A Spacetime Path Formalism for Relativistic Quantum Mechanics

Quantum field theory is the usual solution to the problems inherent in melding quantum mechanics with special relativity. However, there has also been significant literature on an alternate presentation of relativistic quantum mechanics in terms of an “invariant fifth parameter” in addition to the usual four coordinates of spacetime. When this parameter is treated as parameterizing spacetime paths, the formalism can be seen as a relativistic variant on the more familiar Feynman “sum over histories” approach to non-relativistic quantum mechanics. This paper shows how such a spacetime path formalism can be considered to arise naturally from the foundational principles of the Born probability rule, superposition and Poincaré invariance. The resulting approach is similar but not identical to previous parameterized approaches in the literature. It reproduces the results of perturbative quantum field theory for free and interacting, scalar and non-scalar particles (though only massive particles without gauge symmetry are considered in the present work). An important consequence of the formalism presented here is that a clear probabilistic interpretation can be maintained throughout, including for non-scalar particles and scattering. A further payoff is that the formalism allows for a straightforward interpretation in terms of decoherent histories that are coarse-grainings of spacetime paths. This interpretation can then be extended to fully relativistic cosmological “states of the universe,” providing an approach for showing how the classical “real history” of the universe arises from its more precise quantum description.