Sum-over-histories origin of the composition laws of relativistic quantum mechanics and quantum cosmology

Abstract

This paper is concerned with the question of the existence of composition laws in the sum-over-histories approach to relativistic quantum mechanics and quantum cosmology, and its connection with the existence of a canonical formulation. In nonrelativistic quantum mechanics, the propagator is represented by a sum over histories in which the paths move forward in time. The composition law of the propagator then follows from the fact that the paths intersect an intermediate surface of constant time once and only once, and a partition of the paths according to their crossing position may be affected. In relativistic quantum mechanics, by contrast, the propagators (or Green functions) may be represented by sums over histories in which the paths move backward and forward in time. They therefore intersect surfaces of constant time more than once, and the relativistic composition law, involving a normal derivative term, is not readily recovered. The principal technical aim of this paper is to show that the relativistic composition law may, in fact, be derived directly from a sum over histories by partitioning the paths according to their first crossing position of an intermediate surface. We review the various Green functions of the Klein-Gordon equation, and derive their composition laws. We obtain path-integral representations for all Green functions except the causal one. We use the proper time representation, in which the path integral has the form of a nonrelativistic sum over histories but is integrated over time.The question of deriving the composition laws therefore reduces to the question of factoring the propagators of nonrelativistic quantum mechanics across an arbitrary surface in configuration space. This may be achieved using a known result called the path decomposition expansion (PDX). We give a proof of the PDX using a spacetime lattice definition of the Euclidean propagator. We use the PDX to derive the composition laws of relativistic quantum mechanics from the sum over histories. We also derive canonical representations of all of the Green functions of relativistic quantum mechanics, i.e., express them in the form 〈x''\ensuremath{\Vert}x'〉, where the {\ensuremath{\Vert}x〉} are a complete set of configuration-space eigenstates. These representations make it clear why the Hadamard Green function ${\mathit{G}}^{(1)}$ does not obey a standard composition law. They also give a hint as to why the causal Green function does not appear to possess a sum-over-histories representation. We discuss the broader implications of our methods and results for quantum cosmology, and parametrized theories generally. We show that there is a close parallel between the existence of a composition law and the existence of a canonical formulation, in that both are dependent on the presence of a timelike Killing vector. We also show why certain naive composition laws that have been proposed in the past for quantum cosmology are incorrect. Our results suggest that the propagation amplitude between three-metrics in quantum cosmology, as constructed from the sum over histories, does not obey a composition law.