An Elementary Canonical Classical and Quantum Dynamics for General Relativity
aa r X i v : . [ phy s i c s . g e n - ph ] J un
29 May, 2019
An Elementary Canonical Classical and Quantum DynamicsforGeneral 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 consistent (off-shell) canonical classical and quantum dynamics in the framework ofspecial relativity was formulated by Stueckelberg in 1941, and generalized to many- bodytheory by Horwitz and Piron in 1973 (SHP). In this paper, this theory is embedded intothe framework of general relativity (GR), here denoted by SHPGR. The canonical Poissonbrackets of the SHP theory remain valid (invariant under local coordinate transformations)on the manifold of GR, and provide the basis for formulating a canonical quantum theory.A scalar product is defined for constructing the Hilbert space and a Hermitian momentumoperator defined. The Fourier transform is defined, connecting momentum and coordinaterepresentations. The potential which may occur in the SHP theory emerges as a spacetimescalar mass distribution in GR, and electromagnetism corresponds to a gauge field onthe quantum mechanical SHPGR Hilbert space in both the single particle and many-body theory. A diffeomorphism covariant form of Newton’s law is found as an immediateconsequence of the canonical formulation of SHPGR. We compute the classical evolutionof the off shell mass on the orbit of a particle and the force on a particle and its energy atthe Schwarzschild horizon. The propagator for evolution of the one body quantum stateis studied and a scattering theory on the manifold is worked out.
Keywords
Relativistic dynamics, General relativity, Quantum theory on curved space,dynamics at the Schwarzschild horizon, scattering theory in general relativity, U (1) gauge,many-body theory in general relativity. Classification PACS
Running Title: Embedding of SHP Theory into General RelativityIntroduction
The relativistic canonical Hamiltonian dynamics of Stueckelberg, Horwitz and Piron(SHP)[1] with scalar potential and gauge field interactions for single and many body the-ory can, by local coordinate transformation, be embedded into the framework of generalrelativity (GR). Some of the results of this embedding are studied in this paper.1he theory was originally formulated for a single particle by Stueckelberg in 1941[2][3][4]. Stueckelberg envisaged the motion of a particle along a world line in spacetimethat can curve and turn to flow backward in time, resulting in the phenomenon of pairannihilation in classical dynamics. The world line was then described by an invariantmonotonic parameter τ . The theory was generalized by Horwitz and Piron in 1973 [5] (seealso [1][6][7])to be applicable to many body systems by assuming that the parameter τ is universal (as for Newtonian time [8][9]), enabling them to solve the two body problemclassically, and later, a quantum solution was found by Arshanksy and Horwitz [10][11][12],both for bound states and scattering theory.Performing a coordinate transformation to general coordinates, along with the corre-sponding transformation of the momenta (the cotangent space of the original Minkowskimanifold), we obtain, in this paper, the SHP theory in a curved space of general coordi-nates and momenta with a canonical Hamilton-Lagrange (symplectic) structure. We shallrefer to this generalization as SHPGR. We study here the classical dynamics of a particlein the neighborhood of the Schwarzschild radius and obtain the force on a particle and itsenergy in this neighborhood.The invariance of the Poisson bracket under local coordinate transformations pro-vides a basis for the canonical quantization of the theory, for which the evolution under τ is determined by the covariant form of the Stueckelberg-Schr¨odinger equation (see also[13][14][15]). The one particle propagator is worked out, and a scattering theory developed(see also [16][17][18] for a discussion of scattering theory in the framework of general rel-ativity where scattering theory is used to reach GR). The formulation is also generalizedhere to a U (1) Abelian gauge theory (electromagnetism on the manifold), but can be ex-tended to the non-Abelian case. This provides a fundamental derivation of the frameworkassumed by Horwitz, Gershon and Schiffer [19][20] in their discussion* of the Bekenstein-Sanders fields [21] introduced into the TeVeS theory of Bekenstein and Milgrom [22][23][24],a geometrical way of obtaining the MOND theory introduced by Milgrom [25][26][27] toexplain the rotation curves of galaxies. The potential term, entering into the structure ofthe scalar factor introduced by Bekenstein and Milgrom [22][23][24], as pointed out in [19],may provide a representation of “dark energy” as well as a phenomenological descriptionof stars or galaxies in collision.Birrell and Davies [28] have discussed fields on curved spacetime, and considerableprogress has been made, as discussed by Poisson [29], in the formulation of Hamiltoniandynamics of such dynamical fields using Lagrangian functionals associated with the cur-vature of spacetime. The approach used in this paper is fundamentally different in that itstudies a canonical dynamics (both Hamiltonian and Lagrangian) of particles on a curvedspacetime.This method is applied also to the many body case, for which the SHP Hamiltonianis a sum of terms quadratic in four momentum with a many body potential term. Eachparticle is assumed locally to move in a flat Minkowski space, the tangent space of thegeneral manifold of motions at that point; these local motions can then be mapped ateach point x µ by coordinate transformation into the curvilinear coordinates reflecting the* A non-Abelian gauge was discussed there, and then an Abelian limit was taken, leavinga term that could cancel caustic singularities.2urvature induced by the Einstein equations.Throughout most of our development, we assume a τ independent background grav-itational field; the local coordinate transformations from the flat Minkowski space to thecurved space are taken to be independent of τ , consistently with an energy momentumtensor that is τ independent. In a more dynamical setting, when the energy momentumtensor depends on τ , the spacetime evolves nontrivially; the transformations from the lo-cal Minkowski coordinates to the curved space coordinates then depend on τ . We discussthis situtation in an Appendix; many of the results for the τ independent case remain(such as the Poisson bracket relations), but some relations, such as the orbit equations,are modified.
1. Single particle in external potential
We write the SHP Hamiltonian [1] as K = 12 M η µν π µ π ν + V ( ξ ) (1 . η µν is the flat Minkowski metric ( − + ++) and π µ , ξ µ are the spacetime canoni-cal momenta and coordinates in the local tangent space, following Einstein’s use of theequivalence principle.The existence of a potential term (which may be a Lorentz scalar), representing non-gravitational forces, implies that the “free fall” condition is replaced by a local dynamicscarried along by the free falling system (an additional force acting on the particle withinthe “elevator” according to the coordinates in the tangent space).The canonical equations are˙ ξ µ = ∂K∂π µ ˙ π µ = − ∂K∂ξ µ = − ∂V∂ξ µ , (1 . ddτ , with τ the invariant universal “world time”. Since then˙ ξ µ = 1 M η µν π ν , or π ν = η νµ M ˙ ξ µ , (1 . K = M η µν ˙ ξ µ ˙ ξ ν + V ( ξ ) . (1 . . ξ = dtdτ has a sign opposite to π which lies in the cotangent space of the manifold, as we shall see in the Poissonbracket relations below. The energy of the particle for a normal timelike particle shouldbe positive (negative energy would correspond to an antiparticle [2][3][4][5][6][7]). The physical momenta and energy therefore correspond to the mapping π µ = η µν π µ , (1 . . ξ µ = (1 /M ) π µ . This simpleobservation will be important in the discussion below of dynamics of a particle in theframework of general relativity, for which the metric tensor is non-trivial, which we willdiscuss below.We now transform the local coordinates (contravariantly) according to the diffeomor-phism dξ µ = ∂ξ µ ∂x λ dx λ (1 . ξ to corresponding small changes in the coordinates x on thecurved space, so that ˙ ξ µ = ∂ξ µ ∂x λ ˙ x λ . (1 . K = M g µν ˙ x µ ˙ x ν + V ( x ) , (1 . V ( x ) is the potential at the point ξ corresponding to the point x (actually thefunction V ( ξ ) could be labelled V x ( ξ ), a function of ξ in a small neighborhood of the point x ), and g µν = η λσ ∂ξ λ ∂x µ ∂ξ σ ∂x ν (1 . V has significance as the source of a force in the local frame only through its deriva-tives, we can make this pointwise correspondence with a globally defined scalar function V ( x ).* We shall assume in most of the work of this paper that the geometric structure doesnot depend on τ , and is concerned with the study of the covariant dynamical evolution ofa system in a background gravitational field. We study the case of a τ dependent metricin the Appendix.The corresponding Lagrangian is then L = M g µν ˙ x µ ˙ x ν − V ( x ) , (1 . x µ , the symplectic structureof Hamiltonian mechanics ( e.g. da Silva [30] ) implies that the momentum ** π µ , ly-ing in the cotangent space of the manifold { ξ µ } , transforms covariantly under the localtransformation (1 . i.e. , as does ∂∂ξ µ , so that we may define p µ = ∂ξ λ ∂x µ π λ . (1 . V ( x ) has dimension of mass, one can think of this function as a scalar massfield, reflecting forces acting in the local tangent space at each point. It may play the roleof “dark energy” [19][20]. If V = 0, our discussion reduces to that of the usual generalrelativity, but with a well-defined canonical momentum variable.** We shall call the quantity π µ in the cotangent space a canonical momentum , althoughit must be understood that its map back to the tangent space π µ corresponds to the actualphysically measureable momentum. 4his definition is consistent with the transformation properties of the momentum definedby the Lagrangian (1 . p µ = ∂L ( x, ˙ x ) ∂ ˙ x µ , (1 . p µ = M g µν ˙ x ν . (1 . ξ µ and π µ .The second factor in the definition (1 .
