Lorentz (non-)violation in NC space-times

If the not-so-recent paper [1] hep-ph/0603137 is correct, the interesting and nontrivial approach to NC Lorentz invariance of  [2] hep-th/0206035 has some deep problems…

In [2] and subsequent papers (see the references in [1]), NC theories are defined in a way to avoid not only the observer Lorentz violation, but also the particle Lorentz violation (a very good and famous paper on this issue is hep-th/0105082 ). The strongest constraints on the NC phenomenology — if NC is to be seen as a fundamental theory — comes from the very precise measurements of Lorentz invariance; in particular, there is no sign of a fundamental anisotropy in our universe (no privileged direction), while any canonical NC theory in 4D space-time naturaly defines two privileged directions in space (one of them can be assumed to be zero at least in a certain reference frame): one behaves as a vector in space and is given by the components \theta^{0i}, having a "electric-like" nature; the other behaves as a pseudo-vector in space, similarly to a magnectic field, and is given by the Hodge dual of theta (in space).

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