Fourier Transform, Quantum Mechanics and Quantum Field Theory on the Manifold of General Relativity
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20 April, 2020
Fourier Transform, Quantum Mechanics and Quantum Field Theoryon the Manifold of General Relativity
L.P. HorwitzSchool of Physics, Tel Aviv University, Ramat Aviv 69978, IsraelDepartment of Physics, Bar Ilan University, Ramat Gan 52900, IsraelDepartment of Physics, Ariel University, Ariel 40700, Israelemail: [email protected]
Abstract
A proof is given for the Fourier transform for functions in a quantum mechanicalHilbert space on a non-compact manifold in general relativity. In the (configuration space)Newton-Wigner representation we discuss the spectral decomposition of the canonical op-erators and give a proof of the Parseval-Plancherel relation and the Born rule for linearsuperposiion. We then discuss the representations of pure quantum states and their dualvectors, and construct the Fock space and the associated quantum field theory for Bose-Einstein and Fermi-Dirac statistics.Keywords: Fourier transform, manifold, general relativity, Hilbert space, canonical opera-tors, quantum theory, quantum field theory on curved spacetimePACS: 04.20.Cv, 04.20.Fy, 04.20.-q, 04.20.Ds, 04.62.+v
1. Fourier analysis
In a recent paper[1] discussing the embedding of the relativistic canonical classicaland quantum theory of Stueckelberg, Horwitz and Piron[2][3][4] (see also [5][6]) into gen-eral relativity, the Fourier transform on the manifold, necessary for the construction ofa canonical quantum theory, was introduced without proof. In this paper we provide aproof and clarify the conditions for its validity*. We then discuss the configuration spaceNewton-Wigner representation [1][8] and the spectral decomposition of the canonical op-erators. The Dirac form of the quantum theory leads to the notion of a dual space (whichdoes not coincide with the complex conjugate). We give a proof of the Parseval-Plancherettheorem[9][10] and discuss the Born rule for linear superpositions. With the help of thedefinition of the dual states, we then construct the Fock space and formulate the associatedquantum field theory.Since we are working in a canonical framework[1] we do not make use of eigenfunctionsof the Laplace-Beltrami operator [11]. Although there is a translation group along geodesic* Note that Abraham, Marsden and Ratiu [7] apply the formal Fourier transform on amanifold in three dimensions without proof.1urves generated by the canonical momentum, we shall not be concerned here with thegeneral problem of Fourier analysis on group manifolds.We wish to study the construction of the Fourier transform on a manifold with metric g µν ( x ) ( x ≡ x µ , µ = (0 , , , − , + , + , +)) and (compatible) con-nection form Γ λµν ( x ). We shall assume that the manifold is non-compact and geodesicallycomplete.For a function f ( x ) defined almost everywhere on the manifold { x } , we define theFourier transform[1] ˜ f ( p ) = Z d x √ g e − ip µ x µ f ( x ) , (1)where g = − det g µν and the integral is carried out (in the Riemannian sense) in thelimit of the sum over small spacetime volumes with invariant measure d x √ g . Note that p µ x µ ≡ − p x + p x + p x + p x is not local diffeomorphism invariant, and hence nota scalar product, on the manifold. The Fourier transform as we have defined it is carriedout in the framework of a given, arbitrary, coordinatization.Provided that Z d p e − ip µ ( x µ − x ′ µ ) = (2 π ) δ ( x − x ′ ) , (2)so that (2 π ) − Z d x ′ Z d p e − ip µ ( x µ − x ′ µ ) = 1 , (3)to prove consistency of the definition (1) and the requirement (2), we must have **˜ f ( p ) = 1(2 π ) Z d x Z d p ′ e − i ( p µ − p ′ µ ) x µ ˜ f ( p ′ ) . (4)This condition follows by taking the inverse transform of (1), i.e. , Z d p e ip µ x µ ˜ f ( p ) = (2 π ) p g ( x ) f ( x ) , (5)so that f ( x ) = 1(2 π ) p g ( x ) Z d p e ip µ x µ ˜ f ( p ) . (6)Substituting this result into (1), one obtains the condition (4).Exchanging the order of integrations in (4),assuming convergence in either order, wesee that we must study the function (in a particular coordinatization { x } and cotangentspace { p } ) ∆( p − p ′ ) ≡ π ) Z d x e − i ( p µ − p ′ µ ) x µ (7)which should act as the distribution δ ( p − p ′ ).** It is shown in [1] that, in particular, the statement (4) is covariant under local diffeo-morphisms, i.e. , under a local change of coordinatization.2o prove this consistency condition, following the method of Reed and Simon[12] intheir discussion of Lebesgue integration, we represent the integral as a sum over smallboxes around the set of points { x B } that cover the space (which we have assumed to benon-compact), and eventually take the limit as for a Riemann-Lebesgue integral. In eachsmall box, the coordinatization arises from an invertible transformation from the localtangent space in that neighborhood. We write x µ = x B µ + η µ ∈ boxB (8)where η µ = ∂x µ ∂ξ λ ξ λ (9)and ξ λ (small) is in the flat local tangent space at x B .We now write the integral (7) as∆( p − p ′ ) = 1(2 π ) Σ B Z B d η e − i ( p µ − p ′ µ )( x Bµ + η µ ) = 1(2 π ) Σ B e − i ( p µ − p ′ µ ) x Bµ Z B d η e − i ( p µ − p ′ µ ) η µ . (10)Let us call the measure at B ∆ µ ( B, p − p ′ ) = Z B d η e − i ( p µ − p ′ µ ) η µ . (11)In this neighborhood, define ∂x µ ∂ξ λ = ∂η µ ∂ξ λ ≡ a µλ ( B ) , (12)which may be taken to be a constant matrix in each small box. In (11), we then have ∆ µ ( B, p − p ′ ) = det a Z B d ξ e − i ( p µ − p ′ µ ) a µλ ( B ) ξ λ . (13)We now make a change of variables for which ξ ′ µ = a µλ ( B ) ξ λ ; then, since d ξ ′ =det ad ξ , we have ∆ µ ( B, p − p ′ ) = Z B ′ ( B ) d ξ ′ e i ( p µ − p ′ µ ) ξ µ ′ . (14)in each box. We remark that the local coordinate transformation in each box B results,locally, in the metric g λσ ( B ) = η µν ∂ξ µ ∂η λ ∂ξ ν ∂η σ so that det g ( B ) = − (det a ) − , and thereforedet a is well-defined.However, the transformation a µλ ( B ) in the neighborhood of each point B is, in general,different, and therefore the set of transformed boxes may not cover (boundary deficits) thefull domain of spacetime coordinates. It is easy to show, in fact, that the contribution ofthe boundary deficits of this naive partition of the spacetime may diverge.3e may avoid this problem using our assumption of geodesic completeness of themanifold and taking the covering set of boxes along geodesic curves. Parallel transport ofthe tangent space boxes then fills the space in the neighborhood of the geodesic curve weare following, and each infinitesimal box carries an invariant volume (Liouville type flow)transported along a geodesic curve. For succesive boxes along the geodesic curve, sincethe boundaries are determined by parallel transport (rectilinear shift in the succession oflocal tangent spaces), there is no volume deficit between adjacent boxes.We may furthermore translate a geodesic curve to an adjacent geodesic by the mech-anism discussed in [13], so that boxes associated with adjacent geodesics are also