Reply to comments on ‘Backflow in relativistic wave equations’

We present further arguments which show that the backflow has a universal character. It is not restricted to quantum theory and it appears in many theories (quantum or classical). It is a general property of waves propagating in any number of dimensions.

In their Comments [1,2] two groups of authors expressed their strong disagreement with the claim in our paper that there is no fundamental difference between the backflow in quantum systems and classical systems.
Of course, we agree that in the description of particles in quantum theory the backflow has the quantum nature because states of particles are described by quantum wave functions. In other words we agree that in the quantum theory we encounter the quantum backflow. The main purpose of our work, however, was to contradict the statements made in some papers on the backflow, like 'The backflow effect is an intrinsically quantum phenomenon, for which there is no classical analog' [3] or 'The effect is inconceivable from the viewpoint of classical physics' [4]. * Author to whom any correspondence should be addressed.
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The phenomenon of backflow is found when we ask the question: can the probability current of a particle have negative values if the support of the wave function in Fourier space is restricted to positive values? Using this form of the question we can immediately see that it can be applied to any wave (quantum or classical and in any number of dimensions) satisfying a linear evolution equation. In more general terms, the backflow is the result of an interplay between the wave function in the position representation and its Fourier transform. The difference between various cases is only in the definition of density and its flow.
The authors of Comment 2 bring forth their quantum number λ max as a proof of the quantum nature of the backflow. We question this assertion because analogous 'quantum numbers' can be easily obtained in classical theories by using the same approach. In order to prove this, let us consider classical waves (waves in water or in air, even waves on a rubber rope) moving in one direction. These waves are well described in many cases [5] by the d'Alembert equation in one dimension, where v is the wave velocity. The general solution describing the waves running in positive direction of the x axis can be written as a Fourier integral, ) . ( The energy density of the wave field ρ e and the energy flux j e are, They satisfy the continuity equation, ∂ t ρ e + ∂ x j e = 0, like their counterparts in quantum theory. In order to mimic the situation in quantum theory, we will restrict the integration in (2) to positive values of the wave vector k. In this case the general solutions of the wave equation ) .
This expression can be written in terms of two real functions f (k) and g(k), ) .
We can now follow step by step the calculations that led the authors of [6] to their integral equation (43), It is not our aim in this Reply to develop a general theory but we want to show that the eigenvalues of some integral operators (called quantum numbers in Comment 2) arise in a purely classical theory. In order to prove this assertion, we will calculate the integral of the current j e as has been done by the authors of Comment 2. In the case when g(k) = 0 the quantity that directly corresponds to the formula (31) in [6] is equal to, where κ is the dimensionless wave vector κ = kvT. The energy norm of the wave function For normalized wave functions the search for the extremal value of (8) produces the following integral equation, where h(κ) = κf(κ) and the kernel has the form, The similarity with the 'quantum' integral equation (7) is striking. The only difference is in the form of the integral kernel which allows for the analytic determination of λ. This is possible because this kernel is an idempotent operator, Therefore the eigenvalue λ must be equal to 1/2. The same eigenvalue is obtained when both amplitudes f and g do not vanish. The idempotent kernel is then a 2 × 2 matrix and the proof that it is idempotent is more cumbersome.
This very simple example shows that the existence of 'quantum numbers' in the variational principles studied in [6,7] is present also in genuinely classical theories so that this property cannot be used as a proof of the quantum nature of the backflow.
In reply to the question why we have not mentioned in our article previous works on the backflow in the Dirac equation, raised by the authors of Comment 1, we have the following answers. In [5][6][7][8] quoted in this Comment the authors considered two-component wave functions in 1 + 1 dimensions. In this case, the use of the term the Dirac equation is inappropriate. Of course, there are solutions of the true Dirac equation that depend only on one spatial variable and on time. These are the wave packets made of plane waves moving in the same direction but they still must be described by the four-component bispinor. However, two-component wave functions considered in those references cannot accommodate spin degrees of freedom and also the particle and antiparticle components. In turn, in the [6] of Comment 1 the authors made a gross error by considering the superposition of positive and negative energy states. Such a superposition has no physical meaning. In our paper we dealt exclusively with the physical Dirac equation in 3 + 1 dimensions. We have not found it necessary to include in our paper the criticism of the artificial models considered in [5][6][7][8].

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).