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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Explicit illustration of causality violation: Noncausal relativistic wave-packet evolution

B. Rosenstein, Marius Usher

Published Oct 15, 1987• 42 citations
Publisher

Abstract

A class of functions whose relativistic evolution is analytically solvable is found. These wave functions are initially localized (by a Newton-Wigner definition) and their spreading through time is investigated. Using these wave packets an explicit example of the violation of relativistic causality (in the spirit of Hegerfeldt's theorem) is presented. Some detailed qualitative features of relativistic quantum behavior are displayed.

DOI
10.1103/physrevd.36.2381

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A class of functions whose relativistic evolution is analytically solvable is found. These wave functions are initially localized (by a Newton-Wigner definition) and their spreading through time is investigated. Using these wave packets an explicit example of the violation of relativistic causality (in the spirit of Hegerfeldt's theorem) is presented. Some detailed qualitative features of relativistic quantum behavior are displayed.
In relativistic theories the principle of microscopic causality states that ``information cannot travel faster than the speed of light'' \cite{kaku}. In the present work we show that the time evolution of relativistic wave functions violates this principle. We consider here the wave functions of massless and massive particles. In the case of massless particles the wave functions which violate the microscopic causality have an analytic form while in the case of massive particles we have to rely on numerical calculations. In both cases the wave functions which are strictly localized at t=0t=0t=0, at later times do not vanish {\it outside} the future light cone. \end{abstract}
In a series of interesting papers Hegerfeldt has shown that quantum systems with positive energy initially localized in a finite region immediately develop infinite tails. In our paper Hegerfeldt's theorem is analyzed using quantum and classical wave packets. We show that Hegerfeldt's conclusion remains valid in classical physics. No violation of Einstein's causality is ever involved. Using only positive frequencies, complex wave packets are constructed which at t=0t=0t=0 are real and finitely localized and which, furthermore, are superpositions of two nonlocal wave packets. The nonlocality is initially cancelled by destructive interference. However, this cancellation becomes incomplete at arbitrary times immediately afterwards. In agreement with relativity the two nonlocal wave packets move with the velocity of light, in opposite directions.
The causality issues concerning Hegerfeldt's paradox and the localization of relativistic quantum systems are addressed through a proper-time formalism of single-particle operators. The proposed description does not depend on classical parameters connected to an external observer and results in a single-particle formalism in which localization is described by explicitly covariant four-vector operators associated with POVM measurements parametrized by the system's proper-time. As a consequence, it is shown that physically acceptable states are necessarily associated with the existence of a temporal uncertainty and their proper-time evolution is not subject to the causality violation predicted by Hegerfeldt.
Nonlinear modifications of quantum mechanics generically lead to nonlocal effects which violate relativistic causality. We study these effects using the functional Schrödinger equation for quantum fields and identify a type of nonlocality which causes nearly instantaneous entanglement of spacelike separated systems. We describe a simple example involving widely separated wave packet (coherent) states, showing that nonlinearity in the Schrödinger evolution causes spacelike entanglement, even in free field theory.
The implications of the relativistic space-time structure for a physical description by quantum mechanical wave-functions are investigated. On the basis of a detailed analysis of Bell's concept of local causality, which is violated in quantum theory, we argue that this is a subtle, as well as an important effort. A central requirement appearing in relativistic quantum mechanics, namely local commutativity, is analyzed in detail and possible justifications are given and discussed. The complexity of the implications of wave function reduction in connection with Minkowski space-time are illustrated by a quantum mechanical measurement procedure which was proposed by Aharonov and Albert. This procedure and its relativistic implications are explicitly analyzed and discussed in terms of state evolution. This analysis shows that the usual notion of state evolution fails in relativistic quantum theory. Two possible solutions of this problem are given. In particular, it is shown that also a theory with a distinguished foliation of space-time into space-like leafs - accounting for nonlocality - makes the right predictions for the Aharonov-Albert experiment. We will repeatedly encounter that an analysis of the wave-function alone does not suffice to answer the question of relativistic compatibility of the theory, but that the actual events in space time, which are predicted and described by the theory, are crucial. Relativistic versions of quantum mechanical theories which precisely describe actual processes in space-time are briefly described and discussed in the appendix.