Noncommutative Field Theory and Lorentz Violation

Abstract

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...

Lorentz violation bounds on Bhabha scattering

“…The interest in this issue appears in different contexts, such as supersymmetric models [5,6], noncommutative geometry [7], gravity and cosmology [8][9][10][11], high derivative models [12][13][14], renormalization [15][16][17][18] and scattering processes [19,20] in quantum electrodynamics (QED), condensed matter systems [21][22][23], and so on. Following these theoretical developments, many experimental tests on Lorentz-violating (LV) corrections have also been carried out and several constraints on LV parameters were established [24].…”

Noncommutative gravitational quantum well

“…Another interesting feature of noncommutative geometry is its direct connection with the breaking of the Lorentz invariance [7]. The violation of the Lorentz sym- * Email address: orfeu@cosmos.ist.utl.pt † Email address: joaopedrotgr@sapo.pt ‡ Email address: cristiane.aragao@ct.infn.it § Email address: paolo.castorina@ct.infn.it ¶ Email address: dario.zappala@ct.infn.it metry arises directly from the commutator of the coordinates x µ on the space-time manifold, which can be written as:…”

Noncommutative magnetic moment of charged particles

“…However, for studying phenomenological aspects of an NC theory it is advisable [10] to perform the SW map, since it allows one to work with commuting electromagnetic fields A µ that have standard U (1) gauge transformation properties. It is known that in the lowest nontrivial approximation in the NC parameter θ, to which approximation we shall henceforth restrict ourselves, the fieldǍ µ is SW-mapped aš…”