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Physical Review A

Parametrizing relativistic quantum mechanics

John R. Fanchi

Published Sep 1, 1986• 35 citations
Publisher

Abstract

Galilean time in nonrelativistic quantum theory and historical time in parametrized relativistic quantum theories such as the four-space formulation are examined. Clear physical distinctions are drawn between Galilean time, Minkowski time, and historical time. Minkowski time is the temporal coordinate of a space-time four-vector. Galilean time and historical time are evolution (ordering) parameters in their respective theoretical frameworks. Using N-body formulations, a physically unambiguous procedure for measuring evolution parameters in terms of familiar observables is presented. The procedure is illustrated by the design of a simple evolution-parameter clock.

DOI
10.1103/physreva.34.1677

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Galilean time in nonrelativistic quantum theory and historical time in parametrized relativistic quantum theories such as the four-space formulation are examined. Clear physical distinctions are drawn between Galilean time, Minkowski time, and historical time. Minkowski time is the temporal coordinate of a space-time four-vector. Galilean time and historical time are evolution (ordering) parameters in their respective theoretical frameworks. Using N-body formulations, a physically unambiguous procedure for measuring evolution parameters in terms of familiar observables is presented. The procedure is illustrated by the design of a simple evolution-parameter clock.
Relativistic quantum theories are equipped with a background Minkowski spacetime and non‐relativistic quantum theories with a Galilean space‐time. Traditional investigations have distinguished their distinct space‐time structures and have examined ways in which relativistic theories become sufficiently like Galilean theories in a low velocity approximation or limit. A different way to look at their relationship is to see that both kinds of theories are special cases of a certain five‐dimensional generalization involving no limiting procedures or approximations. When one compares them, striking features emerge that bear on philosophical questions, including the ontological status of the wave function and time reversal invariance.
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 first-quantized 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 nonrelativistic quantum mechanics.
The nonrelativistic Schrödinger equation for motion of a structureless particle in four-dimensional spacetime entails a well-known expression for the conserved four-vector field of local probability density and current that are associated with a quantum state solution to the equation. Under the physical assumption that each spatial, as well as the temporal, component of this current is observable, the position in time becomes an operator and an observable in that the weighted average value of the time of the particle's crossing of a complete hyperplane can be simply defined: the theory predicts, and experiment is presumed to be able to observe, the integral over the hyperplane of the normal component of probability current, weighted by the time coordinate. In conventional formulations the hyperplane is always spacelike, i.e. is a time = constant hyperplane in the Galilean relativity, and the result is then trivial. A nontrivial result is obtained if the plane is not of this type. When the spacetime coordinates are ( t , x , y , z ), the paper analyses in detail the case that the hyperplane is of the type z = constant. Particles can cross such a hyperplane in either direction, so it proves convenient to introduce an indefinite metric, and correspondingly a sesquilinear inner product with non-Hilbert space structure, for the space of quantum states on such a surface. Since the metric is indefinite, an uncertainty principle involving the dispersion of the crossing time and the dispersion of its conjugate momentum does not appear to be derivable from the theory. A detailed formalism for computing average crossing times on a z = constant hyperplane, and average dwell times and delay times between a pair of z = constant hyperplanes, is presented.
The theories of special relativity and quantum mechanics both rely on continuous rotation groups to describe physical systems, yet one is formulated on a real spacetime manifold while the other employs a complex Hilbert space. This work investigates whether this distinction may arise from the treatment of time itself. By examining the consequences of describing a relativistic system using a single global time parameter, we explore whether such a description naturally requires a mathematical structure capable of encoding observer-dependent proper times as complex phase information. Within this framework, quantum superposition is investigated as a possible coordinate-level representation of unresolved relativistic frame assignments, while wavefunction collapse is conceptualised as the physical resolution of this ambiguity through interaction. exploring whether aspects of the quantum formalism may be understood as consequences of describing the invariant structure of Minkowski spacetime from a local perspective using a rigid notion of time.
A mathematical framework that unifies the standard formalisms of special relativity and quantum mechanics is proposed. For this a Hilbert space H of functions of four variables x,t furnished with an additional indefinite inner product invariant under Poincaré transformations is introduced. For a class of functions in H that are well localized in the time variable, the usual formalism of nonrelativistic quantum mechanics is derived. In particular, the interference in time for these functions is suppressed; a motion in H becomes the usual Schrödinger evolution with t as a parameter. The relativistic invariance of the construction is proved. The usual theory of relativity on Minkowski space-time is shown to be “isometrically and equivariantly embedded” into H. That is, classical space-time is isometrically embedded into H, Poincaré transformations have unique extensions to isomorphisms of H, and the embedding commutes with Poincaré transformations.