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Noncommutative field theory and violation of translation invariance

Noncommutative field theories with commutator of the coordinates of the form $[x^μ,x^ν]=i Λ_{\quad ω}^{μν}x^ω$ 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.