9) of g µν in (1 .
13) acts on ˙ x ν ; with (1 .
7) we thenhave (as in (1 . p µ = M η λσ ∂ξ λ ∂x µ ˙ ξ σ = ∂ξ λ ∂x µ π λ . (1 . . physical energy andmomenta are given, according to the mapping, p µ = g µν p ν = M ˙ x ν (1 . .
3) and (1 . .
15) that˙ p µ = M ¨ x µ . (1 . p µ , which should be interpreted as the force acting on the particle, isproportional to the acceleration along the orbit of motion (a covariant derivative plus agradient of the potential), described by the geodesic type relation we discuss below. ThisNewtonian type relation in the general manifold reduces in the limit of a flat Minkowskispace to the corresponding SHP dynamics, and in the nonrelativistic limit, to the classicalNewton law. We remark that this result does not require taking a post Newtonian limit,the usual method of obtaining Newton’s law from GR.We now discuss the geodesic equation obtained by studying the condition¨ ξ µ = − M ˙ π µ = − M η µν ∂V ( ξ ) ∂ξ ν . (1 . ξ µ = ddτ (cid:0) ∂ξ µ ∂x λ ˙ x λ (cid:1) = ∂ ξ µ ∂x λ ∂x γ ˙ x γ ˙ x λ + ∂ξ µ ∂x λ ¨ x λ = − M η µν ∂x λ ∂ξ ν ∂V ( x ) ∂x λ (1 . V ( x ) = V ( x ( τ )), so that, after multiplying by ∂x σ ∂ξ µ and summing over µ , we obtain¨ x σ = − ∂x σ ∂ξ µ ∂ ξ µ ∂x λ ∂x γ ˙ x γ ˙ x λ − M η µν ∂x λ ∂ξ ν ∂x σ ∂ξ µ ∂V ( x ) ∂x λ . (1 . .
9) and the usual definition of the connectionΓ σλγ = ∂x σ ∂ξ µ ∂ ξ µ ∂x λ ∂x γ (1 . x σ = − Γ σ λγ ˙ x γ ˙ x λ − M g σλ ∂V ( x ) ∂x λ , (1 . V ( ξ ) is mapped, under this coordi-nate transformation into a force resulting in a modification of the acceleration along thegeodesic-like curves, i.e. , (1 .
16) now reads˙ p µ = M ¨ x ν = − M Γ σλγ ˙ x γ ˙ x λ − g σλ ∂V ( x ) ∂x λ (1 . { x, p } . In the local coordinates { ξ, π } , the τ derivativeof a function F ( ξ, π ) is dF ( ξ, π ) dτ = ∂F ( ξ, π ) ∂ξ µ ˙ ξ µ + ∂F ( ξ, π ) ∂π ν ˙ π µ = ∂F ( ξ, π ) ∂ξ µ ∂K∂π µ − ∂F ( ξ, π ) ∂π µ ∂K∂ξ ν ≡ [ F, K ] P B ( ξ, π ) . (1 . ξ µ locally a function of x ( τ ) only. If the spacetime evolves ( τ depen-dent energy momentum tensor), then it is an explicit function of τ as well, i.e. , ξ ( x ( τ ) , τ ).We show in the Appendix how ∂ξ ( x ( τ ) ,τ ) ∂τ is related to ∂g µν ( x ( τ ) ,τ ) ∂τ . We also recall here that,in the embedding, V ( x ) is assumed to be a world scalar function [20][21].6f we replace in this formula ∂∂ξ µ = ∂x λ ∂ξ µ ∂∂x λ ∂∂π µ = ∂ξ µ ∂x λ ∂∂p λ , (1 . dF ( ξ, π ) dτ = ∂F∂x µ ∂K∂p µ − ∂F∂p µ ∂K∂x ν ≡ [ F, K ] P B ( x, p ) (1 . ξ µ , π ν relation,[ x µ , p ν ] P B ( x, p ) = δ µν . (1 . x µ with the (physical energy-momentum) tangent space variable p µ has then the tensor form [ x µ , p ν ] P B ( x, p ) = g µν . (1 . ξ µ , π ν ] P B ( ξ, π ) = η µν . (1 . p µ (we drop the ( x, p ) label on the Poisson brackethenceforth), [ p µ , F ( x )] P B = − ∂F∂x µ , (1 . p µ acts infinitesimally as the generator of translation along the coordinate curvesand [ x µ , F ( p )] P B = ∂F ( p ) ∂p µ , (1 . x µ is the generator of translations in p µ . In the classical case, if F ( p ) is a generalfunction of p µ , we can write at some point x ,[ x µ , F ( p )] P B = g µν ( x ) ∂F ( p ) ∂p ν , (1 . g µν ( x ) cannot be factored out from polynomials, so,as for Dirac’s quantization procedure [31][32][33], some care is required.This structure clearly provides a phase space which could serve as the basis for statis-tical mechanics (which we leave to a later publication), and lends itself to the constructionof a canonical quantum theory on the curved spacetime, as we discuss below.We now turn to a discussion of the dynamics introduced into the curved space by theprocedure outlined above. Although p µ is not the physical energy momentum as we have7mphasized above, p µ has a simpler Poisson bracket relation with x µ , and this variableand its dynamical evolution will useful in further development of the theory.We start by developing the relation between ˙ p µ and the geodesic equations for x µ ,and show that the result agrees with the direct Hamilonian calculation. Although theresult has been discussed above, the alternative approach below has intrinsic geometricalinterest.Recall from (1 .
13) that p µ = M g µλ ˙ x λ , so that ˙ p µ = M (cid:0) ∂g µλ ∂x γ ˙ x γ ˙ x λ + g µσ ¨ x σ (cid:1) . (1 . . x σ = − Γ σ λγ ˙ x γ ˙ x λ − M g σλ ∂V ( x ) ∂x λ , Eq. (1 . p µ = − ∂V ( x ) ∂x µ + M (cid:0) ∂g µλ ∂x γ ˙ x γ ˙ x λ − g µσ Γ σλγ ˙ x γ ˙ x λ (cid:1) . (1 . σλγ = 12 g ση (cid:0) ∂g ηλ ∂x γ + ∂g ηγ ∂x λ − ∂g λγ ∂x η (cid:1) (1 . p µ = − ∂V ( x ) ∂x µ + M (cid:0) ∂g µλ ∂x γ ˙ x γ ˙ x λ − (cid:0) ∂g µλ ∂x γ + ∂g µγ ∂x λ − ∂g λγ ∂x µ (cid:1) ˙ x γ ˙ x λ (cid:1) . (1 . M , symmetrized under multiplication by˙ x γ ˙ x λ , cancels the first two terms of the contribution from the connection form with theresult* ˙ p µ = − ∂V ( x ) ∂x µ + M ∂g λγ ∂x µ ˙ x γ ˙ x λ . (1 . τ derivative of the relation p µ = g µν p ν . Using theidentity g σγ ∂g µγ ∂x λ g µβ = − ∂g γβ ∂x λ , (1 . p µ = − g µν ∂V ( x ) ∂x ν − M Γ µγλ ˙ x γ ˙ x λ , = M ¨ x µ , (1 . .
36) and (1 .
38) provide a direct interpretation of the geometrical configu-ration as resulting in a very large force on the particle at the black hole horizon as wouldbe seen in coordinates of this type.**We may also write (1 .
36) in terms of the full connection form by noting that with(1 . ∂g λγ ∂x µ = η αβ (cid:0) ∂ ξ α ∂x λ ∂x µ ∂ξ β ∂x γ + ∂ξ α ∂x λ ∂ ξ β ∂x γ ∂x µ (cid:1) . (1 . x γ ˙ x λ , the two terms combine to give a factor of two. We then return tothe original definition of Γ in (1 .
20) in the form ∂ ξ α ∂x λ ∂x µ = ∂ξ α ∂x σ Γ σλµ , (1 . ∂g λγ ∂x µ ˙ x γ ˙ x λ = 2 η αβ ∂ξ α ∂x σ ∂ξ β ∂x γ Γ σλµ ˙ x γ ˙ x λ = 2 g σγ Γ σλµ ˙ x γ ˙ x λ . (1 . p µ = − ∂V ( x ) ∂x µ + M g σγ Γ σλµ ˙ x γ ˙ x λ . (1 . .
8) and carry out the calculation directly. Since˙ x µ is, in general, a function of x µ , we write the Hamiltonian (using (1 . K = 12 M g αβ p α p β + V ( x ) . (1 . p µ = − M ∂g αβ ∂x µ p α p β − ∂V ( x ) ∂x µ . (1 . x again, we have, with p α = M g αλ ˙ x λ ,˙ p µ = M ∂g αβ ∂x µ g αλ g βγ ˙ x γ ˙ x λ − ∂V ( x ) ∂x µ . (1 . t and x in SR. Therefore the force is an effectseen by the observer, and would emerge, for example, in electromagnetic interaction asin (7 . . .
