# The Anomalous Magnetic Moment of a Photon Propagating in a Magnetic Field

## TL;DR

This paper investigates the spectrum and anomalous magnetic moment of a photon in a strong magnetic field near the Schwinger critical field, revealing that the photon’s magnetic moment increases with the field strength and providing numerical expressions for it.

## Contribution

It introduces a detailed analysis of the photon’s Hamiltonian spectrum and computes the anomalous magnetic moment in strong magnetic fields using both analytic and numerical methods.

## Key findings

- The Hamiltonian's expected value is a concave function of magnetic field B.
- The photon’s anomalous magnetic moment increases with B, up to 30 times the critical field.
- At B=30 B_cr, the magnetic moment is 8/3 times its value at B=0.5 B_cr.

## Abstract

We analyze the spectrum of the Hamiltonian of a photon propagating in a strong magnetic field $B\sim B_{\rm{cr}}$, where $B_{\rm cr}= \frac{m^2}{e} \simeq 4.4 \times 10^{13}$ Gauss is the Schwinger critical field . We show that the expected value of the Hamiltonian of a quantized photon for a perpendicular mode is a concave function of the magnetic field $B$. We show by a partially analytic and numerical method that the anomalous magnetic moment of a photon in the one loop approximation is a non - decreasing function of the magnetic field $B$ in the range $0\leq B \leq 30 \, B_{\rm cr}$ We provide a numerical representation of the expression for the anomalous magnetic moment in terms of special functions. We find that the anomalous magnetic moment $\mu_\gamma$ of a photon for $B=30\, B_{\rm cr }$ is $8/3$ of the anomalous magnetic moment of a photon for $B = 1/2 ~ B_{\rm cr}$.