## Abstract

A consistent (off-shell) canonical classical and quantum dynamics in the framework of special relativity was formulated by Stueckelberg in 1941 and generalized to many-body theory by Horwitz and Piron in 1973 (SHP). This theory has been embedded into the framework of general relativity (GR), here denoted by SHPGR. The canonical Poisson brackets 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. The relation between representations based on coordinates and momenta is given by Fourier transform; a proof is given here for this functional relation on a manifold. The potential which may occur in the SHP theory emerges as a spacetime scalar mass distribution in GR. Gauge fields, both Abelian and non-Abelian on the quantum mechanical SHPGR Hilbert space in both the single particle and many-body theory, may be generated by phase transformations. Application to the construction of Bekenstein and Sanders in their solution to the lensing problem in TeVeS is discussed.

### Keywords

- relativistic dynamics
- general relativity
- quantum theory on curved space
- non-Abelian gauge fields
- Bekenstein-Sanders field
- TeVeS

## 1. Introduction

The relativistic canonical Hamiltonian dynamics of Stueckelberg, Horwitz, and Piron (SHP) [1] with scalar potential and gauge field interactions for single- and many-body theories can, by local coordinate transformation, be embedded into the framework of general relativity (GR). This embedding provides a basis for the work of Horwitz et al. [2, 3] in their discussion of the origin of the field introduced by Bekenstein and Sanders [4] to explain gravitational lensing in the TeVeS formulation of modified Newtonian dynamic (MOND) theories [5, 6, 7, 8, 9, 10].

The theory was originally formulated for a single particle by Stueckelberg in [11, 12, 13]. Stueckelberg envisaged the motion of a particle along a world line in spacetime that can curve and turn to flow backward in time, resulting in the phenomenon of pair annihilation in classical dynamics. The world line was then described by an invariant monotonic parameter

Performing a coordinate transformation to general coordinates, along with the corresponding transformation of the momenta (the cotangent space of the original Minkowski manifold), one obtains [20] the SHP theory in a curved space of general coordinates and momenta with a canonical Hamilton-Lagrange (symplectic) structure. We shall refer to this generalization as SHPGR. We discuss the extension of the Abelian gauge theory described in Ref. [20] to the non-Abelian gauge discussed in [2, 3].

The invariance of the Poisson bracket under local coordinate transformations provides a basis for the canonical quantization of the theory, for which the evolution under

In this chapter, we assume a

## 2. Embedding of single particle dynamics with external potential in GR

We write the SHP Hamiltonian [1, 11, 12, 13] as

where

The existence of a potential term (which we assume to be a Lorentz scalar), representing nongravitational forces, implies that the “free fall” condition is replaced by a local dynamics carried along by the free falling system (an additional force acting on the particle within the “elevator” according to the coordinates in the tangent space).

The canonical equations are

where the dot here indicates

the Hamiltonian can then be written as

It is important to note that, as clear from (3), that *opposite* to *physical momenta and energy* therefore correspond to the mapping

back to the tangent space. Thus, equivalently, from (2),

We now transform the local coordinates (contravariantly) according to the diffeomorphism

to attach small changes in

The Hamiltonian then becomes

where

Since ^{1}

The corresponding Lagrangian is then

In the locally flat coordinates in the neighborhood of ^{2}

This definition is consistent with the transformation properties of the momentum defined by the Lagrangian (10):

yielding

The second factor in the definition (9) of

As we have remarked above for the locally flat space in (5), the *physical* energy and momenta are given, according to the mapping,

back to the tangent space of the manifold, which also follows directly from the local coordinate transformation of (3) and (5).

It is therefore evident from (15) that

We see that *acceleration along the orbit of motion* (a covariant derivative plus a gradient of the potential), as described by the geodesic-type relation. This Newtonian-type relation in the general manifold reduces in the limit of a flat Minkowski space to the corresponding SHP dynamics and in the nonrelativistic limit, to the classical Newton 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

To do this, we compute

so that, after multiplying by

Finally, with (9) and the usual definition of the connection

we obtain the modified geodesic-type equation

from which we see that the derivative of the potential

The procedure that we have carried out here provides a canonical dynamical structure for motion in the curvilinear coordinates. We now remark that the Poisson bracket remains valid for the coordinates

If we replace in this formula

we immediately (as assured by the invariance of the Poisson bracket under local coordinate transformations) obtain

In this definition of Poisson bracket, we have, as for the

The Poisson bracket of

In the flat space limit, this relation reduces to the SHP bracket,

Continuing our analysis with

so that *generator of translation* along the coordinate curves and

so that ^{3}

This structure clearly provides a phase space which could serve as the basis for the construction of a canonical quantum theory on the curved spacetime.

We now turn to a discussion of the dynamics introduced into the curved space by the procedure outlined above.

