# Electromagnetic coupling of strongly non-local quantum mechanics

## TL;DR

This paper investigates electromagnetic coupling in non-local quantum mechanics extensions where probability current conservation fails, proposing a modified Maxwell theory with additional scalar fields to address these issues.

## Contribution

It introduces a simple extension of Maxwell theory incorporating scalar degrees of freedom for non-local quantum models with non-conserved probability current.

## Key findings

- Modified Maxwell equations include additional current terms.
- Differences from standard Maxwell theory in solutions and physical predictions.
- Potential applications in non-local quantum systems.

## Abstract

Although standard quantum mechanics has some non-local features, the probability current of the Schr\"odinger equation is locally conserved, and this allows minimal electromagnetic coupling. For some important extensions of the Schr\"odinger equation, however, the probability current is not locally conserved. We show that in these cases the correct electromagnetic coupling requires a relatively simple extension of Maxwell theory which has been known for some time and recently improved by covariant integration of a scalar degree of freedom. We discuss some general properties of the solutions and examine in particular the case of an oscillating dipolar source. Remarkable mathematical and physical differences emerge with respect to Maxwell theory, as a consequence of additional current terms present in the equations for $\nabla \cdot \textbf{E}$ and $\nabla \times \textbf{B}$. Several possible applications are mentioned.