On second quantization on noncommutative spaces with twisted symmetries

TL;DR

This paper develops a covariant framework for second quantization on noncommutative spaces with twisted symmetries, preserving key quantum properties and illustrating with examples in non-relativistic and relativistic QFT.

Contribution

It introduces a twist-induced star-deformation method for second quantization on noncommutative spacetimes, maintaining quantum axioms and Bose-Fermi statistics.

Findings

Deformation of spacetime algebra and symmetries consistent with quantum mechanics

Preservation of field (anti)commutator structure in noncommutative setting

Application to non-relativistic and relativistic QFT examples

Abstract

By application of the general twist-induced star-deformation procedure we translate second quantization of a system of bosons/fermions on a symmetric spacetime in a non-commutative language. The procedure deforms in a coordinated way the spacetime algebra and its symmetries, the wave-mechanical description of a system of n bosons/fermions, the algebra of creation and annihilation operators and also the commutation relations of the latter with functions of spacetime; our key requirement is the mode-decomposition independence of the quantum field. In a conservative view, the use of noncommutative coordinates can be seen just as a way to better express non-local interactions of a special kind. In a non-conservative one, we obtain a covariant framework for QFT on the corresponding noncommutative spacetime consistent with quantum mechanical axioms and Bose-Fermi statistics. One distinguishing feature is that the field commutation relations remain of the type "field (anti)commutator=a distribution". We illustrate the results by choosing as examples interacting non-relativistic and free relativistic QFT on Moyal space(time)s.