On the interpretation of the relativistic quantum mechanics with invariant evolution parameter

John R. Fanchi

Iu. M. Shirokov

James B. Hartle

The `evolving constants' method of defining the quantum dynamics of time-reparametrization-invariant theories is investigated for a particular implementation of parametrized non-relativistic quantum mechanics (PNRQM). The wide range of time functions that are available to define evolving constants raises issues of interpretation, consistency, and the degree to which the resulting quantum theory coincides with, or generalizes, the usual non-relativistic theory. The allowed time functions must be restricted for the predictions of PNRQM to coincide with those of usual quantum theory. They must be restricted to have a notion of quantum evolution in a time parameter connected to spacetime geometry. They must be restricted to prevent the theory from making inconsistent predictions for the probabilities of histories. Suitable restrictions can be introduced in PNRQM but these seem unlikely to apply to a reparametrization-invariant theory like general relativity.

John R. Fanchi

Abstract A common justification for replacing quantum mechanics with quantum field theory (QFT) is that the appearance or disappearance of particles cannot be described using quantum mechanics. We show that this justification for QFT is not generally true by presenting a counterexample: parametrized relativistic quantum mechanics (pRQM). We begin by outlining a pioneering formulation of QFT that includes an invariant evolution parameter. The introduction of an invariant evolution parameter helped guide the development of QFT and is a characteristic feature of pRQM. We then present a probabilistic formulation of pRQM that highlights features of the theory that make it suitable for modelling particle stability. Two examples of particle stability are then presented within the context of pRQM to show that a quantum mechanical theory can be applied to particle stability. The examples considered in this paper are exponential particle decay and neutrino oscillations.

K. V. Roberts

Lawrence Paul Horwitz

A consistent classical and quantum relativistic mechanics can be constructed if Einstein's covariant time is considered as a dynamical variable. The evolution of a system is then parametrized by a universal invariant identified with Newton's time. This theory, originating in the work of Stueckelberg in 1941, contains many questions of interpretation, reaching deeply into the notions of time, localizability, and causality. Some of the basic ideas are discussed here, and as an example, the solution of the two body problem with invariant action-at-a-distance potential is given. A proper generalization of the Maxwell theory of electromagnetic interaction, implied by the Stueckelberg-Schrödinger dynamical evolution equation and its physical implications are also discussed. It is also shown that a similar construction occurs, applying similar ideas, in the theory of gravitation.