2. Off Shell Mass Evolution
In this section, we consider the variation of the measured mass of a particle as it movesalong along its orbit in the manifold.In Eq. (1 . V ( ξ ) generates a force through the Hamilton equa-tions. The mass of the particle that is actually measured in the laboratory is definedby m = − η µν π µ π ν (2 . τ derivative is dm dτ = − η µν π µ ˙ π ν = + M ˙ ξ ν ∂V ( ξ ) ∂ξ . (2 . .
2) and the local coordinate transformation that the sameresult is true on the manifold of SHPGR. To see how this follows within the framework ofSHPGR, we start with the transformation of (2 .
1) to the curved spacetime. With (1 . m = − g λσ p λ p σ ≡ − g λσ p λ p σ . (2 . .
36) and (1 .
13) we see that dm dτ = − ∂g λσ ∂x γ ˙ x γ p λ p σ − g λσ ˙ p λ p σ = − ∂g λσ ∂x γ M g γµ p µ p λ p σ − g λσ p σ [ − ∂V∂x λ + M ∂g σκ ∂x λ M g σα p α g κβ p β ] . (2 . ∂g σκ ∂x λ g σα g κβ = − ∂g αβ ∂x λ , (2 . . dm dτ = 2 g λσ p σ ∂V∂x λ (2 . dm dτ = 2 M ˙ x λ ∂V∂x λ , (2 . dmdτ = Mm ˙ x λ ∂V∂x λ . (2 . m small, so that M/m is large, a small potential gradientcan have a large effect on the mass variation (recall that this potential may represent the“dark energy” distribution, and may occur implicitly in the MOND formulas[19][20]). Inthe absence of a non-constant potential term, the off-shell mass would be striclty conservedalong the orbit.
3. Dynamics of a Particle Near the Schwarzschild Horizon
It is well-known from the geodesic equation, in standard GR, that a particle nearthe Schwarzschild radius undergoes a very large acceleration, which can be thought of asdue to a large force. With the formula (1 .
38) we can compute directly the force, as the τ derivative of the momentum in the radial direction and, by computing ˙ x , the redshift.We shall also compute the particle energy ( E = p ) in this region. In this study, we take V = 0; it may in this sense be considered as a property of the SHPGR phase space in GR.The equations of motion in this case take on the usual geodesic form.*We consider the symmetric case of the Schwarzschild solution [36] for which only the r, t ( µ = 1 ,
0) components are relevant. Let us call ˙ x µ = v µ , so that the geodesic equationfor the 0 component becomes dv dτ = − Γ λσ v λ v σ = − g (cid:8) ∂g σ ∂x λ + ∂g λ ∂x σ − ∂g σλ ∂x (cid:9) v λ v σ . (3 . g = − (cid:0) − M S Gr (cid:1) = − g , (3 . M S is the black hole mass, we obtain dv dτ = − g ∂g ∂x v v . (3 . ∂g ∂x v = dg dτ , we can write this result as dv dτ = − g dg dτ v or, since ( g = g ) ddτ ( g v ) = 0* The large geodesic acceleration of GR is, by (1 . g v = k = constant in τ. (3 . K = 12 M ( g p + g p ) , (3 . dtdτ = v = ∂K∂p = 1 M g p (3 . g v = p M = k. (3 . g <
0, and v >
0, it follows that k <
0. From (3 . v = dtdτ = − (cid:0) − M s Gr (cid:1) − k > . (3 . p µ is in the cotangent bundle of the manifold) isthe set { x µ , p µ } , we recognize, as pointed out above, that it is p that has the interpretationof the energy of the particle and p i , i = 1 , , .
7) that p is a constant of the motion and is negative).The energy of the particle is then E = p = g p = g M k > . (3 . .
8) that in a finite increment of τ , the corresponding increment of t at the horizon undergoes an infinite redshift (as is generally obtained from the structureof the metric)*.Moreover, at r → ∞ ,we see that k = − v ∞ . It follows from the Hamilton equationsthat, since g → − r → ∞ , v ∞ = ∂K∂p | ∞ = g p M | ∞ = E ∞ M (3 . E = (cid:0) − M s Gr (cid:1) E ∞ (3 . ds = − g µν dx µ dx ν . This quantity is, however, dynamical, as discussed in [1],and does not necessarily reflect the invariant evolution of the system as recorded in termsof an ideal universal laboratory clock ( τ ). The redshift we obtain here (explicitly in (3 . r for a given E , we have r ( E ) = 2 M s G E ∞ E . (3 . p (the generator of translations in t = x ) is constant in τ , the energy of theparticle E = p grows rapidly towards the surface of the black hole. Particle productionat high energies ( E >> E ∞ ) could therefore be induced close to the horizon, as assumedby Hawking[38].We now turn to calculate the force on the particle close to the horizon. It is of interest(and useful) to first calculate the rate of change of the cotangent space variable ˙ p , moresingular than the physical ˙ p .From the general relation (1 . V = 0 or constant,˙ p = M ∂g λγ ∂x ˙ x γ ˙ x λ = M { ∂g ∂x ( ˙ x ) + ∂g ∂x ( ˙ x ) } . (3 . v ) . From (2 . m M = 1 g (( v ) − g ( v ) ) , (3 . v ) = g m M + ( g v ) . (3 . .
8) ( ˙ x ≡ v ) and the result k = − v ∞ , we obtain v = v ∞ (1 − M s Gr ) . (3 . .
2) for the metric, we obtain˙ p = − M M s Gr (cid:8) v ∞ ) (cid:0) − M s Gr (cid:1) − (cid:0) − M s Gr (cid:1) m M (cid:9) . (3 . r = 2 M s G + ǫ , for ǫ << M s G ,1 − M s Gr ∼ = ǫ M s G , (3 . m M , ˙ p ∼ = − M M s Gǫ ( v ∞ ) . (3 . p (and therefore p ) grows rapidly, one might expect that the quan-tum mechanical dispersion in p also becomes large, and therefore that (by the uncertaintyrelation) the particle becomes highly localized in the neighborhood of the horizon (see also[39]).On the other hand, for r → ∞ , it follows from (3 .
17) that˙ p ∼ = − M M s Gr { v ∞ ) − m M } . (3 . v ∞ ) = E ∞ M , (3 .
20 can be written as˙ p ∼ = − M M s Gr { ( v ∞ ) + E ∞ − m M } . (3 . ∞ , we have by definition, E ∞ − m = ( p ∞ ) ≥
0, and small, and( v ∞ ) ∼ = 1, we see that ˙ p is close to the Newtonian force.We now study the behavior of the physically observable momentum at the horizon.Since p = g p , we can relate ˙ p to our previous result for ˙ p ,˙ p = dg dτ p + g ˙ p . (3 . g = ( g ) − , and ˙ r ≡ ˙ x ≡ v , we find directly that*˙ p = − M s Gr (cid:0) − M s Gr (cid:1) v p + 1 (cid:0) − M s Gr (cid:1) ˙ p . (3 . v p = M ( v ) , we see that˙ p = − M s Gr (cid:0) − M s Gr (cid:1) ( v ) + 1 (cid:0) − M s Gr (cid:1) ˙ p (3 . p and using (3 .
15) for ( v ) , we obtain˙ p = − M M s Gr (cid:0) − M s Gr (cid:1) ( v ∞ ) + 3 M M s Gr (cid:0) − M s Gr (cid:1) m M (3 . r → ∞ , one finds ˙ p = − M M s Gr ( v ∞ ) + 3 M M s Gr m M . ˙ p (3 . . p and ˙ p . 14urthermore, for ( v ∞ ) = ( E ∞ M ) = p ∞ + m M , we obtain˙ p = − M M s Gr (cid:0) m M (cid:1) − M M s Gr p ∞ M . (3 . p , this result contains small differences from the standard Newtonian force.For r → M s G + ǫ , the general result (3 .
25) yields˙ p ≈ − M ( M s G ) ǫ ( v ∞ ) + 3 m M s GM ǫ , (3 . p . *We now turn to a discussion of the many body problem.
4. The many body system with interaction potential
The many body Hamiltonian of the SHP theory is K = Σ Ni =1 M i η µν π µi π νi + V ( ξ , ξ , . . . ξ N ) , (4 . V ( ξ , ξ , . . . ξ N ) is a function of the locally flat coordinates in theneighborhood of each of the particles at { x i } . Although this function is Lorentz scalar,Poincar´e invariance is, in general, inapplicable (even in the two-body case), unless all ofthe particles are in a sufficiently small neighborhood to be able to neglect the effects ofcurvature.The Hamilton equations are (in the tangent space in the neighborhood of each particleat the point x i ) ˙ ξ µi = ∂K∂π µi ˙ π µi = − ∂K∂ξ µi = − ∂V∂ξ µi , (4 . ξ µi = 1 M i η µν π νi , or π νi = η νµ M i ˙ ξ µi , (4 . K = Σ Ni =1 M i η µν ˙ ξ µi ˙ ξ νi + V ( ξ , ξ , . . . ξ N ) , (4 . x µi of the i th particle, since in this neighborhood, ξ µi is a function locallyof x µi , we can then make a local coordinate transformation dξ σi = ∂ξ σi ∂x µi dx µi . (4 . g µν ( x i ) = η σλ ∂ξ σi ∂x µi ∂ξ λi ∂x ν i , (4 . K = Σ Ni =1 M i g µν ( x i ) ˙ x µi ˙ x νi + V ( x , x , . . . x N ) , (4 . L = Σ Ni =1 M i g µν ( x i ) ˙ x µi ˙ x νi − V ( x , x , . . . x N ) . (4 . .
