Translational Invariance and Noncommutative Field Theories

TL;DR

This paper explores noncommutative field theories with specific coordinate commutators, revealing that while free quantum fields remain unaffected, translation invariance is broken, leading to modified energy-momentum relations with astrophysical implications.

Contribution

It demonstrates how nilpotent noncommutative structures affect translation invariance and energy-momentum conservation in quantum field theories, with applications to astrophysics.

Findings

Free quantum fields are unaffected by noncommutativity.

Translation invariance is broken in these theories.

Modified energy-momentum relations can explain astrophysical phenomena.

Abstract

Implications of noncommutative field theories with commutator of the coordinates of the form $[x^{\mu},x^{\nu}]=i \Lambda_{\quad \omega}^{\mu \nu}x^{\omega}$with nilpotent structure constants are investigated. It is shown that a free quantum field theory is not affected by noncommutativity, but that invariance under translations is broken and hence the energy-momentum conservation is not respected. The new energy-momentum law is expressed by a Poincar\'e-invariant equation and the resulting kinematics is developed and applied to the astrophysical puzzle related with the observed violation of the GZK cutoff.