) D s − s , − s ( U )= ϕ j,mq ,q ( Z ) s(cid:18) s m + s (cid:19)(cid:18) s m + s (cid:19) z m + s z m + s , (75) (with ϕ j,mq ,q in (62) replacing λ → λ s ) provides an orthonormal basis of H s ,s λ ( F ) . Theclosure relation: ∞ X j ∈ N / ∞ X m =0 j X q ,q = − j s X m = − s s X m = − s ˇ ϕ m,m ,m j,q ,q ( Z ′ , z ′ , z ′ ) ˇ ϕ m,m ,m j,q ,q ( Z, z , z )= ( Z ′ , z ′ , z ′ | Z, z , z ) (76) gives the reproducing Bergman’s kernel for spinning particles: K s ,s λ ( Z ′ , z ′ , z ′ ; Z, z , z ) = ( Z ′ , z ′ , z ′ | Z, z , z )= det( σ − Z † Z ′ ) − λ s (1 + ¯ z z ′ ) s (1 + ¯ z z ′ ) s . (77) Proof:
Assuming the orthonormality of (62) (see Appendix B of Ref. [31]), and realizingthat Z C s(cid:18) sm (cid:19) ¯ z m s(cid:18) sm ′ (cid:19) z m ′ s + 1 π ds ( U ) = δ m,m ′ , m, m ′ = 0 , . . . , s, we prove the orthonormality of the functions (75). Moreover, the number of linearlyindependent polynomials Q i,j =1 z n ij ij Q i =1 z n i i with 0 ≤ n i ≤ s i and fixed n = P i,j =1 n ij is (2 s + 1)(2 s + 1)( n + 1)( n + 2)( n + 3) /
6, which coincides with the number of linearlyindependent polynomials (75) with degree of homogeneity n = 2 m + 2 j in the coordinates Z . This proves that the set of polynomials (75) is a basis for analytic functions H s ,s λ ( F ).It just remains to prove the closure relation (76). This proof reduces to that ofTheorem 4.3 when noting the binomial identity P sm =0 (cid:0) sm (cid:1) (¯ zz ′ ) m = (1 + ¯ zz ′ ) s or theWigner D -matrix property s X n = − s D ssn ( U ) D sns ( U ′ ) = D sss ( U U ′ ) . (cid:4) . Remark . At this point it is interesting to compare our construction with others inthe literature like [8], where the proposed basis functionsΦ m,m ,m j,q ,q ( A, D, Z ) = D j j ,m ( A T ) D j m ,j ( D ) ϕ j,mq ,q ( Z ) (78)21o not form an orthogonal set unless a coupling between orbital angular momentum j with spin j , j by means of Clebsch-Gordan coefficients is made:˜Φ m,p ,p j,j ,j = X m ,m ,q ,q C ( j, q ; s , m − s | j , p ) C ( j, q ; s , m − s | j , p )Φ m,m ,m j,q ,q . Moreover, the fact that U = ( AA † ) − / A and U = ( DD † ) − / D introduces a new contri-bution of D j ( A ) and D j ( D ) to the integration measure dµ j ,j λ , with respect to D j ( U )and D j ( U ), such that the square-integrability condition becomes λ ≥ j + 2 j . (cid:3) The Hamiltonian of our spinning particle is ˆ H = − i ∂∂τ with τ given by (33). Itsexpression in terms of right-invariant vector fields R αβ is thenˆ H = X α =1 λ α R αα = ρ D R + ρ Σ R (3)1 + ρ Σ R (3)2 + ρ I, (79)with: ρ = 4 λ ( λ − s − s )( λ − s )( λ − s ) , ρ = − s λ − s , ρ = − s λ − s , ρ = 4 λ ( s − s )( λ − s )( λ − s ) , and Σ R (3)1 , the right-invariant version of (72). In order to compare with the spin-less case,we can always renormalizeˆ H → ˆ H /ρ = ˆ H ′ = D R + ̺ Σ R (3)1 + ̺ Σ R (3)2 + ̺ I, (80)with ̺ α = ρ α /ρ . We can interpret ̺ , as constant “magnetic fields” (oriented along the“ z ” direction) coupled to the spin degrees of freedom Σ R (3)1 , . The set (75) constitutes abasis of eigenfunctions of the Hamiltonian (80) with eigenvalues (energy levels) given by: E λ,m ,m n,q ,q = λ + ̺ + n + ̺ ( m + q ) + ̺ ( m + q ) , n = 2 j + 2 m. (81)Comparing this energy eigenvalues with the energy spectrum (68) of the spin-less Hamil-tonian ˆ H = D R , we realize that the zero-point energy has been shifted from λ to λ + ̺ − s ̺ − s ̺ . Like in the (anomalous) Zeeman effect , the introduction of spinleads to an splitting of a spin-less spectral line E λn into (2 s + 1)(2 s + 1) components inthe presence of a “static magnetic field” ̺ , . In this section we shall translate some expressions obtained