Comment on ‘Backflow in relativistic wave equations’

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.


Introduction
In a recent paper in this journal, Bialynicki-Birula, Bialynicka-Birula and Augustynowicz [1], henceforth referred to as BBA, seek to downplay the exceptional nature of quantum probability backflow, the phenomenon that has been widely discussed in the literature in the many years since it was first indicated by Allcock [2] and then described fully and quantified by us [3]. They also claim new results for backflow associated with Dirac's equation. In addressing their criticisms and claim, we make the following points. * Author to whom any correspondence should be addressed.
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• BBA begin by questioning the statements by numerous authors, including us, that probability backflow is a 'peculiar quantum effect' [3], 'an intriguing quantum-mechanical phenomenon' [4], a 'surprising and clearly non-classical effect with no classical counterpart' [5], 'clearly a nonclassical effect' [6], 'a classically impossible phenomenon' [7], a 'new quantum effect' [8], and 'a generic purely quantum phenomenon' [9]. BBA suggest that these authors 'apparently have not read the paper by Berry [10] who has shown that the backflow effect also exist sic in simple one-dimensional situations in classical optics'. In fact these cited papers represent only a small fraction of the many papers devoted directly or indirectly to quantum probability backflow in the decades since its introduction, as BBA could have seen from the bibliography of the second of our papers that they do cite [8]. Consideration of this extensive literature more fully shows that the statement by BBA creates a false impression and would seriously mislead any reader new to the subject. Indeed we ourselves have referred to Berry [10] in the second of our papers that BBA do cite [8], where we remark that 'a quantum object has wave-like as well as particle-like properties and, from a purely mathematical point of view, probability backflow is one more example of retrogressive wave motion and is not particularly remarkable as such'.
• More important to the present discussion however is the continuation of our remark, which states that 'Nevertheless, in the context of the dynamics of a quantum particle, the effect is strikingly counterintuitive, with no classical analogue'. This brings us to our main criticism of BBA: that they have failed to acknowledge that in the many discussions of quantum probability backflow, context is critically important. Those discussions deal with the behaviour of a quantum object with both wave-like and particle-like properties. The importance attaches not to the mathematics of backflow, which can indeed be regarded as reflecting the wave-like characteristics of the object as has been noted [8,10] prior to BBA. Rather, the importance attaches to the fact that the mathematics in this context describes the behaviour of the probability distribution associated with the particle-like properties of the quantum object. It is that behaviour which has no classical counterpart.
• The situation is similar to that pertaining to other striking features of quantum mechanics that involve the mathematical interplay between the wave-like and particle-like properties of a quantum object. Consider for example Heisenberg's Uncertainty Principle for conjugate position and momentum variables associated with a simple quantum system. The well known uncertainty relation can be derived in several ways, and in particular can be regarded as a consequence of the inverse mathematical relation between the spread of the wave function in coordinate space and the spread of its Fourier transform in momentum space. That inverse Fourier transform relationship appears in many other contexts involving waves, notably in the mathematical description of signals propagated by sound waves. There it leads to an uncertainty relation between the duration of a signal and the frequency of the waves of which it is composed, an uncertainty relation that is important in the theory and performance of musical compositions, if not often made explicit [11]. As a result of the widespread applicability of this property of the Fourier transform, one could say that Heisenberg's Uncertainty Principle is unsurprising from a purely mathematical point of view. But no quantum physicist would suggest that Heisenberg's principle should be regarded as no more than one more example of a general mathematical phenomenon pertaining to wave motion! Of course probability backflow does not have a level of importance to physics comparable to Heisenberg's principle, but the key point is the same in the two cases: it is the physics context that is all-important. The fact that the backflow phenomenon under discussion refers to probability is critical. To make the point again: it is that which has no classical counterpart.
• Reference to Heisenberg's principle, where the appearance of Planck's constant ℏ is central, leads us to another criticism of BBA. Nowhere do they mention the somewhat mysterious dimensionless quantum number λ max ≈ 0.038 452, which characterizes the probability backflow phenomenon in the simplest case of a quantum 'particle' moving freely on the x-axis with a positive momentum. It equals the maximum amount of probability that can flow 'backwards' in any chosen time interval. Again we refer the reader to [8] and references therein for numerous calculations and discussions. Analogous mathematical quantities may well exist for other backflow effects arising in classical physics, but in the context of quantum probability backflow this is a genuine quantum number, with the remarkable property that it is independent of the size of Planck's constant. It is an eigenvalue of an Hermitian operator, and has a significance fundamental to the physical process. • In their paper, BBA go on to describe several processes where backflow appears, the first being Dirac's model for the relativistic electron. They do not note the fact that this has previously been treated in much greater detail by us [12] and others since [13], despite these references being given in the paper [8] which they cite. The discussion by BBA in terms of physically unrealizable plane waves is only indicative of an observable effect. A more detailed treatment [12] involves the introduction of physically realizable, normalized states, and leads inevitably to a generalization of the eigenvalue problem that gives rise to λ max for the free non-relativistic particle. It also leads to a tantalizing suggestion that there may be a connection between relativistic probability backflow and Schrödinger's Zitterbewegung [14]. • When Einstein wrote to Born in 1926 that he was 'convinced that God does not play dice,' he was expressing his unwillingness to accept the fundamental role that probability plays in quantum mechanics through the Born interpretation [15]. The multitude of successful applications of the quantum theory since that time provides convincing evidence of its correctness. But it is also clear that quantum probability behaves differently in important ways from classical probability. (See for example [16,17].) It seems that God does play dice, but they are quantum dice, not classical dice! While there has been much research on the different structural aspects of quantum and classical probability, there has been less effort directed at the quantification of resultant behavioural differences in probabilistic aspects of quantum versus classical systems. The phenomenon of quantum probability backflow provides a striking illustration of those differences, which the quantum number λ max and its generalizations [8,12,18,19] help to quantify.

Data availability statement
No new data were created or analysed in this study.