Noncommutative spaces with twisted symmetries and second quantization

TL;DR

This paper develops a framework for quantum field theory on noncommutative spaces using twist-induced deformations, simplifying the description of multi-particle systems while maintaining key quantum properties.

Contribution

It introduces a twist-based deformation method for second quantization on noncommutative spaces, preserving quantum statistics and covariance.

Findings

Deformation simplifies multi-particle wavefunctions.

Field (anti)commutation relations are preserved.

Framework applies to non-relativistic and relativistic QFTs.

Abstract

In a minimalistic view, the use of noncommutative coordinates can be seen just as a way to better express non-local interactions of a special kind: 1-particle solutions (wavefunctions) of the equation of motion in the presence of an external field may look simpler as functions of noncommutative coordinates. It turns out that also the wave-mechanical description of a system of n such bosons/fermions and its second quantization is simplified if we translate them in terms of their deformed counterparts. The latter are obtained by a general twist-induced *-deformation procedure which deforms in a coordinated way not just the spacetime algebra, but the larger algebra generated by any number n of copies of the spacetime coordinates and by the particle creation and annihilation operators. On the deformed algebra the action of the original spacetime transformations looks twisted. In a non-conservative view, we thus obtain a twisted 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.