On the $\Theta$-term in electrodynamics

TL;DR

This paper investigates the effects of adding a $ heta$-term to the electromagnetic Lagrangian, showing it does not alter the fundamental properties of the electromagnetic field at spatial infinity.

Contribution

It clarifies that the $ heta$-term does not modify the electromagnetic Lagrangian's signature or the boundary conditions at infinity.

Findings

The $ heta$-term does not change the Lagrangian's signature.

The $ heta$-term increases the negative kinetic energy component at infinity.

Magnetic fields remain absent at spatial infinity despite the $ heta$-term.

Abstract

The term $\Theta\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$, when added to the electromagnetic Lagrangian $-{1\over 16\pi}F^{\mu\nu}F_{\mu\nu}$, does not change the signature of the Lagrangian. Actually, it increases the part with negative kinetic energy term at the spatial infinity. For this reason it does not change the conclusion, that at the spatial infinity the magnetic part of the electromagnetic field should be absent.