Translational Invariance and Noncommutative Field Theories

Implications of noncommutative field theories with commutator of the coordinates of the form $[x^μ,x^ν]=i Λ_{\quad ω}^{μν}x^ω$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 Poincaré-invariant equation and the resulting kinematics is developed and applied to the astrophysical puzzle related with the observed violation of the GZK cutoff.

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