6) is valid for each of the particle coordinates,˙ ξ µi = ∂ξ µi ∂x λ i ˙ x λi , (4 . ξ µi = ddτ (cid:0) ∂ξ µi ∂x λi ˙ x λi (cid:1) = ∂ ξ µi ∂x λi ∂x γi ˙ x γi ˙ x λi + ∂ξ µi ∂x λi ¨ x λi = − M i η µν ∂x λi ∂ξ νi ∂V ( x , x , . . . x N ) ∂x λi . (4 . x σi = − ∂x σi ∂ξ µi ∂ ξ µi ∂x λi ∂x γ i ˙ x γi ˙ x λi − M i η µν ∂x λi ∂ξ νi ∂x σi ∂ξ µi ∂V ( x , x , . . . x N ) ∂x λi . (4 . V ( x , x , . . . x N ) is a scalar function under localdiffeomorphisms of any of the variables. 16e can consider the Jacobian for the local mapping (4 .
5) as a field , a mapping definedover all { x µ } , in (4 .
5) evaluated at the point x µi where the i th particle is found.We then define a local connection form in the neighborhood of the point x i asΓ σλγ ( x i ) = ∂x σi ∂ξ µi ∂ ξ µi ∂x λi ∂x γi (4 . field evaluated at the point x µi , so thatthe geodesic equations can be written as¨ x σi = − Γ σ λγ ( x i ) ˙ x γi ˙ x λi − M i g σλ ( x i ) ∂V ( x , x , . . . x N ) ∂x λi . (4 . .
20) at each point x i . Since thisconnection form coincides with Einstein’s, the same method can be used to construct aRicci tensor; the resulting Einstein equations will therefore have the same form, althoughthere will necessarily be differences in the structure of the energy momentum tensor.* Theempty space solution [36] is applicable in this framework as well, providing an interestingexample for application [39](see also [40]), and the homogeneous case of Robertson, Fried-man and Walker[41][42][43][44][45][46][47] would have a similar form to the well-knownsolution. Applications of this type will be investigated in succeeding papers.Following the same procedure as for (1 . F ( x , x , . . . x N , p , p , . . . p N ) with variables ξ , ξ , . . . ξ N , π , π , . . . π N in the cotangentbundle assigned to the points x , x , . . . x N , p , p , . . . p N in the general phase space for the N -body system, the Poisson bracket is defined by dF ( ξ , ξ , . . . ξ N , π , π , . . . π N ) dτ = Σ i (cid:0) ∂F ( { ξ, π } ) ∂ξ µi ˙ ξ µi + ∂F ( { ξ, π } ) ∂π νi ˙ π µi (cid:1) = Σ i (cid:0) ∂F ( { ξ, π } ) ∂ξ µi ∂K∂π µi − ∂F ( { ξ, π } ) ∂π µi ∂K∂ξ µi (cid:1) ≡ [ F, K ] P B ( { ξ, π } ) . (4 . x i ), so the Poisson bracket remains in the same form on the 8 N dimensional phasespace { x i , p i } phase space. We therefore have dF ( x , x , . . . x N , p , p , . . . p N ) dτ = Σ i ∂F ( { x, p } ) ∂x µi ∂K∂p µi − ∂F ( { x, p } ) ∂p µi ∂K∂x µi ≡ [ F, K ] P B ( { x, p } ) . (4 . g µν ( x ) wouldreflect the many body structure of the energy momentum tensor through the Einsteinequations. 17n general, for two functions A ( { x, p } ) and B ( { x, p } ), the many body Poisson bracket isthen [ A, B ] P B = Σ i (cid:0) ∂A ( { x, p } ) ∂x µi ∂B ( { x, p } ) ∂p µi − ∂A ( { x, p } ) ∂p µi ∂B ( { x, p } ) ∂x µi (cid:1) . (4 . x , x , . . . x N , p , p , . . . p N are to be considered as kinematically inde-pendent, we obtain the canonical bracket[ x iµ , p jν ] P B = δ ij δ µν (4 . p µi . At the point x iµ , as we have arguedabove, p µi = ∂ξ λi ∂x iµ ( x i ) π λi . (4 . . p µi = − ∂V ( x , x , . . . x N , p ) ∂x µi + M i g σγ ( x i )Γ σλµ ( x i ) ˙ x γi ˙ x λi = − ∂V ( x , x , . . . x N , p ) ∂x µi + M i (cid:0) g σγ ( x i )Γ σλµ ( x i ) + g σλ ( x i )Γ σγµ ( x i ) (cid:1) ˙ x γi ˙ x λi . (4 . . x i , to the other N −
5. Quantum Theory on the Curved Space
The Poisson bracket formulas (1 .
25) and (1 .
26) can be considered as a basis for defininga quantum theory with canonical commutation relations[ x µ , p ν ] = i ¯ hδ µν , (5 . p µ , F ( x )] = − i ¯ h ∂F∂x µ , (5 . x µ , F ( p )] = i ¯ h ∂F ( p ) ∂p µ . (5 . ψ τ ( x ) canbe taken to be (see also Schwinger and DeWitt [48][49][50][51]) i ∂∂τ ψ τ ( x ) = Kψ τ ( x ) , (5 . .
1) also implies that [ x µ , p ν ] = i ¯ hg µν ( x ), but application topolynomials in p µ would introduce factors of g µν ( x ) and would require some care [31][32].18here the operator valued Hamiltonian can be taken to be the Hermitian form, on a Hilbertspace defined with scalar product (with invariant measure; we write g = − det { g µν } ),( ψ, χ ) = Z d x √ gψ ∗ τ ( x ) χ τ ( x ) . (5 . − i ∂∂x µ isnot Hermitian due to the presence of the factor √ g in the integrand of the scalar product.The problem is somewhat analogous to that of Newton and Wigner [52] in their treatmentof the Klein Gordon equation in momentum space. It is easily seen that the operator p µ = − i ∂∂x µ − i p g ( x ) ∂∂x µ p g ( x ) (5 . . . p µ is Hermitian in the scalar product (5 . K = 12 M p µ g µν p ν + V ( x ) , (5 . . { x } is not a trivial transcription ofthe Euclidean condition on the SHP quantum theory [1]. If we think of the integral (5 . .
5) must be considered as a total volumesum with invariant measure on the whole space, consistent with the notion of Lesbesguemeasure and the idea that the norm is the sum of probability measures on every subsetcontained. The procedure for carrying out such integrals would, of course, depend on thegeometrical structure of the manifold.This construction can be carried over to the many body case directly, i.e , with theoperator properties of the coordinates and momenta[ x µi , p νi ] = i ¯ hδ ij δ µν , (5 . p µi , F ( { x } )] = − i ¯ h ∂F ( { x } ) ∂x µi , (5 . x µi , F ( { p ) } ] = i ¯ h ∂F ( { p } ) ∂p µi . (5 . . { g µν , p ν } , withcommutation relations similar of the form (1 . d x goes over to the localdiffeomorphism invariant d x √ g )( ψ, χ ) = Z Π i (cid:8) d ( x i ) p g ( x i ) (cid:9) ψ ∗ τ ( x , x , . . . x N ) χ τ ( x , x , . . . x N ) . (5 . .
6) for each p µi at x µi ) K = Σ i M i p µi g µν ( x i ) p νi + V ( x , x , . . . x N ) (5 .
