Limits on the Non-commutativity Scale

A non-vanishing vacuum expectation value for an antisymmetric tensor field leads to the violation of Lorentz invariance, controlled by the dimension (-2) parameter, theta_{mu nu}. We assume that the zeroth order term in theta-expansion represents the Standard Model and study the effects induced by linear terms in theta_{mu nu}. If coupling to theta_{mu nu} is realized in strongly interacting sector of the theory, the clock comparison experiments place the limit on the possible size of this background at the level of 1/\sqrt{theta}>~5X10^{14} GeV. If the interaction with theta_{mu nu} is initially present only in the QED sector, this limit can be relaxed to 10^{11}-10^{12} GeV level. The strength of these limits obviates the inferiority of collider physics with regard to experimental checks of Lorentz invariance. Limits of similar strength are expected to hold in the case of mixed non-commutativity between four-dimensional and extra-dimensional coordinates. We also show that in certain models mixed non-commutativity can be interpreted as 4d CPT-violating background.