Spacetime path formalism for massive particles of any spin

TL;DR

This paper extends a spacetime path formalism for relativistic quantum mechanics to include particles of any spin, maintaining a probabilistic interpretation and relevance for quantum gravity and cosmology.

Contribution

It generalizes the spacetime path formalism to nonscalar particles of any spin using Poincare invariance, broadening its applicability.

Findings

Formulation of a Poincare-invariant propagator for particles of any spin.

Retention of a clear probabilistic interpretation in the extended formalism.

Natural reduction to non-relativistic quantum mechanics.

Abstract

Earlier work presented a spacetime path formalism for relativistic quantum mechanics arising naturally from the fundamental principles of the Born probability rule, superposition, and spacetime translation invariance. The resulting formalism can be seen as a foundation for a number of previous parameterized approaches to relativistic quantum mechanics in the literature. Because time is treated similarly to the three space coordinates, rather than as an evolution parameter, such approaches have proved particularly useful in the study of quantum gravity and cosmology. The present paper extends the foundational spacetime path formalism to include massive, nonscalar particles of any (integer or half-integer) spin. This is done by generalizing the principle of translational invariance used in the scalar case to the principle of full Poincare invariance, leading to a formulation for the nonscalar propagator in terms of a path integral over the Poincare group. Once the difficulty of the non-compactness of the component Lorentz group is dealt with, the subsequent development is remarkably parallel to the scalar case. This allows the formalism to retain a clear probabilistic interpretation throughout, with a natural reduction to non-relativistic quantum mechanics closely related to the well known generalized Foldy-Wouthuysen transformation.