6. Fourier Transform, Potential Scattering Theory and the Propagator
In the context of quantum field theory and gravitons, Bjerrum-Bohr et al [16] havediscussed scattering theory to arrive at aspects of classical general relativity, providinginteresting motivation for a scattering theory in general relativity. In this section, wedevelop a potential scattering theory in the framework of the quantum theory we havedescribed in the previous section. In case the potential V is zero (or constant), we alsodiscuss how the “free” particle propagator is affected by the curvature of the manifold.To deal with this problem, we discuss first the formulation of the Fourier transform f ( x ) → ˜ f ( p ) for a scalar function f ( x ) (we shall use x µ and the canonically conjugate p µ in this discussion). Let us define ( g ≡ − det g µν )˜ f ( p ) = Z d x p g ( x ) e ip µ x µ f ( x ) . (6 . Z e − ip µ x µ ˜ f ( p ) d p = Z d pe − ip µ ( x µ − x ′ µ ) f ( x ′ ) p g ( x ′ ) d x ′ = (2 π ) f ( x ) p g ( x ) (6 . f ( x ) = 1(2 π ) p g ( x ) Z e − ip µ x µ ˜ f ( p ) d p. (6 . f ( x ) = f ′ ( x ′ ) , ˜ f ( p ) → ˜ f ′ ( p ) . The Fourier transform of f ′ ( x ′ ) we define as˜ f ′ ( p ) = Z d x ′ p g ( x ′ ) e ip µ x ′ µ f ′ ( x ′ ) , (6 . f ′ ( p ) = Z d x p g ( x ) e ip µ x µ f ′ ( x ) , (6 . f ( x ) = < x | f >, (6 . f ( p ) = < p | f > . (6 . < x | p > = 1(2 π ) p g ( x ) e − ip µ x µ < p | x > = p g ( x ) e ip µ x µ , (6 . e.g. , the usual action of transformation functions Z < x | p >< p | f > d p = < x | f >, (6 . Z < x | p >< p | x ′ > d p = 1(2 π ) p g ( x ) Z d pe − ip µ x µ e ip µ x ′ µ p g ( x ′ )= δ ( x − x ′ ) . (6 . < x | p > and < p | x > are not simple complexconjugates of each other, but require nontrivial factors of p g ( x ) and its inverse to satisfythe necessary transformation laws on the manifold. Conversely, (the factors p g ( x ) and itsinverse cancel) Z < p ′ | x >< x | p > d x = δ ( p ′ − p ) . (6 . p g ( x ) in the integrations over d x is analogous to the Newton-Wignerdiscussion [52] in momentum space, where d p/p is the Lorentz invariant measure forthe Klein-Gordon scalar product; in our case, we are concerned with local diffeomorphisminvarance. We consider, in analogy to the Newton-Wigner construction, the transformationfrom elements of the original Hilbert space, say ψ ( x ), to a new representation, which weshall call the Newton-Wigner representation, ψ NW ( x ) = ( g ( x )) ψ ( x ) . (6 . Z d x | ψ NW ( x ) | = Z d x p g ( x ) | ψ ( x ) | (6 . O acting on ψ ( x ) as O ψ ( x ) = O ( g ( x )) − ψ ′ ( x ) (6 . O should be replaced by O ′ = ( g ( x )) O ( g ( x )) − (6 . g ( x )) (cid:0) − i ∂∂x µ − i p g ( x ) ∂∂x µ p g ( x ) (cid:1) ( g ( x )) − = − i ∂∂x µ , (6 . p µ can be clearly demonstrated in theNewton-Wigner representation (6 . ψ ( p ) ≡ < p | ψ NW > = Z d x p g ( x ) e ip µ x µ ( g ( x )) − ψ NW ( x )= Z d xe ip µ x µ ( g ( x )) ψ NW ( x ) . (6 . ψ NW ( x ) ≡ < x | ψ NW > = 1(2 π ) ( g ( x )) Z d pe − p µ x µ ˜ ψ ( p ) . (6 . < p | x > NW = g ( x ) e ip µ x µ (6 . < x | p > NW = 1(2 π ) g ( x ) e − ip µ x µ (6 . Z < p ′ | x > NW < x | p > NW d x = δ ( p − p ′ ) , (6 . Z < x | p > NW < p | x ′ > NW d p = δ ( x − x ′ ) . (6 . p λ defined in (5 .
6) tomomentum representation, one obtains, for ( p λ ) op the expression (5 . . Z d x < p | x > NW ( p λ ) op < x | p ′ > NW = 1(2 π ) Z d xg ( x ) ( p λ ) op g ( x ) e ip µ − p ′ µ x µ = 1(2 π ) Z d x (cid:0) − i ∂∂x λ (cid:1) e i ( p µ − p ′ µ ) x µ = p λ δ ( p − p ′ ) , . (6 . .
6) becomes translation (alongthe coordinate curves).In the following, we maintain the explicitly covariant form of the theory.We may now formulate the potential scattering problem in interaction picture. Letus write for (4 . K = K + V (6 . V = V ( x , x ) (a scalar function for diffeo-morphisms at x and x for all x and x ) and (for the self-ajoint p µ in x representation) K = 12 M p µ g µν ( x ) p ν + 12 M p µ g µν ( x ) p ν . (6 . g µν may depend on τ . In an adiabaticsense, we shall assume here that there is no explicit τ dependence in the metric.If we then write for the two body wave function ψ τ ( x , x ) = e − iK τ χ τ ( x , x ) , (6 . χ τ becomes, as for the usual interaction picture, i ∂χ τ ( x , x ) ∂τ = V ( x , x , τ ) χ τ ( x , x ) , (6 . V ( x , x , τ ) = e iK τ V ( x , x ) e − iK τ . (6 . τ → −∞ , we shall assume that the wave function moves out of the region of spacetimewhere the two body potential is effective**, so that we have a relation that can be studied,as in usual scattering theory, by iteration: χ τ ( x , x ) = χ −∞ ( x , x ) − i Z τ −∞ dτ ′ V ( x , x , τ ′ ) χ τ ′ ( x , x ) . (6 . N -body problem.** In a model in which the potential term V represents “dark energy”[19][20], modulatedby the proximity of, in this case, two massive systems, this limit would correspond to aconfiguration in which the mutual influence of the two systems becomes negligible andeach is in an environment where the dark energy is distributed, as V ⊕ V , for example,in accordance with the requirements of MOND[20][21][22][23][24][25][26][27].23t then follows that χ τ ( x , x ) can be expressed as a τ ordered product χ τ ( x , x ) = (cid:0) e R τ −∞ − iV ( τ ′ ) dτ ′ (cid:1) + χ −∞ ( x , x ) . (6 . U ( τ ) = e − iKτ and U ( τ ) = e − iK τ ,we assume that for a state ψ that evolves to ψ τ by U ( τ ) = e − iKτ that there is an in statefor which k U ( τ ) ψ − U ( τ ) ψ in k → τ → −∞ (6 . k U ( τ ) ψ − U ( τ ) ψ out k → τ → + ∞ , (6 . ψ in and ψ out define the wave operators Ω+ and Ω − respectivelyas ψ = U † ( τ ) U (0) ψ out → Ω − ψ out τ → + ∞ = U † ( τ ) U (0) ψ out → Ω + ψ in τ → −∞ . (6 . e.g. , Ω + , is that k V U ψ in k vanishes sufficiently rapidly (there are weaker conditions that may apply) so that it isintegrable from −∞ to zero in τ . This conditon must be investigated for any particularmodel, but we shall assume here that it is satisfied.It follows that Ω − ψ out = Ω + ψ in , (6 . ψ out = Ω −† Ω + ψ in ≡ Sψ in , (6 . S -matrix.Now, for χ τ = U ( τ ) † ψ τ = U ( τ ) † U ( τ ) ψ, (6 . χ −∞ = Ω + † ψχ + ∞ = Ω −† ψ, (6 . ψ = Ω + χ −∞ = Ω − χ + ∞ , (6 . χ + ∞ = Sχ −∞ . (6 . .
30) provides, for τ → ∞ , a formula for the S matrix, as in standard scatteringtheory.We now study the first few terms of the iteration of (6 . χ τ ( x , x ) = χ in − i Z τ −∞ dτ ′ V ( τ ′ ) χ in ( x , x )+ ( − i ) Z τ −∞ dτ ′ Z τ ′ −∞ dτ ′′ V ( τ ′ ) V ( τ ′′ ) χ in ( x , x )+ ( − i ) Z τ −∞ dτ ′ Z τ ′ −∞ dτ ′′ Z τ ′′ −∞ dτ ′′′ V ( τ ′ ) V ( τ ′′ ) V ( τ ′′′ ) χ in ( x , x ) + · · · (6 . p , we see that we haveterms of the type < p p | V ( τ ) | p ′ p ′ > = < p p | e iK τ V e − iK τ | p ′ p ′ > = Z dp ′′ dp ′′ dp ′′′ dp ′′′ < p p | e iK τ | p ′′ p ′′ > × < p ′′ p ′′ | V | p ′′′ p ′′′ >< p ′′′ p ′′′ | e − iK τ | p ′ p ′ > . (6 . x representation in intermediate states (through the < x | p > transforma-tion functions) is not useful here since V is not necessarily Poincar´e invariant. Moreover, K is not diagonalized in the p representation, so that the τ integrations cannot be carriedout leading to Feynman propagators in the usual way [54][55][56]. However, we proceed asfollows.The two terms of K , K = K + K = 12 M p µ g µν ( x ) p ν + 12 M p µ g µν ( x ) p ν (6 . < p p | e iK τ | p ′ p ′ > = < p | e iK τ | p ′ >< p | e iK τ | p ′ > (6 . K in momentum eigenstates < p | K | p ′ > = Z d x p g ( x ) 12 M p µ p µ ′ < p | x > g µν ( x ) < x | p ′ > = 12 M p µ p µ ′ Z d x p g ( x ) e ip κ x κ g µν ( x ) 1(2 π ) p g ( x ) e − ip ′ σ x σ = 12 M p µ p µ ′ ˜ g µν ( p − p ′ ) , , (6 . g µν ( p ) is the Fourier transform of g µν ( x ).In coordinate representation, each of the K i is a (essentially self-adjoint) LaplaceBeltrami operator (see, for example, S. Helgason[57]) and has, in general, continuous spec-trum in ( −∞ , ∞ ). Since the two K i commute, the operator K defined in (6 .
