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Journal of Physics A Mathematical and Theoretical

Superluminal speeds for relativistic random waves

Michael Berry

Published Apr 24, 2012• 19 citations
Publisher

Abstract

For Klein?Gordon and Dirac waves representing massive quantum particles, the local group velocity v (weak value of the velocity operator) can exceed c. If the waves consist of superpositions of many plane waves, with different (but subluminal) group velocities u, the superluminal probability Psuper, i.e. that |v| > c for a randomly selected state, can be calculated explicitly. Psuper?depends on two parameters describing the distribution (power spectrum) of u in the superpositions, and lies between 0 and 1/2 for Klein?Gordon waves and 1?1/ and 1/2 for Dirac waves. Numerical simulations display the superluminal intervals in space and regions in spacetime, and support the theoretical predictions for Psuper.

DOI
10.1088/1751-8113/45/18/185308

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For Klein?Gordon and Dirac waves representing massive quantum particles, the local group velocity v (weak value of the velocity operator) can exceed c. If the waves consist of superpositions of many plane waves, with different (but subluminal) group velocities u, the superluminal probability Psuper, i.e. that |v| > c for a randomly selected state, can be calculated explicitly. Psuper?depends on two parameters describing the distribution (power spectrum) of u in the superpositions, and lies between 0 and 1/2 for Klein?Gordon waves and 1?1/ and 1/2 for Dirac waves. Numerical simulations display the superluminal intervals in space and regions in spacetime, and support the theoretical predictions for Psuper.
Abstract Weak values characterize a quantum system in the period of time between preparation and measurement and may lie outside the eigenvalue spectrum of the measured operator. The probability of such “superweak" values for random quantum states has been calculated and applied to Klein–Gordon and Dirac waves, where the maximal probability for superluminal propagation was shown to be 1/2. In a recent paper, a different definition for the velocity of a relativistic quantum particle was proposed in terms of a ratio of two weak values. In this paper, we find the probability distribution of such ratios. With the new definition, the superluminal probability of photons is found to be bounded between 1−1/21-1/\sqrt{2}1−1/2​ and 1/21/\sqrt{2}1/2​ , while for general eigenvalue distributions the superluminal probability can take any value between 0 and 1.
The local group velocity defined as the weak value of the velocity operator in (1 + 1)-dimensional Klein-Gordon and Dirac theory is studied. As shown by Berry [J. Phys. A 45, 185308 (2012)], when the pre- and post-selected states for evaluating the weak value are chosen at random from an ensemble of available states, it gives rise to a universal probability distribution for the local group velocity which can have both subluminal and superluminal components. In this work, we explore the possibility of enhancement of the superluminal fraction of this total probability distribution by applying a Lorentz boost and show that it can indeed be enhanced both in the case of Klein–Gordon and Dirac theories.
The Dirac equation has resided among the greatest successes of modern physics since its emergence as the first quantum mechanical theory fully compatible with special relativity. This compatibility ensures that the expectation value of the velocity is less than the vacuum speed of light. Here, we show that the Dirac equation admits free-particle solutions where the peak amplitude of the wave function can travel at any velocity, including those exceeding the vacuum speed of light, despite having a subluminal velocity expectation value. The solutions are constructed by superposing basis functions with correlations in momentum space. These arbitrary velocity wave functions feature a near-constant profile and may impact quantum mechanical processes that are sensitive to the local value of the probability density as opposed to expectation values. Published by the American Physical Society 2024
We consider rectilinear free-space propagation of electromagnetic wavepackets using electromagnetic field theory, scalar wavepacket propagation, and quantum-mechanical formalism. We demonstrate that spatially localized wavepackets are inherently characterized by a subluminal group velocity and a superluminal phase velocity, whose product equals c2c^2c2. These velocities are also known as the 'energy' and 'momentum' velocities, introduced by K. Milton and J. Schwinger. We illustrate general conclusions by explicit calculations for Gaussian and higher-order beams and wavepackets, and also highlight subtleties of the quantum-mechanical description based on the 'photon wavefunction'.
We consider the tunnelling particle as a pre- and post-selected system and prove that the tunnelling time is the expectation value of the position of a ‘clock’ degree of freedom weakly coupled to it. Such a value, called a ‘weak value’, typically falls outside the eigenvalue spectrum of the operator. The appearance of unusual weak values has been associated with a unique interference structure called ‘superoscillations’ (band-limited functions which on a finite interval, approximate functions with spectra well outside their band). It is proposed that superoscillations play an important role in the interferences which give rise to superluminal effects. To demonstrate that, we consider a certain simple tunnelling barrier which allows a wave packet to travel in zero time and negligible distortion, a distance arbitrarily longer than the width of the wave packet. The peak is shown to result from a superoscillatory superposition at the tail. Similar reasoning applies to the dwell time. For this system, both the Wigner time (related to the group velocity) and a clock time correspond to superluminal velocities.
Wavepackets representing relativistic quantum particles injected into a half-space, from a source that is switched on at a definite time, are represented by superpositions of plane waves that must include negative frequencies. Propagation is causal: it is a consequence of analyticity that at time t no part of the wave has travelled farther than ct , corresponding to the front of the signal. Nevertheless, interference fringes behind the front travel superluminally. For Klein–Gordon and Dirac wavepackets, the spatially integrated density increases because current is injected at the boundary. Even in the simplest causal model, understanding the shape of the wave after long times is an instructive exercise in the asymptotics of integrals, illustrating several techniques at a level suitable for graduate students; different spatial features involve contributions from a pole and from two saddle points, the uniform asymptotics for the pole close to a saddle, and the coalescence of two saddles into the Sommerfeld precursor immediately behind the front.