The exponential of the spin representation of the Lorentz algebra

TL;DR

This paper extends the orthogonal decomposition of Lorentz algebra elements to their spin representation, providing formulas for the exponential and the spin representation of Lorentz transformations.

Contribution

It introduces a novel extension of Lorentz algebra decomposition to spin representations and derives explicit formulas for exponentials and transformations.

Findings

Derived a formula for the exponential of the spin representation.

Obtained a formula for the spin representation of proper orthochronous Lorentz transformations.

Extended the orthogonal decomposition concept to spin representations.

Abstract

As discussed in a previous article, any (real) Lorentz algebra element possess a unique orthogonal decomposition as a sum of two mutually annihilating decomposable Lorentz algebra elements. In this article, this concept is extended to the spin representation of the Lorentz algebra. As an application, a formula for the exponential of the spin representation is obtained, as well as a formula for the spin representation of a proper orthochronous Lorentz transformation.