42) is a directsum on the product Hilbert space. Suppose its generalized eigenfunctions are {| λ , λ > } ,so that K | λ , λ > = ( λ + λ ) | λ , λ > . (6 . | λ λ > ≡ | λ (2) > . (6 . .
40) and introduce the complete set {| λ (2) > } as intermediate25tates: < p p | χ τ > = < p p | χ in > − i Z τ −∞ dτ ′ < p p | V ( τ ′ ) χ in > + ( − i ) Z τ −∞ dτ ′ Z τ ′ −∞ dτ ′′ Z dλ (2) < p p | V ( τ ′ ) | λ (2) >< λ (2) | V ( τ ′′ ) χ in > + ( − i ) Z τ −∞ dτ ′ Z τ ′ −∞ dτ ′′ Z τ ′′ −∞ dτ ′′′ Z dλ (2) dλ (2) ′ < p p | V ( τ ′ ) | λ (2) >< λ (2) | V ( τ ′′ ) | λ (2) ′ >< λ (2) ′ | V ( τ ′′′ ) χ in > + · · · (6 . < p p | λ (2) ′′ > for the first factor to make explicit the matrix element < λ (2) ′′ | V ( τ ′ ) | λ (2) > and in the lastfactor < λ (2) ′′′ | p ′ p ′ > to obtain the matrix element of V ( τ ′′′ ) in the λ (2) representation,for which < λ (2) ′′ | V ( τ ′ ) | λ (2) > = e i ( λ ′′ + λ ′′ − λ − λ ) τ ′ < λ (2) ′′ | V | λ (2) >, (6 . τ integrations to obtain( − Z dλ (2) dλ (2) ′ dλ (2) ′′ dλ (2) ′′′ d p ′ d p ′ < p p | λ (2) >< λ (2) | V | λ (2) ′ >< λ (2) ′ | V | λ (2) ′′ >< λ (2) ′′ | V | λ (2) ′′′ > ( λ + λ − λ ′′′ − λ ′′′ − iǫ )( λ ′ + λ ′ − λ ′′′ − λ ′′′ − iǫ )( λ ′′ + λ ′′ − λ ′′′ − λ ′′′ − iǫ ) × < λ (2) ′′′ | p ′ p ′ >< p ′ p ′ | χ in > . (6 . e ǫτ for convergence as τ → −∞ in the last factor. This procedureis based in the flat space limit on k V e − iK τ ψ k vanishing as τ → −∞ ; if our asymptoticcondition is in flat space, the potential term would have the same features as in the SHPtheory [1], for which this condition can hold. The τ integrations, starting from the last,carry over a factor of e − iǫτ to each successive integration, providing the − iǫ terms inthe denominators, as in the usual scattering theory. The same general structure, withalternating signs, obtains to every order.The structure of the intermediate propagators is similar to the usual (two-body) Feyn-man free propagators, but the evaluation of the vertices < λ (2) | V | λ (2) ′ > involves thetransformation functions < λ (2) | x , x > , known from solutions for the Laplace-Beltramispectral problem. The λ ’s play the same role here as the energy eigenvalues (continu-ous spectrum) for the unperturbed Hamiltonian in the nonrelativistic scattering theory,but in this case they correspond to the spectrum of the Laplace-Beltrami operators thatconstitute the Hamiltonian for evolution on the manifold.It is interesting to compare this result with the emergence of the propagator fromthe Green’s function for a single particle (or, in a simple generalization, to many particle)26ropagation. To see this, let us compute the evolution of a “free” one body state (here K has just one term of (6 . ψ τ ( p ) = Z d p ′ < p | e − iK τ | p ′ > ψ ( p ′ ) (6 . , ∞ ) for Im s > ψ s = i Z d p ′ < p | s − K | p ′ > ψ ( p ′ ) , (6 . G ( s ) = 1 s − K . (6 . η µν = diag ( − , + , + , +)) g µν ( x ) = η µν + h µν ( x ) , (6 . p µ p µ = − m , i.e. , G ( s ) ∼ = 1 s − p µ p µ M − M p µ h µν ( x ) p ν (6 .
7. Electromagnetism
As C.N. Yang [58] wrote, electromagnetism can be thought of as a U (1) fiber bundle.The electromagnetic potential vector field emerges as a section on the fiber bundle inthe gauge transformations of the quantum theory. To illustrate this idea, consider whathappens to Eq. (5 .
4) if we consider, instead of the function ψ τ ( x ), the function ψ ′ τ ( x ) = e i Λ( x,τ ) ψ τ ( x ) resulting from a unitary transformation e i Λ( x,τ ) defined locally (for Λ( x, τ ) ascalar function) on the Hilbert space at each value of τ . Since p µ acts like a derivative on x µ , it differentiates Λ( x, τ ), just as for the corresponding computation in the flat Minkowskispace. As for the flat space case, we must add a gauge compensation term so that( p µ − a ′ µ ( x, τ )) e i Λ( x,τ ) ψ τ ( x ) = e i Λ( x,τ ) ( p µ − a µ ( x, τ )) ψ τ ( x ) , (7 . { λ } representation, for free evolution, this formula would provide the samedenominator as occurs in (6 . .e. , assuring that ( p µ − a µ ( x, τ )) ψ τ ( x ) is an element of the Hilbert space, transformedlocally at every point in the same way, and therefore undergoes the same unitary transfor-mation as ψ τ ( x ). Carrying out the derivative implied by the action of p µ (as in (5 . a ′ µ ( x, τ ) = a µ ( x, τ ) + ∂ Λ( x, τ ) ∂x µ , (7 . a µ ( x, τ ).Unless we restrict ourselves to the so-called “Hamilton gauge” (with Λ independentof τ ), the form of (7 .
4) implies the existence of a fifth field [1][59][60] a ( x, τ ), for whichwe must have (cid:8) i ∂∂τ + a ′ ( x, τ ) (cid:9) ψ ′ τ ( x ) = e i Λ( x,τ ) (cid:8) i ∂∂τ + a ( x, τ ) (cid:9) ψ τ ( x ) (7 . a ′ ( x, τ ) = a ( x, τ ) + ∂∂τ Λ( x, τ ) . (7 . i ∂∂τ ψ τ ( x ) = (cid:8) M ( p µ − a µ ( x, τ )) g µν ( p ν − a ν ( x, τ )) − a ( x, τ )( x ) (cid:9) ψ τ ( x ) , (7 . τ dependent a ( x, τ ).In the usual way, we can define in the flat tangent space, a gauge invariant fieldstrength[1][59][60] ˜ f αβ ( ξ, τ ) = ∂ α a β ( ξ, τ ) − ∂ β a α ( ξ, τ ) , (7 . α, β = (0 , , , , ∂ α ˜ f αβ ( ξ, τ ) = j β ( ξ, τ ) . (7 . τ , and the fifth component is the density ρ ( ξ, τ ) ∝ ψ ∗ τ ( ξ ) ψ τ ( ξ )in the SHP theory (see [1] for details).It is easy to see that a coordinate transformation leads to the rule of replacement ofderivatives by covariant derivatives so that in the curved space f µν ( x, τ ) = a µ ; ν − a ν ; µ = ∂ξ σ ∂x ν ∂ξ λ ∂x µ ˜ f λσ f µ = ∂ µ a − ∂ a µ , (7 . a is a Lorentz scalar. 28or the fourth and fifth components, we have f ; µµν ( x, τ ) + ∂ f ν ( x, τ ) = j ν ( x, τ ); f µ µ ( x, τ ) = j ( x, τ ) = ρ ( x, τ ) , (7 . ∇ · E = ρ . Clearly, the covariantdivergence of j ν ( x, τ ) vanishes.We now study the structure of the corresponding current. To do this, we write anaction for which the variation with respect to ψ ∗ τ ( x ) yields the Stueckelberg-Schr¨odingerequation (7 . S = Z dτ d x √ g (cid:8) iψ ∗ τ ( x ) ∂∂τ ψ τ ( x ) − iψ τ ( x ) ∂∂τ ψ ∗ τ ( x ) + a ( x, τ ) ψ ∗ τ ( x ) ψ τ ( x ) − (cid:8) ψ ∗ τ ( x ) (cid:8) M ( p µ − a µ ( x, τ )) g µν ( p ν − a ν ( x, τ )) − a ( x, τ ) (cid:9) ψ τ ( x ) (7 . f µν ( x, τ ) = g µλ g νσ f λσ ( x, τ ). We add to the action a purely electromagnetic part S em = + 14 √ g (cid:0) f µν f µν + f µ f µ (cid:1) , (7 . a is scalar its covariant derivative is an ordinary derivative) f µ = ∂ µ a − ∂ τ a µ (7 . a = g α a α depending on the metric for the embedding of O (4 ,
1) or O (3 ,
2) chosenfor the 5 D manifold. As for the nonrelativistic theory on 3 D , where the gauge fields makeaccessible the (3 ,
1) manifold of Minkowski space, the gauge fields of the (3 + 1) D theorymake accessible the embedding of the (4 ,
1) or (3 ,
2) manifold. As we shall see, however,in our discussion of the many body problem, the assumption of universality in τ does notadmit such a higher symmetry.In 1995, Land, Shnerb and Horwitz [60] studied the consequences of assuming covari-ant commutation relations between x µ and ˙ x ν on a manifold using a theorem of Hojmanand Sheply [62] extending and generalizing the work of Tanimura [63]. Their results, in-cluding the development of the 5D theory, agree in the one particle sector with what wehave presented here.To fully treat such a development with the methods we have used here, one wouldhave to start with a one degree higher dimensional Stueckelberg equation; its gauge fieldswould open the possibility of a 6 D manifold as a result of gauge invariance. We shall, how-ever, truncate this sequence here at the level of 4 D , retaining τ as the universal invariantparameter of evolution.We now obtain the current by variation of a µ in the action. Integrating by parts inthe kinetic term (for the self-adjoint p µ = − i ∂∂x µ − i √ g ( x ) ∂∂x µ p g ( x )), we have S kin = − M Z dτ d x p g ( x )(( p µ − a µ ) ψ ) ∗ g µν ( p ν − a ν ) ψ = + 12 M Z dτ d x p g ( x )(( p µ + a µ ) ψ ∗ ) g µν ( p ν − a ν ) ψ, (7 . δS kin δa µ = p g ( x )2 M (cid:0) ψ ∗ g µν ( p ν − a ν ) ψ − (( p ν + a ν ) ψ ∗ ) g µν ψ (cid:1) . (7 . ψ ∗ ψ ( x, τ ) is the probability to find the particle (event) in the invariant volumeelement √ gd x , so that ψ ∗ ψ ( x, τ ) must go over to √ g ( x ) δ ( x − x ′ ) in the classical limit(see Weinberg [37]). Therefore, we must define the current as* j µ ( x, τ ) = 1 p g ( x ) 12 M (cid:0) ψ ∗ g µν ( p ν − a ν ) ψ − (( p ν + a ν ) ψ ∗ ) g µν ψ (cid:1) , (7 . g → p µ vanishes). Note further that the integral of thecurrent over a hypersurface with the invariant measure d x √ g has well-defined physicalmeaning.We now study the variation of the action with respect to a . The variation of the fullaction (both S m and S em ) with respect to a then yields the field equation f µ µ ( x, τ ) ≡ p g ( x ) ρ ( x, τ ) = ψ τ ∗ ψ τ ( x, τ ) , (7 . ∇ · E = ρ .Furthermore, the variation with respect to a µ , with the definition of the current (7 . a µ in (7 . , (7 .
