# Relativistic Quantum Backflow

## TL;DR

This paper develops a comprehensive theory of relativistic quantum backflow, deriving eigenfunctions and eigenvalues, and reveals that the relativistic case can be simpler than the non-relativistic limit.

## Contribution

It introduces a new relativistic framework for quantum backflow, providing numerical eigenfunctions, explicit eigenvalue expressions, and improved fitting methods for higher backflow.

## Key findings

- Relativistic backflow can be expressed as a function of a simple fundamental parameter.
- Explicit relativistic eigenvalues are derived in terms of non-relativistic constants.
- Relativistic theory is shown to be intrinsically simpler than the non-relativistic case.

## Abstract

In this paper we discuss relativistic quantum backflow. The general theory of relativistic backflow is written down and it is shown that the backflow can be written as a function of a simple parameter which is defined in terms of fundamental constants and the backflow period. Backflow eigenfunctions are determined numerically for a range of values of and an explicit expression for the relativistic backflow eigenvalue in terms of the non-relativistic backflow constant is presented. Then backflow eigenvectors are fitted with some standard functions which lead to substantially higher backflow than has been found previously with fitting procedures, for some values of. 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.