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Physical Review Letters

Violation of Causality in Relativistic Quantum Theory?

Gerhard C. Hegerfeldt

Published Jun 3, 1985• 145 citations
Publisher

Abstract

We show that states of systems which, in a very general sense, are approximately localized at time $t=0$ in a finite region, with exponentially bounded tails outside, violate Einstein causality at later times. Implications are discussed.

DOI
10.1103/physrevlett.54.2395

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•Jan 1, 2000•0 citations
We show that states of systems which, in a very general sense, are approximately localized at time t=0t=0t=0 in a finite region, with exponentially bounded tails outside, violate Einstein causality at later times. Implications are discussed.
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}
The notion of Einstein causality, i.e. the limiting role of the velocity of light in the transmission of signals, is discussed. We show that under quite general assumptions instantaneous spreading of particle localization occurs in quantum theory, relativistic or not, with fields or without. We discuss if this affects Einstein causality. It is also pointed out that the controversy about recent experiments of Nimtz and coworkers reporting superluminal signal velocities in micro-wave tunneling arise from an indiscriminate use of terminology and that the experimental results are in full agreement with Einstein causality in its ordinary sense.
The notion of Einstein causality, i.e. the limiting role of the velocity of light in the transmission of signals, is discussed. It is pointed out that Nimtz and coworkers use the notion of signal velocity in a different sense from Einstein and that their experimental results are in full agreement with Einstein causality in its ordinary sense. We also show that under quite general assumptions instantaneous spreading of particle localization occurs in quantum theory, relativistic or not, with fields or without. We discuss if this affects Einstein causality.
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.