Abstract

A consistent 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). In this paper, using local coordinate transformations, following the original procedure of Einstein, this theory is embedded into the framework of general relativity (GR) both for potential models (where the potential appears as a spacetime mass distribution with dimension of mass) and for electromagnetism (emerging as a gauge field on the quantum mechanical Hilbert space). The canonical Poisson brackets of the SHP theory remain valid (invariant under local coordinate transformations) on the manifold of GR, and provide the basis, following Dirac’s quantization procedure, for formulating a quantum theory. The theory is developed both for one and many particles.

Highlights

  • The relativistic canonical Hamiltonian dynamics of Stueckelberg, Horwitz and Piron (SHP)[1] with scalar potential and gauge field interactions for single and many body theory can, by local coordinate transformation, be embedded into the framework of general relativity (GR)

  • We have shown that the SHP theory can be embedded by local coordinate transformations into the framework of general relativity

  • The Minkowski spacetime coordinates of the SHP theory are considered to lie in the tangent space of a manifold with metric and connection form derived from the coordinate transformations on the equations of motion for particles moving on the locally flat Minkowski spacetime, parametrized by a universal monotonic world time τ

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Summary

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 theory can, by local coordinate transformation, be embedded into the framework of general relativity (GR). The formulation is generalized here to a U (1) Abelian gauge theory (electromagnetism on the manifold), but can be extended to the non-Abelian case This provides a fundamental derivation of the framework assumed by Horwitz, Gershon and Schiffer [19][20] in their discussion* of the BekensteinSanders 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] to explain the rotation curves of galaxies. In a more dynamical setting, when the energy momentum tensor depends on τ , the spacetime evolves nontrivially; the transformations from the local Minkowski coordinates to the curved space coordinates depend on τ We discuss this 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

Single particle in external potential
M ημν πν
Off Shell Mass Evolution
Dynamics of a Particle Near the Schwarzschild Horizon
The many body system with interaction potential
Quantum Theory on the Curved Space
Electromagnetism
The Many Body Problem for Electromagnetism
Summary and Outlook
Full Text
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