We may also write (22) in terms of the full connection form by noting that with (9),

Multiplying by

so we can write

We therefore have

## 3. Quantum theory on the curved space

The Poisson bracket formulas (25) and (26) can be considered as a basis for defining a quantum theory with canonical commutation relations

so that

and

The transcription of the Stueckelberg-Schrödinger equation for a wave function

where the operator valued Hamiltonian can be taken to be the Hermitian form (42), written below, on a Hilbert space defined with scalar product (with invariant measure; we write

To construct a Hermitian Hamiltonian, we first study the properties of the canonical momentum in coordinate representation. Clearly, in coordinate representation,

is essentially self-adjoint in the scalar product (40), satisfying as well as the commutation relations (36).^{4}

Since

consistent with the local coordinate transformation of (1). The integration (40) must be considered as a total volume sum with invariant measure on the whole space, consistent with the notion of Lebesgue measure and the idea that the norm is the sum of probability measures on every subset contained. We return to this point in our discussion of the Fourier transform below.

## 4. Canonical quantum theory and the Fourier transform

To complete the construction of a canonical quantum theory on the curved space of GR, we discuss first the formulation of the Fourier transform

The inverse is given by

so that

One sees immediately that under diffeomorphisms, for which with the scalar property

By change of integration variables, we have

In Dirac notation,

and we write as well

For

we have, for example, the usual action of transformation functions

where we have used

Note that the transformation functions

The validity of (53) is not obvious on a curved space. We therefore provide a simple, but not trivial, proof of (53). For

we must have

that is, exchanging the order of integrations, on the set

We now represent the integral as a sum over small boxes around the set of points ^{5} In each small box, the coordinatization arises from an invertible transformation from the local tangent space in that neighborhood. We write

where

and

We now write the integral (56) as

Let us call

In this neighborhood, call

which we assume a constant matrix (Lorentz transformation) in each box. In (60), we then have

However, we can make a change of variables; we are left with

in each box.

However, the transformations

We may avoid this problem by assuming geodesic completeness of the manifold and taking the covering set of boxes, constructed of parallel transported edges, along geodesic curves. Parallel transport of the tangent space boxes then fills the space in the neighborhood of the geodesic curve we are following, and each infinitesimal box may carry an invariant volume (Liouville-type flow) transported along a geodesic curve. For successive boxes along the geodesic curve, since the boundaries are determined by parallel transport (rectilinear shift in the succession of local tangent spaces), there is no volume deficit between adjacent boxes.

We may furthermore translate a geodesic curve to an adjacent geodesic by the mechanism discussed in [32], so that boxes associated with adjacent geodesics are also related by parallel transport. In this way, we may fill the entire geodesically accessible spacetime volume.

Let us assign a measure to each point

We may then write (59) as

Our construction has so far been based on elements constructed in the tangent space in the neighborhood of each point ^{6}

In the limit of vanishing spacetime box volume, this approaches the Lebesgue-type integral on a flat space:

If the measure is differentiable, we could write

Since the kernel

so that

or^{7}

It is clear that the assertion (69) requires some discussion. For

for

The function

so that, for

where

## 5. Application to the Bekenstein-Sanders fields

We have discussed the construction of a canonical quantum theory in terms of an embedding of the SHP relativistic classical and quantum theory into general relativity. We show in this section that this systematic embedding provides a framework for the method developed by Bekenstein and Milgrom for understanding the MOND [5, 6, 7, 8, 9, 10] that appeared necessary to explain the galactic rotation curves [35].

The remarkable development of observational equipment and power of computation has resulted in the discovery that Newtonian gravitational physics leads to a prediction for the dynamics of stars in galaxies that is not consistent with observation. It was proposed that there should be a matter present which does not radiate light which would resolve this difficulty, but so far no firm evidence of the existence of such matter has emerged. Milgrom [5, 6, 7, 8, 9, 10] proposed a modification of Newton’s law (MOND) which could resolve the problem. However, since Newton’s law of gravitation emerges in the “post-Newtonian approximation” to the geodesic motion in Einstein’s theory of gravity [35], the modification of Newton’s law must involve a modification of Einstein’s theory.^{8} Such a modification was proposed by Bekenstein and Milgrom [5, 6, 7, 8, 9, 10] in terms of a conformal factor multiplying the usual Einstein metric.

