Maximally localized states and causality in non commutative quantum theories

TL;DR

This paper explores maximally localized states in non-commutative quantum theories, using simple representations and Gaussian functions, and discusses implications for causality in non-commutative quantum field theories.

Contribution

It introduces a simple representation for non-commutative quantum theories and adapts the concept of maximally localized states, with implications for causality in such frameworks.

Findings

Gaussian functions serve as maximally localized states in 2+1D models

Representation reproduces Moyal star product results

Discussion on causality issues in non-commutative QFTs

Abstract

We give simple representations for quantum theories in which the position commutators are non vanishing constants. A particular representation reproduces results found using the Moyal star product. The notion of exact localization being meaningless in these theories, we adapt the notion of ``maximally localized states'' developed in another context . We find that gaussian functions play this role in a 2+1 dimensional model in which the non commutation relations concern positions only. An interpretation of the wave function in this non commutative geometry is suggested. We also analyze higher dimensional cases. A possible incidence on the causality issue for a Q.F.T with a non commuting time is sketched.