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.
- relativistic dynamics
- general relativity
- quantum theory on curved space
- non-Abelian gauge fields
- Bekenstein-Sanders field
The relativistic canonical Hamiltonian dynamics of Stueckelberg, Horwitz, and Piron (SHP)  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  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 . The theory was generalized by Horwitz and Piron in  (see also [15, 16]) to be applicable to many-body systems by assuming that the parameter is universal (as for Newtonian time, enabling them to solve the two-body problem classically, and later, a quantum solution was found by Arshansky and Horwitz [17, 18, 19], both for bound states and scattering theory).
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  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.  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 is determined by the covariant form of the Stueckelberg-Schrödinger equation .
In this chapter, we assume a -independent background gravitational field; the local coordinate transformations from the flat Minkowski space to the curved space are taken to be independent of , consistently with an energy momentum tensor that is independent. In a more dynamical setting, when the energy momentum tensor depends on , the spacetime is evolved nontrivially [20, 21].
2. Embedding of single particle dynamics with external potential in GR
where is the flat Minkowski metric and are the spacetime canonical momenta and coordinates in the local tangent space of a general manifold, following Einstein’s use of the equivalence principle.
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 , with the invariant universal “world time.” Since then
the Hamiltonian can then be written as
It is important to note that, as clear from (3), that has a sign
back to the tangent space. Thus, equivalently, from (2), . This simple observation will be important in the discussion below of the dynamics of a particle in the framework of general relativity, for which the metric tensor is nontrivial.
We now transform the local coordinates (contravariantly) according to the diffeomorphism
to attach small changes in to the corresponding small changes in the coordinates on the curved space, so that
The Hamiltonian then becomes
where is the potential at the point corresponding to the point (a function of in a small neighborhood of the point ) and
Since has significance as the source of a force in the local frame only through its derivatives, we can make this pointwise correspondence with a globally defined scalar function .1
The corresponding Lagrangian is then
In the locally flat coordinates in the neighborhood of , the symplectic structure of Hamiltonian mechanics implies that the momentum2, lying in the cotangent space of the manifold , transforms covariantly under the local transformation (5), that is, as does , so that we may define
This definition is consistent with the transformation properties of the momentum defined by the Lagrangian (10):
As we have remarked above for the locally flat space in (5), the
It is therefore evident from (15) that
We see that , which should be interpreted as the force acting on the particle, is proportional to the
We now discuss the geodesic equation obtained by studying the condition
To do this, we compute
so that, after multiplying by and summing over , we obtain
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 is mapped, under this coordinate transformation into a force resulting in a modification of the acceleration along the geodesic-like curves, that is, (16) now reads
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 . In the local coordinates , the derivative of a function is
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 relation,
The Poisson bracket of with the (physical energy momentum) tangent space variable has then the tensor form
In the flat space limit, this relation reduces to the SHP bracket,
Continuing our analysis with (we drop the label on the Poisson bracket henceforth),
so that acts infinitesimally as the
so that is the generator of translations in . In the classical case, if is a general function of , we can write at some point ,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.
Multiplying by , the two terms combine to give a factor of two. We then return to the original definition of in (20) in the form
so we can write
We therefore have
3. Quantum theory on the curved space
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 not Hermitian due to the presence of the factor in the integrand of the scalar product. The problem is somewhat analogous to that of Newton and Wigner  in their treatment of the Klein-Gordon equation in momentum space. It is easily seen that the operator
Since is Hermitian in the scalar product (41), we can write the Hermitian Hamiltonian as
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 for a scalar function (we shall use and the canonically conjugate in this discussion). Let us define ()
The inverse is given by
One sees immediately that under diffeomorphisms, for which with the scalar property , . The Fourier transform of is
By change of integration variables, we have
In Dirac notation,
and we write as well
we have, for example, the usual action of transformation functions
where we have used
Note that the transformation functions and are not simple complex conjugates of each other, but require nontrivial factors of and its inverse to satisfy the necessary transformation laws on the manifold. Conversely (the factors and its inverse cancel), we should obtain
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 that cover the space and eventually take the limit as for a Riemann integral.5 In each small box, the coordinatization arises from an invertible transformation from the local tangent space in that neighborhood. We write
and is in the flat local tangent space at .
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 in the neighborhood of each point may be different, and therefore the set of transformed boxes may not cover (boundary deficits) the full domain of spacetime coordinates (one can easily estimate that the deficit from an arbitrarily selected set can be infinite in the limit).
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 , 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 . Relating all points along a geodesic to the corresponding tangent spaces and putting each patch in correspondence by continuity, we may consider the set to be in correspondence with an extended flat space , for which to obtain6
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 is to act on elements of a Hilbert space , the support for vanishes, so that is essentially bounded, as we discuss below. In the small box, say, size ,
so that , and we have
It is clear that the assertion (69) requires some discussion. For we must be sure that does not become too large, so that our local measure is equivalent to . In one of the dimensions, what we want to find are conditions for which
for , where we have written for . As a distribution, on functions , the left member of (72) acts as
The function is analytic in the neighborhood of if has a Fourier transform for all and the series is convergent in this neighborhood, since is identically zero and successive derivatives correspond to the Fourier transforms of (differentiating under the integral). This implies that if the (usual) Fourier transform of is a function (as a simple sufficient condition) in the local tangent space and we have appropriate convergence properties, we can reliably use the first order term in the Taylor expansion;
so that, for ,
where is the Fourier transform of . As a distribution on such functions , the assertion (3.39) then follows.
