Quantum Backflow: General Relativistic Formulation

• The paper introduces a general formulation of relativistic quantum backflow using the 1D Dirac equation, defining an eigenvalue problem for the backflow operator.
• It employs a single dimensionless parameter, epsilon, to encode all relativistic effects, connecting the relativistic and non-relativistic regimes through numerical and analytic methods.
• The work demonstrates enhanced numerical stability and practical analytic trial functions, deepening our understanding of quantum-classical transitions in nonlocal transport phenomena.

Quantum backflow is a nonclassical effect in which, despite a quantum particle’s momentum being supported entirely on positive values, the probability current can be negative—leading to a net flow of probability density oppositely to the momentum direction. The general formulation of this effect has been extended to relativistic contexts, in particular to the 1D Dirac equation for spin-1/2 particles, where the mathematical structure, its eigenvalue problem, numerical and analytic solutions, and non-relativistic limits have been precisely characterized (Ashfaque et al., 2019). This formulation reveals both similarities and qualitative differences with its non-relativistic analog, especially regarding the structure of the backflow kernel and the emergence of a single dimensionless parameter encoding all relativistic effects.

1. Relativistic Backflow: Operator Formalism and Eigenvalue Problem

Consider a spinless, zero-charge Dirac particle of mass $m$ in one space dimension, governed by the two-component Dirac equation

$$i\hbar\partial_t\psi(x,t) = \hat H\, \psi(x,t), \quad \hat H = c\,\sigma_1 \hat p + \sigma_3 m c^2,\quad \hat p = -i\hbar \partial_x.$$

The probability current at spatial point $x=0$ is

$$j(0, t) = c\,\psi^\dagger(0, t) \sigma_1 \psi(0, t).$$

A general positive-energy, positive-momentum wave packet is constructed as

$$\Psi(x, t) = \frac{1}{\sqrt{2\pi\hbar}} \int_0^\infty \Phi(p)\, e^{i(px - E(p)t)/\hbar}\, dp,$$

with energy dispersion $E(p) = \sqrt{p^2 c^2 + m^2 c^4}$ and a two-component spinor

$$\Phi(p) = f(p) \begin{pmatrix} U_1(p) \ U_2(p) \end{pmatrix}, \quad U_1(p) = \sqrt{\frac{\gamma(p) + 1}{2\gamma(p)}}, \quad U_2(p) = \sqrt{\frac{\gamma(p) - 1}{2\gamma(p)}},$$

where $\gamma(p) = \sqrt{1 + p^2/(m^2 c^2)}$ and normalization is $\int_0^\infty |f(p)|^2 dp = 1$.

The total (integrated) probability flux across $x=0$ over a period $T$ is

$\Delta = \int_0^T j(0, t)\,dt.$

The maximal (most negative) possible value of $\Delta$, subject to normalization of $f$, captures the maximal relativistic backflow.

To find this extremum, introduce dimensionless variables

$$p = \sqrt{\frac{4m\hbar}{T}}\,r,\quad q = \sqrt{\frac{4m\hbar}{T}}\,s,\quad \epsilon = \sqrt{\frac{4\hbar}{mc^2 T}},$$

such that $E(p) = m c^2 \gamma(r)$, $\gamma(r) = \sqrt{1 + (\epsilon r)^2}$, and

$$\eta(r) = \exp\left[-2i \frac{\gamma(r)}{\epsilon^2}\right] f(mc\epsilon r).$$

The maximal negative flux becomes the most negative eigenvalue $\lambda$ of the integral equation

$$\int_0^\infty K_\epsilon(r, s)\, \eta(s)\, ds = \lambda\, \eta(r),$$

where the kernel is

$$K_\epsilon(r, s) = -\frac{1}{\pi} \frac{r[\gamma(s) + 1] + s[\gamma(r) + 1]} {\sqrt{\gamma(r)(\gamma(r) + 1)\gamma(s)(\gamma(s) + 1)}} \frac{\sin\left[2(\gamma(r) - \gamma(s))/\epsilon^2\right]} {2(\gamma(r) - \gamma(s))/\epsilon^2}.$$

This establishes the general relativistic backflow operator and its eigenproblem.

2. Single-Parameter Structure and Non-relativistic Limit

All relativistic effects in the backflow problem are encapsulated in the single dimensionless parameter

$\epsilon = \sqrt{\frac{4\hbar}{mc^2 T}},$

where $T$ is the backflow period (time interval), $c$ is the speed of light, $m$ the particle mass, and $\hbar$ Planck’s constant. The magnitude of the largest negative eigenvalue, denoted $c_{rbf}(\epsilon) = |\lambda(\epsilon)|$, interpolates between zero (ultrarelativistic or short $T$, large $c$) and the universal non-relativistic constant $c_{bf}$,

$$c_{bf} = \lim_{\epsilon\to 0} c_{rbf}(\epsilon) \approx 0.0384517.$$

An explicit fit for $c_{rbf}(\epsilon)$ over the range $0 \leq \epsilon \lesssim 2.5$ is given by

$$c_{rbf}(\epsilon) \approx c_{bf}\, \exp\left[ -\frac{4\,\epsilon}{9}(1 - 4\alpha\epsilon) \right],$$

with $\alpha = e^2/(4\pi\hbar c)$ the fine-structure constant. In the non-relativistic limit $c \to \infty$ or $T \to \infty$, $\epsilon \to 0$ and $c_{rbf} \to c_{bf}$.

