Validity of the Gell-Mann formula for sl(n,R) and su(n) algebras

TL;DR

This paper investigates the conditions under which the Gell-Mann formula accurately reconstructs generators of sl(n,R) and su(n) algebras, clarifying its applicability and providing explicit matrix element expressions.

Contribution

The paper clarifies the validity constraints of the Gell-Mann formula for specific algebras and supplies explicit generator expressions for applicable representations.

Findings

Validity constraints for Gell-Mann formula are identified.

Complete list of applicable representation spaces is provided.

Explicit generator matrix elements are derived for certain cases.

Abstract

The so called Gell-Mann formula, a prescription designed to provide an inverse to the Inonu-Wigner Lie algebra contraction, has a great versatility and potential value. This formula has no general validity as an operator expression. The question of applicability of Gell-Mann's formula to various algebras and their representations was only partially treated. The validity constraints of the Gell-Mann formula for the case of sl(n,R) and su(n) algebras are clarified, and the complete list of representations spaces for which this formula applies is given. Explicit expressions of the sl(n,R) generators matrix elements are obtained, in these cases, by making use of the Gell-Mann formula.