Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and $c \rightarrow \infty $ Limit

TL;DR

This paper develops a Lorentz covariant quantum mechanics framework using group theory and pseudo-Hermitian operators, providing explicit wavefunctions and recovering classical and Galilean limits through algebraic contractions.

Contribution

It introduces a novel group theoretical formulation of Lorentz covariant quantum mechanics based on pseudo-unitary representations and pseudo-Hermitian operators, extending the algebraic structure of observables.

Findings

Explicit wavefunction description with finite inner product

Covariant harmonic oscillator basis with standard properties

Rigorous derivation of Galilean and classical limits

Abstract

We present in the article the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg-Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors under the Lorentz symmetry. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group $C^*$-algebra giving the algebra of observables. The key feature of the formulation is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators of any `dimension'. Galilean limit of the Lorentz symmetry and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.