relatedby parallel transport. In this way, we may fill the entire (geodesically complete) spacetimevolume.We may then write (10) as∆( p − p ′ ) = 1(2 π ) Σ B e − i ( p µ − p ′ µ ) x Bµ ∆ µ ( B, p − p ′ ) , (15)Our construction has so far been based on elements constructed in the tangent spacein the neighborhood B of each point x B . Relating all points along a geodesic to the corre-sponding local tangent spaces, and putting each patch in correspondence by continuity, wemay consider the set { x B } to be in correspondence with an extended flat space { ξ ( x B ) } ,to obtain*** ∆( p − p ′ ) = 1(2 π ) Σ B e − i ( p µ − p ′ µ ) ξ B µ ∆ µ ( ξ B , p − p ′ ) . (16)In the limit of small spacetime box volume, this approaches a Lebesgue type integral on aflat space ∆( p − p ′ ) = 1(2 π ) Z e − i ( p µ − p ′ µ ) ξ µ dµ ( ξ, p − p ′ ) . (17)If the measure is differentiable, we could write, dµ ( ξ, p − p ′ ) = m ( ξ, p − p ′ ) d ξ. (18)In the small box, say, size ǫ , ∆ µ ( B, p − p ′ ) = Z ǫ/ − ǫ/ dξ dξ dξ dξ e − i ( p µ − p ′ µ ) ξ µ = (2 i ) Π j =3 j =0 sin( p j − p ′ j ) ǫ ( p j − p ′ j ) → ǫ ∼ d ξ, (19)so that for ǫ sufficiently small, m ( ξ, p − p ′ ) = 1, and we have∆( p − p ′ ) = 1(2 π ) Z e − i ( p µ − p ′ µ ) ξ µ d ξ, (20)*** This procedure is somewhat similar to the method followed in the simpler case ofconstant curvature by Georgiev[14] who, however, used eigenvalues of the Laplace-Beltramioperator. 4r ∆( p − p ′ ) = δ ( p − p ′ ) . (21)It is clear that the assertion (19) requires some discussion. For ǫ → p ′ does not become too large. In each of the dimensions, what we want to find areconditions for which, in (19), sinpǫp → ǫ (22)for ǫ →
0, where we have written p for p − p ′ . Since the kernel ∆( p − p ′ ) is to act onelements of a Hilbert space { ˜ f ( p ) } , the support for p ′ → ∞ vanishes, so that p − p ′ isessentially bounded.As a distribution, on smooth functions g ( p ), the left member of (22) acts as G ( ǫ ) ≡ Z ∞−∞ dp sinpǫp g ( p ) . (23)The function G ( ǫ ) is analytic if p n g ( p ) has a Fourier transform for all n , since G (0) isidentically zero, and successive derivatives correspond to the Fourier transforms of p n g ( p )(differentiating under the integral). This implies, as a simple sufficient condition, that the(usual) Fourier transform of g ( p ) is a C ∞ function in the local tangent space { ξ } . In thiscase we can reliably use the first order term in the Taylor expansion, ddǫ G ( ǫ ) | ǫ =0 = Z dp cosǫp g ( p ) | ǫ =0 (24)so that, for ǫ → G ( ǫ ) → ǫ ˜ g (0) , (25)where ˜ g ( ξ ) is the Fourier transform of g ( p ). As a distribution on such functions g ( p ), theassertion (19) then follows.The structure of the proof outlined above emerges due to the factorization possiblein the exponential function. For example, in a simpler case, applying the same methodto the integration of an arbitrary well-behaved function on the manifold, not necessarilycompact, we could write Z d x √ g f ( x ) = Σ B Z B f ( x B + η ) √ g ( x B ) d η, (26)where we again cover the spacetime, assumed geodesically complete, with small boxesrelated by parallel transport.Since for a small interval ξ λ in B , η µ = ∂x µ ∂ξ λ ξ λ = a µλ ξ λ , (27)as above, d η = det ad ξ and we have (in each box B , g = − det g ( B ) = (det a ) − ) Z d x √ g f ( x ) = Σ B Z B f ( x B + a µλ ξ λ ) d ξ. (28)5o lowest order, this is Z d x √ g f ( x ) = Σ B Z B f ( x B ) d ξ, (29)just our usual understanding of the meaning of R d x √ g f ( x ) as a sum over the wholespace with local measure d x √ g .
