PT symmetry, Cartan decompositions, Lie triple systems and Krein space related Clifford algebras

TL;DR

This paper explores the mathematical structures underlying PT-symmetric quantum mechanics, revealing Lie and Clifford algebra frameworks that help understand gauge potentials, symmetries, and models like generalized Jaynes-Cummings in Krein spaces.

Contribution

It introduces new Lie triple and Clifford algebra structures in PTQM models, connecting them to gauge potentials, symmetries, and cavity QED systems with transmon states.

Findings

Lie triple structures in PTQM models with non-Abelian gauge potentials

Clifford algebra structures in models with Abelian gauge potentials

Identification of fundamental symmetries for Krein space J-selfadjoint extensions

Abstract

Gauged PT quantum mechanics (PTQM) and corresponding Krein space setups are studied. For models with constant non-Abelian gauge potentials and extended parity inversions compact and noncompact Lie group components are analyzed via Cartan decompositions. A Lie triple structure is found and an interpretation as PT-symmetrically generalized Jaynes-Cummings model is possible with close relation to recently studied cavity QED setups with transmon states in multilevel artificial atoms. For models with Abelian gauge potentials a hidden Clifford algebra structure is found and used to obtain the fundamental symmetry of Krein space related J-selfadjoint extensions for PTQM setups with ultra-localized potentials.