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Journal of Physics A Mathematical and Theoretical

Comment on ‘Backflow in relativistic wave equations’

Anthony J. Bracken, G.F. Melloy

Published Mar 3, 2023• 4 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8121/acbd70

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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.
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.
Quantum backflow is a counterintuitive phenomenon in which the probability density of a quantum particle propagates opposite to its momentum. Experimental observation of backflow has remained elusive due to two main challenges: (i) the effect is intrinsically small, with less than 4% of the probability able to flow backward, and (ii) it requires wave packets with a well-defined momentum direction, which are difficult both to prepare and to verify under realistic, noisy conditions. Here, we overcome these challenges by introducing a general formulation of quantum backflow applicable to arbitrary momentum distributions. The framework recovers the standard backflow limit for unidirectional states and identifies general backflow as probability flow exceeding that predicted by the particle's momentum distribution alone. We show that this excess can reach nearly 13%, surpassing the standard backflow bound by more than a factor of three. Furthermore, we extend the framework to the closely related phenomenon of quantum reentry, provide explicit examples of quantum states exhibiting large general backflow and reentry, and discuss the foundational implications of these nonclassical effects. Our results open a pathway toward the experimental observation of quantum backflow in realistic settings.