from the complex Cartandomain (16) into the forward tube domain (17), where we enjoy more (Minkowskian)intuition. We shall restrict ourselves to the scalar case, since it is representative of themore general case. 22 .1 Tube domain as a homogeneous space of SU (2 , As we have already said, the forward tube domain T is naturally homeomorphic to thequotient G/H in the realization of G in terms of matrices f = (cid:18) R iS − iT Q (cid:19) (82)which preserve Γ = γ , instead of Γ = γ ; that is, f † γ f = γ . Both realizations of G arerelated by the map (19), which can be explicitly written as g = (cid:18) A BC D (cid:19) = Υ − f Υ = 12 (cid:18) R + iS − iT + Q − R + iS + iT + Q − R − iS − iT + Q R − iS + iT + Q (cid:19) . (83)The identification of T with the quotient G/H is given through W ( f ) = i ( R − iS )( Q + iT ) − . (84)Hence, the left translation f ′ → f f ′ of G on itself induces a left action of G on T givenby: W = W ( f ′ ) → W ′ = W ( f f ′ ) = ( RW + S )( T W + Q ) − . (85)Setting W = x µ σ µ , and making use of the standard homomorphism (spinor map) between SL (2 , C ) and SO + (3 ,
1) given by: W ′ = RW R † ↔ x ′ µ = Λ µν x ν , R ∈ SL (2 , C ) , Λ µν ∈ SO + (3 , x ′ µ = Λ µν ( ω ) x ν , correspond to T = S = 0 and R = Q − † ∈ SL (2 , C ).ii) Dilations correspond to T = S = 0 and R = Q − = ρ / I iii) Spacetime translations equal R = Q = σ and S = b µ σ µ , T = 0.iv) Special conformal transformations correspond to R = Q = σ and T = a µ σ µ , S = 0by noting that det( σ + T W ) = 1 + 2 ax + a x . Let us see the expression of the wave functions (41) in the tube domain T . Performingthe change of variables (83) in (41) we get ψ λ ( f ) = det( Q + iT ) − λ λ det( σ − iW ) − λ φ ( Z ( W )) ≡ Ω λ ( f ) ˜ φ ( W ) , (86)where we have defined a new ground state Ω λ and a new function ˜ φ as:Ω λ ( f ) ≡ det( Q + iT ) − λ , ˜ φ ( W ) ≡ λ det( σ − iW ) − λ φ ( Z ( W )) . (87)23n the same manner, the coherent-state overlap (56) can be cast as h f ′ | f i = det( Q † − iT † ) − λ det( i W † − W ′ )) − λ det( Q ′ + iT ′ ) − λ (88)Since the ground state Ω λ is a fixed common factor of all the wave functions (86), wecan factor it out (as we did in (57) with W λ ) and define the restricted action[ ˜ U λf ′ ˜ φ ]( Z ) ≡ Ω − λ ( f )[ U Lf ′ ψ λ ]( f )= det( R ′† − T ′† W ) − λ ˜ φ (( Q ′† W − S ′† )( R ′† − T ′† W ) − ) ≡ ˜ φ ′ ( W ) (89)of G on the arbitrary (holomorphic) part ˜ φ of ψ λ . The Radon-Nicodym derivative isnow det( R ′† − T ′† W ) − λ . The representation (89) of G on holomorphic functions ˜ φ ( W ) isunitary with respect to the re-scaled integration measure d ˜ µ λ ( W, W † ) ≡ | Ω λ ( f ) | d ˜ µ L ( f ) (cid:12)(cid:12) G/H = c λ det( i W † − W )) λ − | dW | , (90)where we are using | dW | as a shorthand for the Lebesgue measure V i,j =1 d ℜ w ij d ℑ w ij on T . To arrive at (90), firstly, we have performed the Cayley transformation (18) in theprojected integration measure: dµ L ( g ) (cid:12)(cid:12) G/H = c λ det( σ − ZZ † ) − | dZ | → d ˜ µ L ( f ) (cid:12)(cid:12) G/H = c λ det( i W † − W )) − | dW | , (91)taking into account that det( σ − ZZ † ) = det(2 i ( W † − W )) | det( σ − iW ) | − and theJacobian determinant | dZ | / | dW | = 2 | det( σ − iW ) | − , and secondly, we have written | Ω λ ( f ) | = det( Q † − iT † ) − λ det( Q + iT ) − λ = det( i W † − W )) λ by making use of (84) and its hermitian conjugate.As in (59), we could also introduce a modified bracket notation ˜ φ ( W ) ≡ ( W | ˜ φ ) and anew set {| W ) , W ∈ T } of coherent states in the Hilbert space H λ ( T ) of analytic measur-able holomorphic functions ϕ on T . The new coherent state overlap ( W | W ′ ) is the newBergman’s kernel ˜ K λ ( W ′ , W ). It is related to (88) by:˜ K λ ( W ′ , W ) = ( W ′ | W ) = h f ′ | f i Ω λ ( f ′ )Ω λ ( f ) = det( i W † − W ′ )) − λ . (92)We again notice that, unlike | f i , the coherent state | W ) is not normalized. Now, theK¨ahler potential is ln ( W | W ), which defines T as a K¨ahler manifold too.The identification (87) actually provides an isometry between the spaces of analyticholomorphic functions H λ ( D ) and H λ ( T ). Let us formally state it.24 roposition 5.1. The correspondence S λ : H λ ( D ) −→ H λ ( T ) φ λ φ ≡ ˜ φ, with ˜ φ ( W ) = 2 λ det( I − iW ) − λ φ ( Z ( W )) (93) and Z ( W ) given by the Cayley transformation(18), is an isometry, that is: h φ | φ ′ i H λ ( D ) = hS λ φ |S λ φ ′ i H λ ( T ) . (94) Moreover, S λ is an intertwiner (equivariant map) of the representations (47) and (89),that is: U λ = S − λ ˜ U λ S λ . (95) Proof:
The isometry property is proven by construction from (87). The intertwiningrelation (95) can be explicitly written as:[ U λ φ ]( Z ) = det( D † − B † Z ) − λ φ (cid:0) ( A † Z − C † )( D † − B † Z ) − (cid:1) = h S − λ ˜ U λ ˜ φ i ( Z ) = det( I − iW ) λ det( R † − T † W ) − λ det( I − iW ′ ) − λ φ ( Z ( W ′ )) , (96)where W ′ = ( Q † W − S † )( R † − T † W ) − . On the one hand, we have that the argument of φ is: Z ( W ′ ) = ( I + iW ′ )( I − iW ′ ) − = (cid:0) ( R † − T † W ) + i ( Q † W − S † ) (cid:1) (cid:0) ( R † − T † W ) − i ( Q † W − S † ) (cid:1) − = (cid:0) ( R † − iS † ) + i ( Q † + iT † ) W (cid:1) (cid:0) ( R † + iS † ) − i ( Q † − iT † ) W (cid:1) − . Taking now into account the map (83) we have: Z ( W ′ ) = (cid:0) ( A † − C † ) + i ( A † + C † ) W (cid:1) (cid:0) ( D † − B † ) − i ( D † + B † ) W (cid:1) − = (cid:0) A † ( I + iW ) − C † ( I − iW ) (cid:1) (cid:0) D † ( I − iW ) − B † ( I + iW ) (cid:1) − = (cid:0) A † Z − C † (cid:1) (cid:0) D † − B † Z (cid:1) − , as desired. On the other hand, we have that( I − iW ′ )( R † − T † W ) = ( R † − T † W ) − i ( Q † W − S † ) = ( R † + iS † ) − i ( Q † − iT † ) W = ( D † − B † ) − i ( D † + B † ) W = D † ( I − iW ) − B † ( I + iW ) = ( D † − B † Z )( I − iW )which impliesdet( I − iW ) λ det( R † − T † W ) − λ det( I − iW ′ ) − λ = det( D † − B † Z ) − λ (cid:4) As a direct consequence of Proposition 5.1, the set of functions defined by˜ ϕ j,mq ,q ( W ) ≡ λ det( I − iW ) − λ ϕ j,mq ,q ( Z ( W )) , (97)with ϕ j,mq ,q defined in (62), constitutes an orthonormal basis of H λ ( T ) and the closurerelation X j ∈ N / ∞ X m =0 j X q,q ′ = − j ˜ ϕ j,mq ′ ,q ( W ) ˜ ϕ j,nq ′ ,q ( W ′ ) = det( i W † − W ′ )) − λ , (98)renders again the reproducing Bergman kernel (92). As we said for the Cartan domain D in (60) and (61), the K¨ahler potential˜ K λ ( W, W † ) ≡ ln ( W | W ) = − ln | Ω λ ( f ) | = − λ ln( ℑ ( w )) = − λ ln y (99)defines T as a K¨ahler manifold with local complex coordinates W = w µ σ µ , w µ = x µ + iy µ ,an Hermitian Riemannian metricg µν ≡ ∂ ˜ K λ ∂w µ ∂ ¯ w ν = − λ y (cid:18) η µν − y µ y ν y (cid:19) . (100)and a corresponding closed two-form ωω = − i g µν dw µ ∧ d ¯ w ν . (101)The line element ds = g µν dw µ d ¯ w ν = − λ y (cid:18) η µν − y µ y ν y (cid:19) ( dx µ dx ν + dy µ dy ν ) (102)turns out to be positive and provides a conformal counterpart of the Born’s line element(2). The two-form (101) defines the Poisson bracket: { a, b } ≡ i g µν (cid:18) ∂a∂w µ ∂b∂ ¯ w ν − ∂b∂w µ ∂a∂ ¯ w ν (cid:19) (103)for the inverse metric g µν = − λ (cid:0) η µν y − y µ y ν (cid:1) . (104)so that g µν g νρ = δ ρµ . In particular, we have that: { x µ , y ν } = −