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 .
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 , 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  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.  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
that is, a conformal factor on the original metric. Applying the same idea to the Hamiltonian (8), with the of Einstein replaced by the conformal form
with a point in the spectrum of , so that
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 could represent the so-called dark energy [2, 3], establishing a relation between the MOND picture and the anomalous expansion of the universe, a question presently under study.
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  proposed the introduction of a
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  could explain the lensing effect. In the following, we show that this new field may arise from a
To preserve the normalization condition (83), it is clear that we have the possibility of moving the field on a hyperbola with a Lorentz transformation, which we can perform by a gauge transformation.
A Lorentz transformation on is noncommutative, and therefore the gauge field is non-Abelian .
An analogy can be drawn to the usual Yang-Mills gauge on , where there is a two-valued index for the wave function . The gauge transformation is a two-by-two matrix function of and acts only on the indices . The condition of invariant absolute square (probability) is
Generalizing this structure, one can take the indices to be infinite dimensional, and even continuous, so that (84) becomes (in the spectral representation for )
implying that (at each point ) is a unitary operator on a Hilbert space . Since we are assuming that lies on a hyperbola determined by (83), the measure is
that is, a three-dimensional Lorentz invariant integration measure.
We now examine the gauge condition:
Since the Hermitian operator acts as a derivative under commutation relations, we obtain
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 is a Lorentz transformation, the numerical-valued field would be carried, at least in the first term, to a new value on a hyperbola. However, the second term is operator valued on , and thus, as in the Yang-Mills theory, would become operator valued. Therefore, in general, the gauge field is operator valued.
It follows from (87) that the “field strengths”
Under a gauge transformation , the new fields create field strengths in the transformed form
just as in the finite-dimensional Yang-Mills theories.
where is infinitesimal, (87) becomes
Let us take
where symmetrization is required since is a function of as well as and
Our investigation in the following will be concerned with a study of the infinitesimal gauge neighborhood of the Abelian limit, where the components of do not commute and therefore still constitute a Yang-Mills-type field. We shall show in the limit that the corresponding field equations acquire nonlinear terms and may therefore nullify the difficulty found by Contaldi et al.  demonstrating a dynamical instability for an Abelian vector-type TeVeS gauge field. They found that nonlinear terms associated with a non-Maxwellian-type action, such as , could nullify this caustic singularity, so that the nonlinear terms we find as a residue of the Yang-Mills structure induced by our gauge transformation might achieve this effect in a natural way.
Now, the second term of (94), which is the commutator of with , vanishes, since this product is Lorentz invariant (the symmetrization in does not affect this result).
We now consider the third term in (94).
There are two terms proportional to
If we take
where , then
For the remaining two terms,
The commutators contain only terms linear in and they cancel; the remaining terms are zero, and therefore the condition is invariant under this gauge transformation. It involves the coefficient which is a function of the projection of onto a hyperplane orthogonal the . The vector of course depends on . We take, for definiteness, , for some .
We now consider the derivation of field equations from a Lagrangian constructed with the and . We take the Lagrangian to be of the form
In carrying out the variation of , the contributions of varying the s with respect to vanish due to the field equations (Stueckelberg-Schrödinger equation) obtained by varying (or ), and therefore in the variation with respect to , only the explicit presence of in (103) need be taken into account.
Note that for the general case of generally operator valued, we can write
since the Lagrangian density (108) contains an integration over (in spectral representation, considered in lowest order, as well as an integration over in the action). In the limit in which is evaluated in the spectral representation, and noting that is represented by an imaginary differential operator, we can write this as
that is, replacing explicitly by (since it acts by commutator with the fields); we have
where has the usual form of a gauge invariant current.
For the calculation of the variation of , we note that the commutator term in (89) is, in lowest order, a c-number function.
we compute the variation of
With our choice of ,
so we see that
where the quantity depends on the first and second derivatives of , in general, nonlinear in . We therefore have
In the limit that , its derivative and higher derivatives which appear in may not vanish (somewhat analogous to the case in gravitational theory when the connection form vanishes, but the curvature does not), so that this term can contribute in the limit of the an Abelian gauge.
Returning to the variation of , we see that
where we have taken into account the fact that is a c-number function and integrated by parts the derivatives of . We then obtain
Since the coefficient of must vanish, we obtain the Yang-Mills equations for the fields given the source currents:
which is nonlinear in the fields , as we have seen, even in the Abelian limit, where, from (106),
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  theory into general relativity . 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  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.  which can arise in a purely Abelian gauge theory.
- 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 , to the form −i∂∂xμ by the Foldy-Wouthuysen transformation  gx14pμgx−14.
- We follow here essentially the method discussed in Reed and Simon  in their discussion of the Lebesgue integral.
- Similar to the method followed in the simpler case of constant curvature by Georgiev .
- Note that Abraham et al.  apply the formal Fourier transform on a manifold in three dimensions without proof.
- Yahalom  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.