Formulation, Interpretation and Application of non-Commutative Quantum Mechanics

TL;DR

This paper develops a formal framework for non-commutative quantum mechanics, extending standard interpretations and analyzing physical implications for symmetries and specific systems like free particles and oscillators.

Contribution

It introduces a Hilbert space formulation for non-commutative quantum mechanics and discusses its interpretation, symmetry properties, and physical signatures.

Findings

Formalism based on Hilbert-Schmidt operators is consistent with quantum interpretation.

Non-commutativity affects rotational and time reversal symmetries.

Physical signatures of non-commutativity are identified in specific systems.

Abstract

In analogy with conventional quantum mechanics, non-commutative quantum mechanics is formulated as a quantum system on the Hilbert space of Hilbert-Schmidt operators acting on non-commutative configuration space. It is argued that the standard quantum mechanical interpretation based on Positive Operator Valued Measures, provides a sufficient framework for the consistent interpretation of this quantum system. The implications of this formalism for rotational and time reversal symmetry are discussed. The formalism is applied to the free particle and harmonic oscillator in two dimensions and the physical signatures of non commutativity are identified.