Invariant scalar product on extended Poincare algebra

TL;DR

This paper calculates the invariant scalar product on the extended Poincare algebra using two methods, confirming the orthogonality of Poincare and gauge generators despite the algebra's infinite-dimensional nature.

Contribution

It provides a consistent calculation of the Killing form on an infinite-dimensional extended Poincare algebra using both trace and invariance methods.

Findings

Scalar product calculated consistently by both methods

Poincare generators are orthogonal to gauge generators

Handling infinities in the algebra was carefully managed

Abstract

Two methods can be used to calculate explicitly the Killing form on the Lie algebras. The first one is a direct calculation of the traces of the generators in a matrix representation of the algebra, and the second one is the usage of the group invariance of the scalar product. We use both methods in our calculation of the scalar product on the extended Poincare algebra in order to have a cross check of our results. The algebra is infinite-dimensional and requires careful treatment of the infinities. The scalar product on the extended algebra found by both methods coincides and the important conclusion which follows is that Poincare generators are orthogonal to the gauge generators.