The existence of manifestly breaks Lorentz invariance and hence the size of is constrained by tests of Lorentz violation. However, in order to match a non-commutative theory to low energy observations, we must have the appropriate low energy theory, which implies that the infamous UV/IR mixing problem of non-commutative field theory must be tamed enough to create a well-defined low energy expansion. No general method for doing this is known, although supersymmetry  can perhaps do the trick.9 If the UV/IR mixing is present but regulated by a cutoff, then the resulting field theory can be re-expressed in terms of the mSME [31, 75]. In order to see how constraints come about, consider for the moment non-commutative QED. The Seiberg-Witten map  can be used to express the non-commutative fields in terms of ordinary gauge fields. At lowest order in the effective action for low energy is then. A stronger bound of  on the non-commutativity scale can be derived from clock comparison experiments with Cs/Hg magnetometers  (see Section 5.2). Similarly, the possibility of constraints from synchrotron radiation in astrophysical systems has been analyzed in .
Other strong constraints can be derived by noting that without a custodial symmetry loop effects with the dimension six operators will induce lower dimension operators. In , the authors calculated what dimension four operators would be generated, assuming that the field theory has some cutoff scale . The dimension six operators induce dimension four operators of the form and , where are dimensionless numbers that depend on . There are two different regimes of behavior for . If then are (up to loop factors and coupling coefficients), independent of the scale . Such strong Lorentz violation is obviously ruled out by current experiment, implying that in this perturbative approach such a limit is observationally not viable. If instead one takes then . The resulting field theory becomes a subset of the standard model extension; specifically the new operators have the form of the term in Equation (35). It has been argued  that any realistic non-commutative theory must eventually reduce to part of the mSME. The approach of  shows this is possible, although the presence of such a low energy cutoff must be explained.
All of the above approaches use an expansion in to get some low energy effective field theory. In terms of Lorentz tests, the results are all based upon this EFT expansion and not on the full non-commutative theory. Therefore we will restrict ourselves to discussing limits on various terms in effective field theories rather than directly quoting limits on the non-commutative scale. We leave it up to the reader to translate this value into a constraint (if any) on and or .
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