The Exponential Map for the Conformal Group 0(2,4)

TL;DR

This paper develops a general, metric-independent method to compute the exponential map for orthogonal groups, specifically applying it to the conformal group SO(2,4), aiding in advanced gauge transformation theories.

Contribution

It introduces a novel, explicit formula for the exponential of generators of SO(2,4), extending the mathematical tools for conformal and spin gauge transformations.

Findings

Derived a closed-form exponential map for SO(2,4) generators.

Provided new expressions for secular equation coefficients.

Enabled generalization of gauge transformations in theoretical physics.

Abstract

We present a general method to obtain a closed, finite formula for the exponential map from the Lie algebra to the Lie group, for the defining representation of the orthogonal groups. Our method is based on the Hamilton-Cayley theorem and some special properties of the generators of the orthogonal group, and is also independent of the metric. We present an explicit formula for the exponential of generators of the $SO_+(p,q)$ groups, with $p+q = 6$, in particular we are dealing with the conformal group $SO_+(2,4)$, which is homomorphic to the $SU(2,2)$ group. This result is needed in the generalization of U(1) gauge transformations to spin gauge transformations, where the exponential plays an essential role. We also present some new expressions for the coefficients of the secular equation of a matrix.