11) leads to f µν ; µ ( x, τ ) = j ν ( x, τ ) , (7 . ∂ µ ( √ gf µν ) = j ν (7 . F µ = f µν dx ν dτ (7 . In the following, we generalize this structure to the many body problem.* Note that if we follow the method of Jackson[61], defining the macroscopic current J µ ( x ) = 1 p g ( x ) Z dτ ˙ x µ δ ( x − x ( τ )) , then ∂ µ p g ( x ) J µ ) = 0. 30he many-body wave function can be written as the span of the direct product of wavefunctions associated with isomorphic one particle Hilbert spaces (which also may be usedin the construction of the Fock space on the manifold). The norm and orthogonality followfrom the properties of the one particle spaces as above (with the rule that correspondingelements of the sequences are contracted by scalar product). We may therefore write ψ τ ( x , x , . . . x N ) = Σ a α ,α ...α N φ α ,τ ( x ) φ α ,τ ( x ) · · · φ α N ,τ ( x N ) . (8 . e i Λ( x,τ ) should act, with the same function Λ( x, τ ) in each of the factor spaces.* This construction provides a convenientmechanism for the gauge transformations of the Bose-Einstein or Fermi-Dirac Fock spaces(and, in general, for linear combinations). Furthermore, as we shall see below, it enablesus to define a field a µ ( x, τ ). We therefore define the gauge transformation ψ → ψ ′ as ψ ′ τ ( x , x , . . . x N ) = Σ a α ,α ...α N e i (Λ( x ,τ )+Λ( x ,τ )+ ··· Λ( x N ,τ )) × φ α ,τ ( x ) φ α ,τ ( x ) · · · , φ α N ,τ ( x N )= e i (Λ( x ,τ )+Λ( x ,τ )+ ··· Λ( x N ,τ )) ψ τ ( x , x , . . . , x N ) (8 . x i ofthe wave function associated with the i th factor, we have (cid:0) − i ∂∂x iµ − a ′ µ ( x i , τ ) (cid:1) ψ ′ τ ( x , x , . . . x N ) = e i (Λ( x ,τ )+Λ( x ,τ )+ ··· Λ( x N ,τ )) (cid:0) − i ∂∂x iµ − a µ ( x i , τ ) (cid:1) ψ τ ( x , x , . . . x N ) , (8 . a ′ µ ( x i , τ ) = a µ ( x i , τ ) + ∂ Λ( x i , τ ) ∂x iµ , (8 . field for the electromag-netic potential vector.If we call the compensation function for the τ evolution a ( x , x , . . . x N , τ ) andΛ( x , x , . . . x N , τ ) = Λ( x , τ ) + Λ( x , τ ) + · · · Λ( x N , τ ) (8 . a ′ ( x , x , . . . x N , τ ) = a ( x , x , . . . x N , τ ) + ∂∂τ Λ( x , x , . . . x N , τ ) . (8 . x , x , . . . x N )).* One can think of this procedure as the action of an operator (cid:0) e i Λ (cid:1) N = (cid:0) e i Λ N (cid:1) actingon the N particle state. 31he field strengths associated with the fifth field form a set f iµ ( x , x , . . . x N ) = ∂ µi a ( x , x , . . . x N ) − ∂ τ a µ ( x i ) . (8 . ∂ µi ( a ( x , x , . . . x N , τ ) + ∂∂τ Λ( x , x , . . . x N , τ )) − ∂ τ ( a µ ( x i , τ ) + ∂∂x iµ Λ( x i τ ))= ∂ µi a ( x , x , . . . x N , τ ) − ∂ τ a µ ( x i , τ ) , (8 . ∂∂x iµ selects the term in the sum that cancels ∂∂x iµ Λ( x i τ ).To be able to write the elements of this set in a uniform way in the arguments, wedefine a field on x , x , . . . x N for each τ such that the projection a iµ ( x , x , . . . x N , τ ) = a µ ( x i , τ ) . (8 . f iµ ( x , x , . . . x N ) = ∂ µi a ( x , x , . . . x N ) − ∂ τ a iµ ( x , x , . . . x N ) . (8 . N body case.We write the action for an N particle system in the presence of electromagnetism as S = Z dτ Π i d x i p g ( x i ) (cid:8) iψ ∗ τ ( x , x , . . . x N ) ∂∂τ ψ τ ( x , x , . . . x N ) − iψ τ ( x , x , . . . x N ) ∂∂τ ψ ∗ τ ( x , x , . . . x N )+ a ( x , x , . . . x N , τ ) ψ ∗ τ ( x , x , . . . x N ) ψ τ ( x , x , . . . x N ) − ψ ∗ τ ( x , x , . . . x N )Σ i (cid:8) ( p µ − a µ ( x i , τ )) g µν ( x i )( p ν − a ν ( x i , τ )) − a ( x , x , . . . x N , τ ) (cid:9) ψ τ ( x , x , . . . x N ) (cid:9) (8 . S em = + Z dτ Σ i Z d x i √ g (cid:0) f µν ( x i , τ ) f µν ( x i , τ ) (cid:9) + f iµ ( x , x , . . . x N ) f iµ ( x , x , . . . x N ) (cid:1) . (8 . a (taking into account the factor g µν in raising the index)yields the equation of motion f µ µ ( x , x , . . . x N , τ ) = ψ τ ∗ ( x , x , . . . x N ) ψ τ ( x , x , . . . x N )= Π i g ( x i ) − ρ ( x , x , . . . x N , τ ) , (8 . x , x , . . . x N ) at each τ .32inally, the variation with respect to a µ ( x i , τ ) of the interaction term, since f µν ( x, τ )is a one particle quantity, the field equations f µν ; ν ( x i , τ ) = j µ ( x i , τ ) , (8 . a µ ( x i , τ ) fixes τ but not the coordinates except for x i ) j µ ( x i , τ ) = Π j = i d x j q g ( x j )12 M i p g ( x i ) (cid:0) ψ τ ∗ ( x , x , . . . x N ) g µν ( x i )( p νi − ia ν ( x i )) ψ τ ( x , x , . . . x N ) − (( p νi + a ν ( x i )) ψ τ ∗ ( x , x , . . . x N )) g µν ( x i ) ψ τ ( x , x , . . . x N ) (cid:1) . (8 . f µν ; µ ( x i , τ ) = j ν ( x i , τ ) (8 . ∂ µi ( p g ( x i ) f µν ( x i , τ )) = p g ( x i ) j ν ( x i , τ ) . (8 . F µ ( x i , τ ) = f µν ( x i , τ ) dx iν dτ , (8 .