2. Consequences for the Quantum Theory
The scalar product for the SHPGR Hilbert space [1] is Z d x p g ( x ) ψ ∗ ( x ) χ ( x ) = < ψ | χ > . (30)As pointed out in [1], the operator − i ∂∂x µ is not self adjoint in this scalar product. However,the operator p µ = − i ∂∂x µ − i p g ( x ) ∂∂x µ p g ( x ) (31)is essentially self-adjoint. It was furthermore pointed out that in the representationobtained by replacing all wave functions ψ ( x ) by g ( x ) ψ ( x ), which we call the Foldy-Wouthuysen representation in coordinate space[1][15], the operator (31) becomes just − i ∂∂x µ .To cast our results in the familiar form of the quantum theory, we write the scalarproduct (30) as < ψ | χ > = Z d x < ψ | x >< x | χ >, (32)where < x | χ > = g ( x ) χ ( x ) < x | ψ > = g ( x ) ψ ( x ) , (33)(and < ψ | x > = < x | ψ > ∗ ) consistently with (32). This definition coincides with theFoldy-Wouthuysen representation as defined in ref.[1]. We now wish to show that theParseval-Plancherel relation[9][10] holds for the momentum representation for the integral(32).As in the definition of Fourier transforms given in ref.[1], we define * < x | p > = 1(2 π ) g ( x ) e ip µ x µ (34)and < p | x > = g ( x ) e − ip µ x µ , (35)* We have changed the signs in the exponents appearing in ref.[1] to conform with theusual convention. 6hich also follows from considering the ket | p > as a limiting case of a sharply definedfunction ˜ f ( p ) in (6) (but in Foldy-Wouthuysen representation). With (35) we have Z d p < x | p >< p | x ′ > = δ ( x − x ′ ) . (36)It then follows from (35) that < p | χ > = Z d x < p | x >< x | χ > = Z d x g ( x ) e − ip µ x µ g ( x ) χ ( x )= Z d x e − ip µ x µ p g ( x ) χ ( x ) = ˜ χ ( p ) . (37) . Moreover, from (34), < ψ | p > = Z d x < ψ | x >< x | p > = Z d x g ( x ) ψ ∗ ( x ) 12 π g ( x ) e ip µ x µ = Z d x (2 π ) e ip µ x µ ψ ∗ ( x ) = ˜ ψ ∗ ( p ) . (38)Note that this is the complex conjugate of < p | ψ > only in the flat space limit, reflectingthe structure of (34) and (35). This function, however, serves as the dual of the function < p | ψ > for the construction of the scalar product contracting, for example, with < p | ψ > to give the squared norm.From (37) and (38), we have Z d p < ψ | p >< p | χ > = Z d p Z d x (2 π ) × e ip µ x µ ψ ∗ ( x ) Z d x ′ e − ip µ x ′ µ p g ( x ′ ) χ ( x ′ )= Z d x p g ( x ) ψ ∗ ( x ) χ ( x ) . (39)This completes our explicit proof of the Parseval relation Z d x p g ( x ) | ψ ( x ) | = Z d p < ψ | p >< p | ψ > . (40)Note that < ψ | p >< p | ψ > is not necessarily a positive number; only the integral assurespositivity and unitarity of the Fourier transform, since in this representation, < ψ | p > isnot the complex conjugate of < p | ψ > . 7s pointed out above, the operator p µ = − i ∂∂x µ is essentially self-adjoint in the Foldy-Wouthuysen representation. We now examine its spectrum. We use the notation { X } and { P } to distinguish the canonical operators from the numerical parameters.Since, by definition, we should have < x | P µ | ψ > = − i ∂∂x µ < x | ψ > (41)we have, by completeness of the spectral family of X , P µ | ψ > = Z d x | x > (cid:0) − i ∂∂x µ (cid:1) < x | ψ >, (42)giving P in operator form in the x − representation. In p − representation, we have* Z d x < p | x > P λ < x | p ′ > = Z d x e − ip µ x µ (cid:0) g ( x ) P λ g ( x ) − (cid:1) e ip ′ µ x µ = p λ δ ( p − p ′ ) , (43)where we recognize the central factor in parentheses as the Foldy-Wouthuysen form of themomentum operator.Finally,in the same way, for the canonical coordinate, we should have < p | X µ | ψ > = i ∂∂p µ < p | ψ > . (44)Then, in the x representation ( X λ commutes with g ( x )), Z d p < x | p > X λ < p | x ′ > = Z d p π ) e ip µ x µ X λ e − ip µ x ′ µ = x λ δ ( x − x ′ ) . (45)We now turn to linear superpositions, which have the same form as in the flat spacetheory. Orthogonal sets (on the measure d x p g ( x )) can be generated using the scalarproduct (1) for the Schmidt orthogonalization process (or the method of Murray[16] usingminimal distance, here defined by choice of a in k ψ − aχ k ) to define the orthonormalproperty < φ n | φ m > = δ mn . (46)Then, for any linear superposition ψ = Σ a n φ n (47)* In (43), < x | p ′ > corresponds to a wave function which we might call ψ p ′ ( x ) and < p | x > to the dual function ψ † p ( x ); in (45), < p | x ′ > corresponds to what we might call ψ x ′ ( p ) and < x | p > to the dual function ψ † x ( p ).8e have, as usual, a n = < φ n | ψ >, (48)and, if k ψ k = 1 , Σ | a n | = 1 , (49)and | a n | is the probability (Born)to find the system in the state φ n .
