# Wigner Rotation: Theory and Application to Practical Relativistic Engineering Problems

## TL;DR

This paper corrects the common misconceptions about the Wigner rotation in relativistic physics, showing that traditional expressions are incorrect and clarifying their implications for accelerator physics and the BMT equation.

## Contribution

It provides a detailed analysis of the correct Wigner rotation expression and explains why previous textbook derivations were physically incorrect.

## Key findings

- Standard expressions overestimate the Wigner rotation angle by a factor of γ.
- Incorrect derivations in textbooks are based on flawed physical arguments.
- Corrected Wigner rotation expressions impact relativistic accelerator physics.

## Abstract

The standard expression for the Wigner rotation leads to incorrect result for the wavefront rotation. The results for the Wigner rotation in the Lorentz lab frame obtained by many experts on special relativity such as Moeller and Jackson, are incorrect. They overestimate the angle of the Wigner rotation by a factor $\gamma$ compared to its real value, and the direction of the rotation is also determined incorrectly. In 1959, Bargman, Michel, and Telegdi (BMT) proposed a consistent relativistic theory for the dynamics of the spin as observed in the lab frame, which was successfully tested in experiments. The BMT equation naturally involves the Wigner rotation as the purely kinematic addition to the Larmor rotation which, in turn, is a consequence of interaction of the intrinsic magnetic moment with the external magnetic field. It is commonly believed that the BMT equation contains the standard (and incorrect) result for the Wigner rotation in the Lorentz lab frame. The existing textbooks then suggest that the experimental test of the BMT equation is a direct proof of validity for the standard expression for Wigner rotation. Here we focus on the analysis of the reason why authors of textbooks obtained an incorrect expression for the Wigner rotation. We demonstrate that the notion that the standard (incorrect) result for the Wigner rotation as an integral part of the BMT equation in most texts is based, in turn, on an incorrect physical argument. The aim of the present paper is to analyze the complicated situation relating to the use of the theory of relativity and, in particular, Wigner rotation theory, in accelerator physics.