Foundations of a spacetime path formalism for relativistic quantum mechanics

Abstract

Quantum field theory is the traditional solution to the problems inherent in melding quantum mechanics with special relativity. However, it has also long been known that an alternative firstquantized formulation can be given for relativistic quantum mechanics, based on the parametrized paths of particles in spacetime. Because time is treated similarly to the three space coordinates, rather than as an evolution parameter, such a spacetime approach has proved particularly useful in the study of quantum gravity and cosmology. This paper shows how a spacetime path formalism can be considered to arise naturally from the fundamental principles of the Born probability rule, superposition, and Poincaré invariance. The resulting formalism can be seen as a foundation for a number of previous parametrized approaches in the literature, relating, in particular, "off-shell" theories to traditional on-shell quantum field theory. It reproduces the results of perturbative quantum field theory for free and interacting particles, but provides intriguing possibilities for a natural program for regularization and renormalization. Further, an important consequence of the formalism is that a clear probabilistic interpretation can be maintained throughout, with a natural reduction to non-relativistic quantum mechanics.

Spacetime path formalism for massive particles of any spin

“…As discussed there, such an approach is particularly suited for further study of quantum gravity and cosmology, and it can be given a natural interpretation in terms of decoherent histories [2]. However, the formalism as given in [1] is limited to scalar particles. The present paper extends this spacetime path formalism to non-scalar particles, although the present work is still limited to massive particles.…”

“…II of some background for this approach, Sec. III generalizes the postulates from [1] to the non-scalar case, leading to a path integral over an appropriate Lagrangian function on the Poincaré group variables.…”

“…The approach to be considered here extends the approach from [1] to non-scalar particles by expanding the configuration space of a particle to be the Poincaré group (also known as the inhomogeneous Lorentz group). That is, rather than just considering the position of a particle, the configuration of a particle will be taken to be both a position and a Lorentz transformation.…”

“…However, rather than using the usual Wigner approach of reduction along the momentum vector [22], we will reduce along an independent time-like four-vector [23,24]. This allows for a very parallel development to [1] for antiparticles in Sec. VI and for a clear probability interpretation in Sec.…”

“…Following a similar approach to [1], the form of the path integral for non-scalar particles will be deduced from the fundamental principles of the Born probability rule, superposition, and Poincaré invariance. After a brief overview in Sec.…”