12 g µν , { x µ , y ν } = η µν ; that is, x µ and y ν are not “canonical” coordinates.However, we can define a proper conjugate four-momentum p µ ≡ λy µ /y which gives thedesired (canonical) Poisson bracket { x µ , p ν } = η µν , (105)as can be checked by direct computation. The line element (102) then becomes: ds = − λ (cid:0) η µν p − p µ p ν (cid:1) ( dx µ dx ν + λ p dp µ dp ν ) . (106)Note the close resemblance between the coordinates, dx µ ( K ν ) = − x µ x ν + x η µν , of thevector field K ν in (5) and the metric coefficients ( − p µ p ν + p η µν ) in (106) under theinterchange x µ ↔ p µ . The line element (106) of the (curved) manifold T is the conformalcounterpart of the Born’s line element (2) in the (flat) complex Minkowski space C , ,both of them considered as phase spaces of relativistic (conformal) particles. Concerningthe extension of BRP to the case of curved spacetimes, see also [54] for the constructiona reciprocal general relativity theory as a local gauge theory of the quaplectic group of[21, 22].Remember that one could deduce the existence of a maximal acceleration from thepositivity of the Born’s line element (3). The existence of a maximal acceleration insidethe conformal group does not seem to be apparent from (106), although there are otherarguments supporting the existence of a bound a max for proper accelerations. One of themwas given time ago in Ref. [55], where the authors analyzed the physical interpretationof the singularities, 1 + 2 ax + a x = 0, of the conformal transformations to a uniformlyaccelerating frame [last transformation in (4)]. When applying the transformation to anextended object of size ℓ , an upper-limit to the proper acceleration, a max ≃ c /ℓ , is shownto be necessary in order that the tenets of special relativity not be violated (see [55] formore details).In a coming paper [27], we shall provide an alternative proof of the existence of amaximal acceleration inside the conformal group. It is related to the Unruh effect (vacuumradiation in uniformly accelerated frames) and turns out to be a consequence of thefiniteness of the radiated energy (black body spectrum). Contrary to other approaches tothe Unruh effect, a bound for the proper acceleration does not necessarily imply a boundfor the temperature. We have revised the use of complex Minkowski 8-dimensional space (more precisely, thedomains D and T ) as a base for the construction of conformal-invariant quantum (field)theory, either as a phase space or a configuration space [the last case related to La-grangians of type (29)]. We have followed a gauge-invariant Lagrangian approach (ofnonlinear sigma-model type) and we have used a generalized Dirac method for the quan-tization of constrained systems, which resembles in some aspects the particular approachto quantizing coadjoint orbits of a group G developed in, for instance, [9].27ne could think of these 8-dimensional domains as the replacement of space-time atshort distances or high momentum transfers, as it is implicit in the original BRP [15, 16],the standard relativity theory being then the limit ℓ min →
0. Group-theoretical revisionsof the BRP, replacing the Poincar´e by the Canonical (or Quaplectic) group of reciprocalrelativity, have been proposed in [21, 22]. In this article we put a (conformal) BRP-likeforward, as a natural symmetry inside the conformal group SO (4 ,
2) and the replacementof space-time by the 8-dimensional conformal domain D or T at short distances. Ac-tually, we feel tempted to establish a connection between holomorphicity ↔ chirality andBRP ↔ CPT symmetry inside the conformal group. Indeed, the definition of P µ and K µ in(10) is linked to the right- and left-handed projectors (1+ γ ) / − γ ) /