3. Integral Equations, Numerical Results, and Analytic Trial Functions

Non-relativistic Limit

For $\epsilon \to 0$, the kernel reduces to the Bracken–Melloy sine-kernel, yielding the canonical integral equation for standard backflow: $$\frac{1}{\pi}\int_0^\infty \frac{\sin(r^2 - s^2)}{r - s}\, \eta(s)\,ds = \lambda\, \eta(r).$$ The largest negative eigenvalue here defines the Bracken–Melloy constant $c_{bf}$.

Numerical Maximal Backflow

Sample numerical values of $c_{rbf}(\epsilon)$ are:

$\epsilon$	0.10	0.50	0.80	1.00	1.60	2.00	2.50
$c_{rbf}$	0.0369	0.0309	0.0272	0.0250	0.0195	0.0166	0.0137

These results rigorously establish that relativistic effects reduce the possible maximal backflow as $\epsilon$ grows.

Analytic Trial Function Fits

High-fidelity analytic fits to the numerically obtained eigenfunctions are possible via two ansätze:

• Airy-type:

$$f_{\rm Ai}(r) = \frac{\mathrm{Ai}\left[a_1(r + a_2)^{a_3}\right]}{(a_4 r + a_5)^{a_6}}$$

• Bessel-type:

$$f_{J}(r) = \frac{J_0\left[a_1(r + a_2)^{a_3}\right]}{(a_4 r + a_5)^{a_6}}$$

For $\epsilon \approx 0.9$, Bessel fits with parameters $$(a_1, a_2, a_3, a_4, a_5, a_6) \approx (-1.347, 0.603, 0.986, 0.341, 0.435, 0.715)$$ attain over $99\,\%$ of numerically exact $c_{rbf}$, while Airy fits consistently recover over $90\,\%$ in the same regime. For smaller $\epsilon$, the problem becomes highly non-convex in parameter space and analytic fits detoriate due to oscillatory structure in the eigenfunctions.

4. Mathematical and Numerical Properties: Simplicity of the Relativistic Kernel

While the non-relativistic Bracken–Melloy problem involves an oscillatory sine-kernel with no small parameter to regularize high-frequency oscillations, the relativistic kernel contains a $$\sin[\Delta\gamma/\epsilon^2]/(\Delta\gamma/\epsilon^2)$$ factor. For intermediate $\epsilon$, this acts as an effective regularization, smoothing the eigenproblem and suppressing large-$r, s$ fluctuations.

Consequently:

• Numerical eigenvalue routines demonstrate enhanced stability and faster convergence for $\epsilon \gtrsim 0.5$ compared to $\epsilon \to 0$.
• The relativistic backflow integral operator is, practically, more numerically tractable than its non-relativistic counterpart.
• Despite being quantum in origin, the Bracken–Melloy constant survives even in the naive classical limit $\hbar \to 0$, as $\epsilon \to 0$ with $c_{rbf} \to c_{bf} \neq 0$.

5. Physical and Foundational Implications

The relativistic generalization demonstrates that quantum backflow persists in the Dirac theory, with the maximal effect controlled precisely by the parameter $\epsilon$. Approaching the non-relativistic regime retrieves the classic backflow constant. The existence of analytic, nearly optimal trial states in the relativistic case offers practical advantages for theoretical proposals and numerical studies.

The counterintuitive result that the relativistic problem is more tractable arises from the presence of the dimensionless regularizing parameter. In the non-relativistic limit, this regularization is lost, and one is left with the more singular sine-kernel characteristic of the Bracken–Melloy problem.

This structure also raises questions about the connection between operator orderings, spectral properties of quantum mechanical currents, and quantum–classical correspondence. The invariance of the backflow constant under $\hbar \to 0$—until practical measurements (with finite precision) are considered—highlights the subtlety of quantum-to-classical transitions for nonlocal dynamical observables.

6. Summary

The general formulation for relativistic quantum backflow is based on the spectral theory of a one-parameter integral operator defined via the 1D Dirac equation. The problem reduces to finding the most negative eigenvalue of an explicitly constructed kernel, with all dependence on system parameters entering through $\epsilon$. Analytic fits using Airy and Bessel functions provide efficient approximate eigenstates, and numerical results across the full $\epsilon$ range quantitatively map the transition from relativistic depletion of backflow to the non-relativistic universal bound. The relativistic theory’s superior numerical conditioning, the presence of a regularizing parameter, and the resulting insights into the quantum-classical interface position this formulation as a central reference for both foundational studies and practical modeling of nonclassical transport phenomena in relativistic quantum mechanics (Ashfaque et al., 2019).