# The Exact Foldy-Wouthuysen Transformation for a Dirac Theory Revisited

## TL;DR

This paper generalizes the Exact Foldy-Wouthuysen transformation for Dirac Hamiltonians, introducing a new involution operator to handle complex Hamiltonians with CPT-Lorentz breaking terms, enabling explicit transformations and physical analysis.

## Contribution

It presents a novel form of the involution operator allowing the EFWT to be applied to more complex Hamiltonians with CPT-Lorentz breaking terms.

## Key findings

- Successfully constructed the EFWT for a Hamiltonian with 160 CPT-Lorentz breaking terms.
- Derived equations of motion and analyzed their physical implications.
- Demonstrated the applicability of the new transformation technique.

## Abstract

The Exact Foldy-Wouthuysen transformation (EFWT) method is generalized here. In principle, it is not possible to construct the EFWT to any Hamiltonian. The transformation conditions are the same but the involution operator has a new form. We took a special example and constructed explicitly the new involution operator that allows one to perform the transformation. We treat the case of the Hamiltonian with 160 possible CPT-Lorentz breaking terms, using this new technique. The transformation was performed and physics analysis of the equations of motion is shown.