2, respectively.According to the (conformal) BRP-like symmetry (7), conformal physics is symmetric un-der the interchange P µ ↔ K µ , as long as we perform a proper-time reversal D → − D .On the other hand, P µ ↔ K µ entails a swapping of chirality (1 + γ ) / ↔ (1 − γ ) /
2, acomplex conjugation ψ λ ( g ) ↔ ˇ ψ λ ( g ) = ψ λ ( g ) (remember the discussion in Remark 4.2)and a parity inversion σ µ ↔ ˇ σ µ = σ µ . Nevertheless, at this stage, a BRP ↔ CPT connec-tion inside the conformal group is just conjectural and it is still premature to draw anyphysical conclusions based on it. It is not either the main objective of this paper.In this article we have considered a particular class of representations (discrete series)of the conformal group, although other possibilities could also be tackled. For example,we could consider the new (vector and pseudo-vector) combinations˜ P µ ≡
12 ( P µ + K µ ) , ˜ K µ ≡
12 ( P µ − K µ ) , with new commutation relations: h ˜ P µ , ˜ K ν i = η µν D, h ˜ P µ , ˜ P ν i = M µν , h ˜ K µ , ˜ K ν i = − M µν . (107)Unlike in formulas (37) and (40), the fact that now h D, ˜ K µ i = − ˜ P µ precludes the impo-sition of D L , M Lµν and ˜ K Lµ as a compatible set of constraints on wave functions. Instead,we could impose M Lµν ψ = 0 , ˜ K Lµ ψ = 0together with the Casimir (8) constraint C L ψ = m ψ , which leads to(( D L ) + ( ˜ P L ) ) ψ = m ψ, This equation could be seen as a generalized
Klein-Gordon equation ( P ψ = m ψ ), with D replacing P as the (proper) time generator and m replacing the Poincar´e-invariantmass m , as a “conformally-invariant mass” (see e.g.[56] for the formulation of otherconformally-invariant massive field equations of motion in generalized Minkowski space).This means that Cauchy hypersurfaces have dimension 4. In other words, the Poincar´etime is a dynamical variable, on an equal footing with position, the usual Poincar´e Hamil-tonian P suffering Heisenberg indeterminacy relations too. Instead of the proper time28dilation) generator D , one could also consider the new combination ˜ P = ( P + K ) / X µ in (39) giving spin generators [48, 49], can be seen as asign of the granularity (non-commutativity) of space-time in conformal-invariant theories,along with the existence of a minimal length or, equivalently, a maximal acceleration.The appearance of a maximal acceleration inside the conformal group will be manifestin analyzing the Unruh effect from a group-theoretical perspective [27]. In a previouspaper [28], vacuum radiation in uniformly accelerated frames was related to a sponta-neous breakdown of the conformal symmetry. In fact, in conformally-invariant quan-tum field theory, one can find degenerated pseudo-vacua (which turn out to be coherentstates of conformal zero-modes) which are stable (invariant) under Poincar´e transforma-tions but are excited under accelerations and lead to a black-body spectrum. The samespontaneous-symmetry-breaking mechanism applies to general U ( N, M )-invariant quan-tum field theories, where an interesting connection between “curvature and statistics”has emerged [59]. We hope this is just one of many interesting physical phenomena thatremain to be unravelled inside conformal-invariant quantum field theory.
Acknowledgements
Work partially supported by the Fundaci´on S´eneca (08814/PI/08), Spanish MICINN(FIS2008-06078-C03-01) and Junta de Andaluc´ıa (FQM219). M.C. thanks the “Uni-versidad Polit´ecnica de Cartagena” and C.A.R.M. for the award “Intensificaci´on de laActividad Investigadora”. We all thank V. Aldaya for stimulating discussions.
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