Canonical Coherent States for the Relativistic Harmonic Oscillator

TL;DR

This paper develops covariant relativistic coherent states for the harmonic oscillator, introduces higher-order creation and annihilation operators with canonical commutation relations, and constructs a minimal configuration space representation.

Contribution

It introduces a new set of relativistic coherent states with canonical commutation relations and a minimal configuration space representation, advancing the understanding of relativistic quantum systems.

Findings

Constructed covariant relativistic coherent states on the complex plane.

Introduced higher-order relativistic creation and annihilation operators with canonical commutation.

Established a minimal configuration space representation using eigenstates of a canonical position operator.

Abstract

In this paper we construct manifestly covariant relativistic coherent states on the entire complex plane which reproduce others previously introduced on a given $SL(2,R)$ representation, once a change of variables $z\in C\rightarrow z_D \in $ unit disk is performed. We also introduce higher-order, relativistic creation and annihilation operators, $\C,\Cc$, with canonical commutation relation $[\C,\Cc]=1$ rather than the covariant one $[\Z,\Zc]\approx$ Energy and naturally associated with the $SL(2,R)$ group. The canonical (relativistic) coherent states are then defined as eigenstates of $\C$. Finally, we construct a canonical, minimal representation in configuration space by mean of eigenstates of a canonical position operator.