3. Quantum Field Theory
To define a quantum field theory on the curved space, we shall construct a Fockspace for the many body theory in terms of the direct product of single particle statesin momentum space [4], and define creation and annihilation operators [17]. The Fouriertransform of these operators is then used to construct the quantum fields. We have seen inthe previous section that for the state ψ ( x ) of a one particle system, the complex conjugateof the state (we suppress the tilde in following) in momentum space ψ ( p ) = Z d x e − ip µ x µ p g ( x ) ψ ( x ) (50)is not equal to the dual < ψ | p >ψ † ( p ) = 1(2 π ) Z d x e ip µ x µ ψ ∗ ( x ) . (51)Here, the dagger is used to indicate the vector dual to ψ ( x ), necessary, as in Eq. (39), toform the scalar product. In this form, (39) can be written as Z d pψ † ( p ) ψ ( p ) = Z d x p g ( x ) ψ ∗ ( x ) ψ ( x ) . (52)In this sense, < p | x > (of (35)) is the dual of the (generalized) momentum state < x | p > in the Foldy-Wouthuysen representation. The operator representations (43) and (45) aretherefore bilinears in the states and their duals, and, as shown below, correspond to thesecond quantized form of the operators, as in the usual form of “second quantization”.Note that the linear functional L ( ψ ) of the Riesz theorem[18] that reaches a maximum fora given ψ is given uniquely by the scalar product (52), L ( ψ ) = R d pψ † ( p ) ψ ( p ).The many-body Fock space is constructed [17][19] by representing the N -body wavefunction for identical particles on the basis of states of the form, here suitably symmetrizedfor Bose-Einstein or Fermi-Dirac statistics at equal τ ,* In the folowing we work out the* Other approaches to quantum field theory on the manifold of general relativity, suchas in [20][21][22][23], introduce a timelike foliation of space time to describe the fields andtheir evolution. There is no necessity for us to do this since we have available the universalinvariant parameter τ . The usual specification of a spacelike surface on which a completeset of local observables {O τ ( x ) } commute is then correlated to this τ .9ermi-Dirac case explicitly;the Bose-Einstein formulation is similar. We define, for theFermi-Dirac case,Ψ N,τ ( p N , p N − , . . . p ) = 1 N ! Σ( − P P ψ N ⊗ ψ N − ⊗ . . . ⊗ ψ )( p N , p N − . . . p ) , (53)where all states in the direct product are at equal τ (with e.g. Ψ = ( ψ ⊗ ψ − ψ ⊗ ψ )( p , p ) = ( ψ ( p ) ψ ( p ) − ψ ( p ) ψ ( p ))). We work initially in momentum space,since in this representation the structure is most similar in form to the usual construction.The Fock space consists of span of the set of the form (53), for every N = (0 , , . . . ∞ ),where N = 0 is the vacuum state. We now define the creation operator a † ( ψ N +1 on thisspace with the property that*Ψ N +1 ( p N +1 .p N , . . . p ) = a † ( ψ N +1 )Ψ N )( p N , p N +1 , . . . p ) , (54)which carries out as well the appropriate antisymmetrization. In order to define the anni-hilation operator we take the scalar product of this state with an N + 1 particle stateΦ N +1 ,τ ( p N +1 , p N , . . . p ) = 1( N + 1)! Σ( − P P φ N +1 ⊗ φ N ⊗ . . . ⊗ ψ )( p N +1 , p N . . . p ) , (55)for which (Φ N +1 , a † ( ψ N +1 )Ψ N ) = ( a ( ψ N +1 )Φ N +1 , Ψ N ) (56)where a ( ψ N +1 ), the Hermtian conjugate of a † ( ψ N +1 ) in the Fock space, is an annihilationoperator that removes the particle in the state ψ N +1 . This scalar product is defined onthe momentum space by (52) term by term, using the dual vectors ψ † , as in (52), thusdefining the adjoint on the Fock space. For example, for N = 2,Ψ = 12! ( ψ ⊗ ψ − ψ ⊗ ψ ) . (57)Then, a † ( ψ )Ψ = 13! ( ψ ⊗ ψ ⊗ ψ + ψ ⊗ ψ ⊗ ψ + ψ ⊗ ψ ⊗ ψ − ψ ⊗ ψ ⊗ ψ − ψ ⊗ ψ ⊗ ψ − ψ ⊗ ψ ⊗ ψ ) . (58)We then take the scalar product withΦ = 13! Σ P ( − P P φ ⊗ φ ⊗ φ , (59)with conjugate states in the dual space, for which, by the Parseval result,( φ, ψ ) = Z d p φ † ( p ) ψ ( p ) = Z d x p g ( x ) φ ∗ ( x ) ψ ( x ) . (60)* Here the dagger indicates the Hermitian conjugate in the Fock space scalar product, asin Eq. (61). We use this notation because, as in (62), its Fock space Hermitian conjugatecarries the dual vector to the scalar product (62).10t then follows, by carrying out the scalar product and selecting terms proportional to thetwo-body states Ψ( φ i , φ j ), that the action of the operator a ( ψ ) on the state Φ( φ , φ , φ )is given by a ( ψ )Φ( φ , φ .