Complementarity and phases in SU(3)

TL;DR

This paper explores phase operators in su(3), revealing non-uniqueness in their Hermitian form and proposing two approaches—SU(2) invariance and complementarity—to define them, with potential extensions to SU(n).

Contribution

It introduces phase operators for su(3) representations, highlighting their non-uniqueness and proposing two methods to define them, extending the concept beyond su(3).

Findings

su(3) polar decomposition does not yield a unique Hermitian phase operator

Two approaches to define phase operators: SU(2) invariance and complementarity

Potential generalization of results to SU(n) groups

Abstract

Phase operators and phase states are introduced for irreducible representations of the Lie algebra su(3) using a polar decomposition of ladder operators. In contradistinction with su(2), it is found that the su(3) polar decomposition does not uniquely determine a Hermitian phase operator. We describe two possible ways of proceeding: one based in imposing SU(2) invariance and the other based on the idea of complementarity. The generalization of these results to SU(n) is sketched.