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arXiv:1112.1324

On the Apparent Superluminal Motion of a Damped Gaussian Pulse

Norman Hugh Redington

Published Dec 4, 2011• 0 citations
arXivHTMLPDF

Abstract

Alicki has demonstrated that a travelling Gaussian pulse subject to damping is indistinguishable from an undamped pulse moving with greater speed; such an effect could create the illusion of a pulse moving faster than light. In this note, an alternative derivation of the same result is presented. However, it is unlikely that this particular illusion could explain the superluminal neutrino-velocities reported by OPERA.

DOI
10.48550/arxiv.1112.1324

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Alicki has demonstrated that a travelling Gaussian pulse subject to damping is indistinguishable from an undamped pulse moving with greater speed; such an effect could create the illusion of a pulse moving faster than light. In this note, an alternative derivation of the same result is presented. However, it is unlikely that this particular illusion could explain the superluminal neutrino-velocities reported by OPERA.
The result of the OPERA experiment revealed that the velocity of muon-neutrinos was larger than the speed of light. We argue that this apparent superluminal velocity can be interpreted as a weak value, which is a new concept recently studied in the context of quantum physics. The OPERA experiment setup forms a scheme that manifests the neutrino velocity as a weak value. The velocity defined in the scheme of weak measurement can exceed the speed of light. The weak velocity is not a concept associated to a single phenomenon but it is a statistical concept defined by accumulating data at separated places and by comparing the data. Neither information nor physical influence is conveyed at the weak velocity. Thus the superluminal velocity in the sense of weak value does not contradict the causality law. We propose also a model for calculating the neutrino velocity with taking neutrino oscillation into account.
There have been many recent theoretical and experimental reports on the propagation of light pulses at speeds exceeding the speed of light in vacuum ccc within media with anomalous dispersion, either opaque or with gain. Superluminal propagation has also been reported within vacuum, in the case of inhomogeneous pulses. In this paper we show that the observations of superluminal and non-causal propagation of evanescent pulses under the conditions of frustrated internal reflection are only apparent, and that they can be simply explained employing an explicitly (sub)luminal causal theory. However, the usual one-dimensional approach to the analysis of pulse propagation has to be abandoned and the spatial extent of the incoming pulse along the directions normal to the propagation direction has to be accounted for to correctly interpret the propagation speed of these evanescent waves. We illustrate our theory with animations of the time development of a pulse built upon the Huygen's construction.
An explanation for superluminal phenomena based on wave-particle duality of photons is suggested. A single photon may be regarded as a wave packet, whose spatial extension is its coherence volume. As a photon propagates as a wave train in vacuum, its velocity is just the speed of light. When it tunnels through a barrier as a particle, its wave function collapses and it will travel faster than light. Superluminal motion can occur only within the coherence length and the time constrained by uncertainty principle. A massive particle cannot be superluminal during the tunneling process. So superluminality does not violate causality. As for the superluminal and negative group velocities in anomalously dispersive medium, they are merely reshaping effect of the pulse, and they will become subluminal at large distances. A couple of experiments are proposed to test the superluminal phenomena.
The apparent superluminal propagation of electromagnetic signals seen in recent experiments is shown to be the result of simple and robust properties of relativistic field equations. Although the wave front of a signal passing through a classically forbidden region can never move faster than light, an attenuated replica of the signal is reproduced ``instantaneously'' on the other side of the barrier. The reconstructed signal, causally connected to the forerunner rather than the bulk of the input signal, appears to move through the barrier faster than light.