The origin of such a conformal factor can be found in the potential term of the special relativistic SHP theory. The embedding of this theory in GR [20] brings this potential term as a world scalar. The Hamiltonian for the general relativistic case then has the form (8). It was shown by Horwitz et al. [37] that a very sensitive test by geodesic deviation can be formulated by to study stability by transforming a standard nonrelativistic Hamiltonian of the form

to the form

with

that is, a conformal factor on the original metric. Applying the same idea to the Hamiltonian (8), with the

where

with

We see that we can in this way achieve the structure proposed by Bekenstein and Milgrom [5, 6, 7, 8, 9, 10] systematically. Moreover, in addition to providing a mechanism for achieving a realization of the MOND theory, in the original form (8), the world scalar term

The theory proposed by Bekenstein and Milgrom [5, 6, 7, 8, 9, 10] did not, however, account for the lensing of light observed when light passes a galaxy. To solve this problem, Bekenstein and Sanders [4] proposed the introduction of a *vector field*

so that the vector is timelike.

This vector field can then be used to construct a modified meric of the form

With this modification, Bekenstein and Sanders [4] could explain the lensing effect. In the following, we show that this new field may arise from a *non-Abelian* gauge transformation [38, 39] on the quantum theory that we have discussed in Section 3. Although Contaldi et al. [40] point out that a gauge field in this context can have caustic singularities due to the presence of a massive system, Horwitz et al. [2, 3] show that in the limit in which the gauge field approaches the Abelian limit, as required by Bekenstein and Sanders [4], there is a residual term that can cancel the caustic singularities.

To preserve the normalization condition (83), it is clear that we have the possibility of moving the

A Lorentz transformation on

An analogy can be drawn to the usual Yang-Mills gauge on

Generalizing this structure, one can take the indices

implying that

that is, a three-dimensional Lorentz invariant integration measure.

We now examine the gauge condition:

Since the Hermitian operator

in the same form as the Yang-Mills theory [38, 39]. It is evident in the Yang-Mills theory, due to the operator nature of the second term, the field will be algebra-valued, and thus we have the usual structure of the Yang-Mills non-Abelian gauge theory. Here, if the transformation

It follows from (87) that the “field strengths”

Under a gauge transformation

according to

just as in the finite-dimensional Yang-Mills theories.

For

where

Then,

Let us take

where symmetrization is required since

Our investigation in the following will be concerned with a study of the infinitesimal gauge neighborhood of the Abelian limit, where the components of

Now, the second term of (94), which is the commutator of

We now consider the third term in (94).

There are two terms proportional to

If we take

where

For the remaining two terms,

The commutators contain only terms linear in

We now consider the derivation of field equations from a Lagrangian constructed with the

where

and

In carrying out the variation of

Note that for the general case of

since the Lagrangian density (108) contains an integration over

that is, replacing explicitly

or

where

For the calculation of the variation of

Calling

we compute the variation of

Then, for

we compute

With our choice of

so that

Here,

so we see that

where the quantity

In the limit that

Returning to the variation of

where we have taken into account the fact that

Since the coefficient of

which is nonlinear in the fields

## 6. Summary

In this chapter, we have shown that the formulation of MOND theory by Bekenstein and Milgrom [5, 6, 7, 8, 9, 10] can have a systematic origin within the framework of the embedding of the SHP [1] theory into general relativity [20]. The SHP formalism admits a scalar potential term that appears both in the conformal factor giving rise to the MOND functions in the galaxy and, in the original form of the Hamiltonian, to a possible candidate for “dark energy.” The solution of the lensing problem by Bekenstein and Sanders [4] by introduction of a local vector field was also shown to arise in a natural way in terms of a non-Abelian gauge field, for which, in the Abelian limit, there is a residual term that can cancel the caustic singularity found by Contaldi et al. [40] which can arise in a purely Abelian gauge theory.

## Notes

- Since Vx has the dimension of mass, one can think of this function as a scalar mass field, reflecting forces acting in the local tangent space at each point. It may play the role of “dark energy” [2, 3]. If V=0, our discussion reduces to that of the usual general relativity, but with a well-defined canonical momentum variable.
- We shall call the quantity πμ in the cotangent space as canonical momentum, although it must be understood that its map back to the tangent space πμ corresponds to the actual physically measureable momentum.
- In the quantized form, the factor gμνx cannot be factored out from polynomials, so, as for Dirac’s quantization procedure [22, 23, 24, 25], some care is required.
- The physically observable momentum can be defined, as in (15), as 12gμνpν, with commutation relations of the form (27). This operator can be transformed, as for the Newton-Wigner operator [29], to the form −i∂∂xμ by the Foldy-Wouthuysen transformation [30] gx14pμgx−14.
- We follow here essentially the method discussed in Reed and Simon [31] in their discussion of the Lebesgue integral.
- Similar to the method followed in the simpler case of constant curvature by Georgiev [33].
- Note that Abraham et al. [34] apply the formal Fourier transform on a manifold in three dimensions without proof.
- Yahalom [36] has proposed an alternative view involving the retardation effects associated with gravitational waves, presently being tested and developed. We do not discuss this approach further here.