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arXiv:2204.10880 · Journal of Physics A Mathematical and Theoretical

Backflow in relativistic wave equations

Iwo Białynicki‐Birula, Zofia Białynicka-Birula, Szymon Augustynowicz

Published Apr 8, 2022• 26 citations
arXivHTMLPDF

Abstract

Abstract We show that, contrary to the statements made by many authors, the backflow is not a nonclassical effect. The backflow is a characteristic feature of solutions of the wave equations: quantum and classical. We present simple solutions of the Dirac equation, Maxwell equations and equations of linearized gravity where the backflow phenomenon is clearly seen. In this work we describe backflow in relativistic theories but this phenomenon can occur in the solutions of all kinds of wave equations: quantum and classical.

DOI
10.1088/1751-8121/ac65c1

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Comment on ‘Backflow in relativistic wave equations’

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Abstract Criticisms and a claim in the recent paper Backflow in relativistic wave equations by Bialynicki–Birula et al 2022 J. Phys. A: Math. Theor. 55 255702 are addressed, and it is emphasized again that the widely discussed phenomenon of quantum probability backflow has no classical counterpart. It is pointed out that backflow for the relativistic Dirac Equation has been treated in depth by us some years ago and by others since.

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We have calculated normalized wave function solutions to the time-dependent Dirac equation on a space-time lattice grid. Using these solutions we systematically explore phenomena that are consistent with the classical relativistic Liouville phase space approach and those that do not have a direct classical counterpart. The examples reviewed are the paradoxical Zitterbewegung, the time-resolved Klein-paradox, the dynamics of spin in an inhomogeneous magnetic field, the suppression of the spatial spreading rate and new dynamical features that occur close to relativistic resonances.

AIP conference proceedings•Jan 1, 2000•1 citation
Abstract We show that, contrary to the statements made by many authors, the backflow is not a nonclassical effect. The backflow is a characteristic feature of solutions of the wave equations: quantum and classical. We present simple solutions of the Dirac equation, Maxwell equations and equations of linearized gravity where the backflow phenomenon is clearly seen. In this work we describe backflow in relativistic theories but this phenomenon can occur in the solutions of all kinds of wave equations: quantum and classical.
It was known that a free, non-relativistic particle in a superposition of positive momenta can, in certain cases, bear a negative probability current — hence termed quantum backflow. Here, it is shown that more variations can be brought about for a free Dirac particle, particularly when negative-energy solutions are taken into account. Since any Dirac particle can be understood as an antiparticle that acts oppositely (and vice versa), quantum backflow is found to arise in the superposition (i) of a well-defined momentum but different signs of energies, or more remarkably (ii) of different signs of both momenta and energies. Neither of these cases has a counterpart in non-relativistic quantum mechanics. A generalization by using the field-theoretic formalism is also presented and discussed.
Abstract In this paper we discuss relativistic quantum backflow. The general theory of relativistic backflow is presented and it is shown that the backflow can be written as a function of a simple parameter ϵ, which is defined in terms of fundamental constants and the backflow period. Backflow eigenfunctions are determined numerically for a range of values of ϵ, and an explicit expression for the relativistic backflow eigenvalue in terms of the non-relativistic backflow constant is presented. Backflow eigenvectors are then fitted with some standard functions, which lead to substantially higher backflow than has been found previously with fitting procedures, for some values of ϵ . In analysing the non-relativistic limit of the theory we show that this problem is one of those rare cases where the relativistic theory is intrinsically more simple than the non-relativistic theory.
We have calculated normalized wave function solutions to the time-dependent Dirac equation on a space-time lattice grid. Using these solutions we systematically explore phenomena that are consistent with the classical relativistic Liouville phase space approach and those that do not have a direct classical counterpart. The examples reviewed are the paradoxical Zitterbewegung, the time-resolved Klein-paradox, the dynamics of spin in an inhomogeneous magnetic field, the suppression of the spatial spreading rate and new dynamical features that occur close to relativistic resonances.