φ ) = ( ψ , φ )Φ ( φ , φ ) − ( ψ , φ )Φ ( φ , φ ) + ( ψ , φ )Φ ( φ , φ ) , (61) i.e. , the annihilation operator acts like a derivation with alternating signs due to itsfermionic nature.This calculation has a direct extension to the N -body case. For bosons, the proceduremay be carried out in a similar way.Applying these operators to the N and N + 1 particle states, one finds directly thecommutation relations [ a ( ψ ) , a † ( φ )] ∓ = ( ψ, φ ) , (62)so that for orthonormal states (with scalar product (60)),[ a ( φ n ) , a † ( φ m )] ∓ = δ nm . (63)Based on the Dirac form (34) (in Foldy-Wouthuysen representation), consider the distorted“plane wave” ˆ φ p ( x ) = 1(2 π ) g ( x ) e ip µ x µ (64)and its dual ˆ φ † p ( x ) = g ( x ) e − ip µ x µ (65)so that ( ˆ φ p , ˆ φ p ′ ) = δ ( p − p ′ ) (66)and [ a ( ˆ φ p ) , a † ( ˆ φ p ′ )] ∓ = δ ( p − p ′ ) . (67)We may call these operators, as is usually done, a ( p ) , a † ( p ′ ). Then, the Fourier transformwith kernel (34) to transform a ( p ) and (35) to transform the dual operator a † ( p ), we findfor the corresponding quantum fields ψ ( x ) = 1(2 π ) g ( x ) Z d p e ip µ x µ a ( p ) ψ † ( x ) = g ( x ) Z d p e − ip µ x µ a † ( p ) . (68)We have used here the same symbol as for the wave functions and their dual (with thefactor g ( x ) for the Foldy-Wouthuysen representation) to maintain a close analogy to theusual form of second quantization (as in (43), (45) and the associated footnote) for therepresentation of operators on the Fock space. With the commutation relations (62) andthe result (2) proven above, it is easy to see that these fields satisfy (at equal τ )[ ψ ( x ) , ψ † ( x ′ )] ∓ = δ ( x − x ′ ) , (69)11s for the commutation-anti-commutation relations of the usual quantum field theory onMinkowski space.
4. Conclusions
We have constructed a proof of the Fourier transform used in ref.[1], valid for anynon-compact geodesically complete manifold. This proof is valid, for example, for theexterior region of the Schwarzschild solution since there is an infinite redshift at the sin-gularity, and in the interior region as well, since the geodesics approach the singularityonly asymptotically. We have, furthermore, extended the discussion of ref.[1] to provethe Parseval-Plancherel theorem, assuring the equality of the norm in both coordinateand momentum representations using the Foldy-Wouthuysen configuration representationdiscussed there, for convenience, and for the sake of the correspondence of the resultingformulas with those of the usual quantum theory. Following our formulation of the Diracform of the quantum theory, we identify the functions corresponding to the state of thesystem and their dual vectors permitting us to construct the Fock space and the quantumfields for Bose-Einstein and Fermi-Dirac particles on the curved space. Although the theoryis not manifestly diffeomorphism covariant (due to the structure of the Fourier transform),it is indeed invariant in form under arbitrary coordinate transformations as well as in anyarbitrary coordinatization of the manifold.We treat the structure of the theory with representations of particles with spin in asucceeding paper.
Acknowledgements
I wish to thank Moshe Chaichian, Asher Yahalom, and Gil Elgressy for fruitful dis-cussions.
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