Noncommutative field theory and violation of translation invariance

TL;DR

This paper explores noncommutative field theories with Lorentz-invariant coordinate commutators, revealing violations of translation invariance and energy-momentum conservation, with implications for quantum field behavior and astrophysical phenomena.

Contribution

It introduces a Lorentz-invariant noncommutative framework that demonstrates how translation invariance is broken, affecting quantum field theory and leading to novel IR divergences and modified dispersion relations.

Findings

Energy-momentum conservation is violated in the noncommutative setting.

Radiative corrections exhibit IR divergences and altered dispersion relations.

Application to astrophysics suggests potential explanations for GZK cutoff violations.

Abstract

Noncommutative field theories with commutator of the coordinates of the form $[x^{\mu},x^{\nu}]=i \Lambda_{\quad \omega}^{\mu \nu}x^{\omega}$ are studied. Explicit Lorentz invariance is mantained considering $\Lambda $ a Lorentz tensor. It is shown that a free quantum field theory is not affected. Since invariance under translations is broken, the conservation of energy-momentum is violated, obeying a new law which is expressed by a Poincar\'e-invariant equation. The resulting new kinematics is studied and applied to simple examples and to astrophysical puzzles, such as the observed violation of the GZK cutoff. The $\lambda $$\Phi ^{4}$ quantum field theory is also considered in this context. In particular, self interaction terms violate the usual conservation of energy-momentum and, hence, the radiative correction to the propagator is altered. The correction to first order in $\lambda $ is calculated. The usual UV divergent terms are still present, but a new type of term also emerges, which is IR divergent, violates momentum conservation and implies a correction to the dispersion relation.