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On the Principle of Causality and Superluminal Signal Propagation Velocities

  • ELECTRODYNAMICS AND WAVE PROPAGATION
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Abstract

This paper considers the question of superluminal (or negative) group velocities of signals in dispersive media, as well as negative group delay times of signals passing through linear filters. It is shown that information can be transmitted only with the help of jumps in the time dependence of a function or its derivatives and, therefore, the information propagation velocity in any (including dispersive, absorbing, or amplifying) medium exactly coincides with the speed of light in vacuum. As for the group velocity of a wave packet, it is only the propagation velocity of its infinitely differentiable envelope, by means of which the transmission of information, strictly speaking, is impossible. Therefore, in regions with anomalous dispersion, the group velocity can be superluminal or negative without violating the “light limitation” of the theory of relativity or the principle of causality. Examples of situations in which superluminal or negative group velocity occurs are given.

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Notes

  1. This article is not a review; therefore the list of references is, of course, incomplete; for a review on the topic, see [37, 39].

  2. More precisely, a function that can be expanded in a power series with a nonzero radius of convergence.

  3. In other words, all information about the time dependence of such a signal beyond any particular finite time interval is redundant: it can be reconstructed from the time dependence within this interval.

  4. The growth of the precursors, clearly visible in the graphs, is actually apparent: the precursors will just retain their amplitude. It’s just that, as the path length grows, they seem to get larger against the background of an exponentially decaying signal.

  5. It would be more correct to call them analytic (in the sense of the theory of analytic functions). But the term “analytic signal” is already engaged, and the mathematical terms “holonomic function” and “analytic function” are practically identical.

  6. This conclusion is sometimes criticized: see, e.g., [74].

  7. In essence, we are talking about the Heisenberg uncertainty principle \(\Delta E\Delta t \approx h\), where \(\Delta E\) is the pulse energy and \(\Delta t\) is the minimum possible duration of its front.

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  1. Samara State Technical University, 443100, Samara, Russia

    N. S. Bukhman

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Translated by E. Chernokozhin

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Bukhman, N.S. On the Principle of Causality and Superluminal Signal Propagation Velocities. J. Commun. Technol. Electron. 66, 227–241 (2021). https://doi.org/10.1134/S1064226921030049

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