Dirac field on Moyal-Minkowski spacetime and non-commutative potential scattering

TL;DR

This paper develops a framework for quantizing the Dirac field on Moyal-deformed Minkowski spacetime, analyzing scattering processes and the resulting quantum field operators within a non-commutative geometric setting.

Contribution

It introduces a novel approach to quantum field theory on Moyal-deformed spacetime, linking spectral geometry with non-commutative potential scattering.

Findings

Scattering transformation is unitarily implementable in the vacuum representation.

Functional derivatives of scattering operators induce quantum field operators.

Framework connects non-commutative geometry with quantum field theory on curved spacetime.

Abstract

The quantized free Dirac field is considered on Minkowski spacetime (of general dimension). The Dirac field is coupled to an external scalar potential whose support is finite in time and which acts by a Moyal-deformed multiplication with respect to the spatial variables. The Moyal-deformed multiplication corresponds to the product of the algebra of a Moyal plane described in the setting of spectral geometry. It will be explained how this leads to an interpretation of the Dirac field as a quantum field theory on Moyal-deformed Minkowski spacetime (with commutative time) in a setting of Lorentzian spectral geometries of which some basic aspects will be sketched. The scattering transformation will be shown to be unitarily implementable in the canonical vacuum representation of the Dirac field. Furthermore, it will be indicated how the functional derivatives of the ensuing unitary scattering operators with respect to the strength of the non-commutative potential induce, in the spirit of Bogoliubov's formula, quantum field operators (corresponding to observables) depending on the elements of the non-commutative algebra of Moyal-Minkowski spacetime.