The role of Lorentz symmetry in noncommutative field theory is considered. Any realistic noncommutative theory is found to be physically equivalent to a subset of a general Lorentz-violating standard-model extension involving ordinary fields. Some theoretical consequences are discussed. Existing experiments bound the scale of the noncommutativity parameter to ͑10 TeV͒ 22 . DOI: 10.1103/PhysRevLett.87.141601 PACS numbers: 11.30.Cp, 02.40.Gh, 12.20.Fv The idea that spacetime may intrinsically involve noncommutative coordinates has undergone a recent revival following the realization that this occurs naturally in string theory [1]. In this framework, the commutator of the coordinates x m in the spacetime manifold iswhere u mn is real and antisymmetric. It is of interest to speculate that the physical world might involve noncommutative coordinates and to ask about current experimental sensitivity to putative realistic noncommutative quantum field theories.The primary goal of this work is to study a physical issue that is central to any realistic noncommutative theory: the role of Lorentz symmetry. Violations of Lorentz symmetry are intrinsic to noncommutative theories by virtue of nonzero u mn in Eq.(1). Our study of these violations is motivated partly by theoretical progress in understanding the physics associated with Lorentz violation in ordinary quantum field theory and partly by recent experimental advances that make Lorentz tests among the most sensitive null experiments in existence [2].One approach to constructing a noncommutative quantum field theory is to promote an established ordinary theory to a noncommutative one by replacing ordinary fields with noncommutative fields and ordinary products with Moyal products, defined byFor gauge theories such as quantum electrodynamics (QED), ordinary gauge transformations must be modified to noncommutative generalizations. For noncommutative QED [3], the Hermitian Lagrangian is . This poses difficulties for a noncommutative generalization of the standard model, which would require other values for hypercharge assignments. In fact, noncommutative QED is similar to U͑N͒ gauge theory as N !`, and the allowed representations are the adjoint, fundamental, and antifundamental. In D-brane physics, adding two D-branes of charge 1 under a noncommutative U(1) leads to noncommutative U(2) gauge theory, which has non-Abelian U(2) gauge theory as its commutative limit instead of U (1) with charge 2. The implementation of Lorentz transformations in a noncommutative theory is more involved than usual because the parameter u mn carries Lorentz indices. Two distinct types of Lorentz transformation exist [4]. For example, Eq. (3) is fully covariant under observer Lorentz transformations: rotations or boosts of the observer inertial frame leave the physics unchanged because both the field operators and u mn transform covariantly. However, these coordinate changes differ profoundly from rotations or boosts of a particle or localized field configuration within a fixed observer frame...

Constraints from clock-comparison experiments on violations of Lorentz and CPT symmetry are investigated in the context of a general Lorentz-violating extension of the standard model. The experimental signals are shown to depend on the atomic and ionic species used as clocks. Certain experiments usually regarded as establishing comparable bounds are in this context sensitive to different types of Lorentz violation. Some considerations relevant to possible future measurements are presented. All these experiments are potentially sensitive to Lorentz-violating physics at the Planck scale.

We show that the general Lorentz-and CPT-violating extension of quantum electrodynamics is one-loop renormalizable. The one-loop Lorentz-violating beta functions are obtained, and the running of the coefficients for Lorentz and CPT violation is determined. Some implications for theory and experiment are discussed.

A method is presented for deriving the nonrelativistic quantum Hamiltonian of a free massive fermion from the relativistic Lagrangian of the Lorentz-violating standard-model extension. It permits the extraction of terms at arbitrary order in a Foldy-Wouthuysen expansion in inverse powers of the mass. The quantum particle Hamiltonian is obtained and its nonrelativistic limit is given explicitly to third order.

Precision experiments with muons are sensitive to Planck-scale CPT and Lorentz violation that is undetectable in other tests. Existing data on the muonium ground-state hyperfine structure and on the muon anomalous magnetic moment could be analyzed to provide dimensionless figures of merit for CPT and Lorentz violation at the levels of 4 3 10 221 and 10 223 . PACS numbers: 11.30.Er, 11.30.Cp, 13.40.Em, 14.60.Ef The minimal standard model of particle physics is CPT and Lorentz invariant. However, spontaneous breaking of these symmetries may occur in a more fundamental theory incorporating gravity [1,2]. Minuscule low-energy signals of CPT and Lorentz breaking could then emerge in experiments sensitive to effects suppressed by the ratio of a low-energy scale to the Planck scale. At presently attainable energies, the resulting effects would be described by a general standard-model extension [3] that allows for CPT and Lorentz violation but otherwise maintains conventional properties of quantum field theory, including gauge invariance, renormalizability, and energy conservation.In the present work, we study the sensitivity of different muon experiments to CPT and Lorentz violation. Planck-scale sensitivity to possible effects is known to be attainable in certain experiments without muons. These include, for example, tests with neutral-meson oscillations [4,5], searches for cosmic birefringence [3,6,7], clockcomparison experiments [8,9], comparisons of particles and antiparticles in Penning traps [10,11], spectroscopic comparisons of hydrogen and antihydrogen [12], measurements of the baryon asymmetry [13], and observations of high-energy cosmic rays [14]. However, in the context of the standard-model extension, dominant effects in the muon sector would be disjoint from those in any of the above experiments because the latter involve only photons, hadrons, and electrons. Moreover, if the size of CPT and Lorentz violation scales with mass, high-precision experiments with muons would represent a particularly promising approach to detecting lepton-sector effects from the Planck scale.The standard CPT test involving muons compares the g factors for m 2 and m 1 , with a bound [15,16] given by the figure of meritWe show here that data from experiments normally not associated with CPT or Lorentz tests, including muonium microwave spectroscopy [17] and g 2 2 experiments on m 1 alone [18], can indeed provide Planck-scale sensitivity to CPT and Lorentz violation.For the experiments considered here, it suffices to consider a quantum-electrodynamics limit of the standardmodel extension incorporating only muons, electrons, and photons. Other terms in the full standard-model extension would be irrelevant or lead only to subdominant effects. In natural units withh c 1, the Lorentz-violating Lagrangian terms of interest areHere, the lepton fields are denoted by l A with A 1, 2 corresponding to e 2 , m 2 , respectively, and iD l ϵ i≠ l 2 qA l with charge q 2jej. To avoid confusion with fourvector indices, the symbol m is reserved i...