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4.2 Non-commutative spacetime

A common conjecture for the behavior of spacetime in quantum gravity is that the algebra of spacetime coordinates is actually noncommutative. This idea has led to a large amount of research in Lorentz violation and we would be remiss if we did not briefly discuss Lorentz violation from non-commutativity. We will look at only the most familiar form of spacetime non-commutativity, “canonical” non-commutativity, where the spacetime coordinates acquire the commutation relation
--1-- [xa,xb] = i/\2 Qab. (43) NC
Qab is an O(1) tensor that describes the non-commutativity and /\NC is the characteristic noncommutative energy scale. /\NC is presumably near the Planck scale if the non-commutativity comes from quantum gravity. However, in large extra dimension scenarios /\ NC could be as low as 1 TeV. For discussions of other types of non-commutativity, including those that preserve Lorentz invariance or lead to DSR-type theories, see [187225]. The phenomenology of canonical non-commutativity as it relates to particle physics can be found in [14798].

The existence of Qab manifestly breaks Lorentz invariance and hence the size of /\NC 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 [216Jump To The Next Citation Point] 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 [31Jump To The Next Citation Point75Jump To The Next Citation Point]. In order to see how constraints come about, consider for the moment non-commutative QED. The Seiberg-Witten map [254] can be used to express the non-commutative fields in terms of ordinary gauge fields. At lowest order in /\NC the effective action for low energy is then

1 -- <--> -- 1 S = -iygm Dm y - m yy - --FmnF mn 2 4 1- hab-- -- m <--> 1- hab-- --m <--> - 8 iq/\2 Fab yg Dm y + 4 iq/\2 Fam yg Db y NCab NC + 1-mq h---F yy 4 /\2NC ab 1 hab 1 hab - --q----FamFbnF mn + -q ----FabFmnF mn. (44) 2 /\2NC 8 /\2NC
Direct constraints on the dimension six non-renormalizable operators from cosmological birefringence and atomic clocks have been considered in [75Jump To The Next Citation Point]. A stronger bound of 14 /\NC > 5 × 10 GeV [218] on the non-commutativity scale can be derived from clock comparison experiments with Cs/Hg magnetometers [46Jump To The Next Citation Point] (see Section 5.2). Similarly, the possibility of constraints from synchrotron radiation in astrophysical systems has been analyzed in [77].

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 [31Jump To The Next Citation Point], 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 B(Q2)abFanF nb and AQ Q F amF bn ab mn, where A, B are dimensionless numbers that depend on /\ ,/\ NC. There are two different regimes of behavior for A, B. If /\ » /\NC then A, B are O(1) (up to loop factors and coupling coefficients), independent of the scale /\NC. 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 /\ « /\NC then A,B oc /\2//\2NC. The resulting field theory becomes a subset of the standard model extension; specifically the new operators have the form of the ab gd (kF)abgdF F term in Equation (35View Equation). It has been argued [75] that any realistic non-commutative theory must eventually reduce to part of the mSME. The approach of [31] 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 ab Q ,/\NC 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 /\NC and or /\.

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