Noncommutative field theory and violation of translation invariance

O. Bertolami; L. Guisado

doi:10.1088/1126-6708/2003/12/013

paper

Noncommutative field theory and violation of translation invariance

arXiv:hep-th/0306176 · doi:10.1088/1126-6708/2003/12/013

Abstract

Noncommutative field theories with commutator of the coordinates of the form $[x^{μ}, x^{ν}] = i Λ_{ω}^{μν} x^{ω}$ are studied. Explicit Lorentz invariance is mantained considering $Λ$ 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 Poincaré-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 $λ$ $Φ^{4}$ 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 $λ$ 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.

Final version published at JHEP

Noncommutative field theory and violation of translation invariance

Abstract

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