9. Summary and Outlook
We have shown that the SHP theory can be embedded by local coordinate transfor-mations into the framework of general relativity. The Minkowski spacetime coordinates ofthe SHP theory are considered to lie in the tangent space of a manifold with metric andconnection form derived from the coordinate transformations on the equations of motionfor particles moving on the locally flat Minkowski spacetime, parametrized by a universalmonotonic world time τ . The four momentum is well-defined on the manifold, and a for-mula for its τ derivative, which may be understood as a “force”, is obtained, displayingthe effect of the potential as well as the curvature (through the connection form). Thecanonical momentum vector p µ in the cotangent space has simple canonical Poisson brack-ets with the coordinates x µ , but we show that it is the mapping p µ = g µν ( x ) p ν back to thetangent space which corresponds to the measured energy and momentum, and computethe energy and momentum of a particle near the Schwarzschild radius (horizon) of a blackhole.For the many body system, each particle, at the points { x iµ } , is assumed to movelocally on a flat Minkowski space, which is then transformed by local coordinate transfor-mation to the manifold of GR with coordinates x iµ . Since particles with (flat Minkowski)33oordinates ξ , ξ . . . ξ N lie in different local tangent spaces at the points x , x . . . x N ofthe curvilinear coordinatization of GR, Poincar´e invariance of the potential function is notapplicable.Since the Poisson bracket of the SHP theory is unchanged in form under local dif-feomorphisms, it forms the basis of a quantum theory in which the momentum operatorgenerates infinitesimal translations along the local coordinates. However, the operator − i ∂∂x µ is not Hermitian on a Hilbert space of functions ψ τ ( x ), square integrable over theinvariant measure d x p g ( x ). It was necessary to define the Hermitian momentum opera-tor p µ = − i ∂∂x µ − i ∂∂x µ p g ( x ), in analogy to the operator defined by Newton and Wigner[52] in momentum space. We showed, in the discussion of Fourier transforms, that thisoperator generates infinitesimal translations.We then developed the basic scattering theory in this quantum mechanical framework.The interaction picture expansion for a potential model is worked out, similar to theFeynman type expansions, but with more complicated vertices due to the curvature ofspace time. We also showed that the propagator for “free” evolution (Green’s function)has a Laplace-Beltrami operator in the denominator which, for small curvature, reducesto the flat space SHP Hamiltonian with the addition of an effective mass shift due to thecurvature.This Hilbert space provides a basis for a local U (1) gauge, for which the compensationfields (sections on the bundle) correspond to classical 5 D electromagnetic fields [59][60].We obtain field equations for the electromagnetic fields and associated currents from anaction ( τ integrated Lagrangian).The many body quantum theory is treated by constructing a tensor product spaceand the associated electromagnetic theory is developed assuming that each factor in thetensor product carries the same gauge transformations. This enables us to define a gaugecompensation field a µ ( x ) for the four components which can be evaluated on each particle,but due to the universality of τ , the fifth component must depend on the coordinates ofall of the particles as a locally defined function on the full configuration space, similar tothe function V ( x , x , . . . x N ) of the potenial model.The work of this paper is primarily restricted to describing a relativistic dynamics ina τ independent gravitational field, i.e. , the metric is assumed independent of τ . Since theconnection has the same structure as in GR, one can write Einstein’s equations in the sameform ( e.g. [64]). Therefore, in this case,the energy momentum tensor, determining g µν ( x ),should be independent of τ . To achieve this, one may use partially integrated currents,taking into account correlations [65], or the zero modes extracted from full integrationyielding 4 D conserved currents [1][59][60]. In the more general case, where the structure ofspacetime is dynamical (for example, star formation, collision between stars or black holes,or for unstable stars such as supernova) the energy momentum tensor would depend on τ . We show in the Appendix how, in such cases, the corresponding explicit dependence ofthe local transformations from the Minkowski space coordinates to the curved coordinatescan be expressed in terms of such a τ dependent metric tensor.The classical results of this paper provide an eight dimensional phase space for generalrelativity, just as the SHP theory provides for special relativity, and therefore a generalrelativistic statistical mechanics can be formulated. The assumptions necessary to con-34truct Gibbs ensembles [66][67] in this context must be carefully examined; we leave thissubject for a future publication.The many body problem, generalizing the work of Horwitz and Arshansky [68], canbe formulated in this framework. The many body Hilbert space can be used to constructa Fock space as the basis for a quantum field theory.We finally remark that the results of the work of this paper provide a basis for thevector field approach of Bekenstein and Sanders[21] and the associated discussion of non-abelian gauge fields given by Horwitz, Gershon and Schiffer [19]; it will therefore be ofinterest to follow the development of the U (1) gauge theory given here with a study ofnon-Abelian gauge theories (see [60]). Appendix
We study here the effect of a τ evolving spacetime, a situation which would occur ifthe energy momentum tensor depends on τ .Generally, at a point x λ , the velocity of a particle is ˙ x λ , just a motion on the coor-dinates { x } . If the spacetime is changing, we think of the tangent space as reflecting thischange. Therefore, the local coordinatization ξ changes as the world coordinates evolve.At each τ , it is still true that dξ µ = ∂ξ µ ∂x λ dx λ , ( A. ∂ξ µ ∂x λ changes as the particle moves and as the spacetime evolves. We can write ∂ξ µ ∂x λ = ∂ξ µ ∂x λ ( x ( τ ) , τ ) ( A. ddτ ∂ξ µ ∂x λ = ∂ ξ µ ∂x λ ∂x σ (( x ( τ ) , τ ) ˙ x σ + ∂∂τ ∂ξ µ ∂x λ ( x ( τ ) , τ ) , ( A. ξ µ coordinates in τ .The canonical structure postulated in (1 .
1) and (1 . g µν remainthe same but, as a function of ∂ξ µ ∂x λ ( x, τ ), it now becomes a function of τ . We thereforehave K = M g µν ( x ( τ ) , τ ) ˙ x µ ˙ x ν + V ( x ) . ( A. A.
1) the total τ derivative¨ ξ µ = ddτ (cid:0) ∂ξ µ ∂x λ ˙ x λ (cid:1) = ∂ ξ µ ∂x λ ∂x γ ˙ x γ ˙ x λ + ∂∂τ ∂ξ µ ∂x λ ˙ x λ + ∂ξ µ ∂x λ ¨ x λ = − M η µν ∂x λ ∂ξ ν ∂V ( x ) ∂x λ , ( A. ∂x σ ∂ξ µ , and solving for ¨ x σ , we find a geodesic type equation as before but withan additional (velocity dependent) term¨ x σ = − Γ σ λγ ˙ x γ ˙ x λ − M g σλ ∂V ( x ) ∂x λ − ∂x σ ∂ξ µ ∂∂τ (cid:0) ∂ξ µ ∂x λ (cid:1) ˙ x λ . ( A. ∂∂τ (cid:0) ∂ξ µ ∂x λ (cid:1) can be expressed in terms of ∂∂τ g µν ( x ( τ ) , τ ), which thencarries the information, from the Einstein equations, about the evolution of the spacetime.From the definition (1 .
9) we compute ∂g µν ∂τ ( x ( τ ) , τ ) = η σγ (cid:2) ∂∂τ (cid:0) ∂ξ σ ∂x µ (cid:1) ∂ξ γ ∂x ν + ∂ξ γ ∂x µ ∂∂τ (cid:0) ∂ξ σ ∂x ν (cid:1)(cid:3) , ( A. σ, γ symmetry of η σγ in the second term.Now, define ∂∂τ (cid:0) ∂ξ µ ∂x λ (cid:1) ≡ t µλ . ( A. A.
7) as ∂g µν ∂τ ( x ( τ ) , τ ) = η σγ (cid:2) t σµ ∂ξ γ ∂x ν + t σν ∂ξ γ ∂x µ (cid:3) = η σγ (cid:0) δ µλ ∂ξ γ ∂x ν + δ ν λ ∂ξ γ ∂x µ (cid:1) t σλ ≡ M µν λσ t σλ . ( A. N λσµν satisfying N λ ′ σ ′ µν M µν λσ = δ λ ′ λ δ σσ ′ . ( A. N λσµν could be singular; this would correspond to asingular development of the transformation function ∂ξ σ ∂x ν in τ , which we do not treat here.Multiplying ( A.
9) by N λ ′ σ ′ µν , we obtain N λ ′ σ ′ µν ∂g µν ∂τ ( x ( τ ) , τ ) = t σ ′ λ ′ = ∂∂τ (cid:0) ∂ξ σ ′ ∂x λ ′ (cid:1) . ( A. ∂ξ γ ∂x µ ; as pointed out by Weinberg [64], these functionsenter quadratically in g µν , and are therefore determined up to Lorentz transformation, atany given τ , by g µν . Acknowledgements
36 wish to thank Asher Yahalom, Yossi Strauss, Igal Aharonovich, Gil Elgressy andMartin Land for helpful discussions.
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