Towards Quantum Noncommutative $\kappa$-deformed Field Theory

TL;DR

This paper develops a new $$-star product for quantized $$-deformed free fields, linking space-time noncommutativity with oscillator algebra deformation, and demonstrates its consistency with microcausality, bosonic statistics, and classical momentum conservation in a noncommutative quantum field theory.

Contribution

It introduces a novel $$-star product for $$-deformed fields, relating space-time noncommutativity to oscillator algebra deformation, and constructs a consistent perturbative framework.

Findings

The $$-deformed microcausality condition is satisfied.

Bosonic statistics are preserved in the $$-deformed Fock space.

Classical momentum conservation holds at interaction vertices.

Abstract

We introduce new $\kappa$-star product describing the multiplication of quantized $\kappa$-deformed free fields. The $\kappa$-deformation of local free quantum fields originates from two sources: noncommutativity of space-time and the $\kappa$-deformation of field oscillators algebra - we relate these two deformations. We demonstrate that for suitable choice of $\kappa$-deformed field oscillators algebra the $\kappa$-deformed version of microcausality condition is satisfied, and it leads to the deformation of the Pauli-Jordan commutation function defined by the $\kappa$-deformed mass shell. We show by constructing the $\kappa$-deformed Fock space that the use of $\kappa$-deformed oscillator algebra permits to preserve the bosonic statistics of n-particle states. The proposed star product is extended to the product of $n$ fields, which for $n=4$ defines the interaction vertex in perturbative description of noncommutative quantum $\lambda\phi^4$ field theory. It appears that the classical fourmomentum conservation law is satisfied at the interaction vertices.