aa r X i v : . [ h e p - ph ] O c t IUHET 550, August 2010
TOPICS IN LORENTZ AND CPT VIOLATION
V. ALAN KOSTELECK ´Y
Physics Department, Indiana UniversityBloomington, IN 47405, USA
This talk given at the CPT’10 meeting provides a brief introduction to Lorentzand CPT violation and outlines a few recent developments in the subject.
1. Introduction
The possibility that Lorentz violation might be manifest in nature, perhapswith attendant CPT violation, continues to attract attention from experi-mentalists and theorists alike. In the CPT’07 Proceedings, I outlined howthe triple requirements of coordinate independence, realism, and generalitylead to the conclusion that effective field theory is the appropriate frame-work for studying Lorentz and CPT violation. The present CPT’10 talkprovides some introductory comments about this framework.The comprehensive effective field theory incorporating General Rel-ativity (GR) and the Standard Model can be constructed by combin-ing all Lorentz-violating operators together with controlling coefficients toform observer-invariant terms in the Lagrange density. This theory is theStandard-Model Extension (SME).
A useful limit is the miminal SME,which restricts operators to mass dimension d ≤ the SME also describes general CPT violation.Many observable effects arise from the interactions of particles with thecoefficients, varying with velocity, spin, flavor, and couplings. Numeroussearches have been performed, but no compelling positive measurementexists to date. Some intriguing current prospects for signals include, amongothers, oscillations of neutrinos and neutral mesons. Additional effects occur for spontaneous Lorentz violation because thecoefficients can then fluctuate, yielding massless Nambu-Goldstone (NG)modes for the broken generators and also massive modes. The NG modes can be identified directly with the photon in Einstein-Maxwell the-ory, the graviton in GR, a spin-dependent force, or a spin-independentforce, or they can generate composite photons or gravitons.
2. Nonminimal terms
In the full SME with nonminimal terms, infinitely many possible Lorentz-violating operators become candidates for inclusion in the Lagrange density.As a result, enumerating these operators and determining their physicaleffects becomes challenging.For operators of arbitrary mass dimension d , a systematic investigationhas so far been performed only in the photon sector. This investigationstudied all operators quadratic in the photon field A µ , allowing for arbitraryspacetime derivatives. The resulting explicit gauge-invariant action revealsthat the number of Lorentz-violating operators grows rapidly: the minimalSME has 4 operators at d = 3 and 19 at d = 4, but 36 nonminimal onesappear at d = 5, 126 at d = 6, and the growth is cubic with d at large d .Each of these numerous operators produces a distinct Lorentz-violatingeffect on photon propagation. In some respects, the behavior of SME pho-tons is analogous to Maxwell photons moving in an anisotropic dispersivecrystal. For example, Lorentz violation can cause light to exhibit modeseparation (birefringence), pulse deformation (dispersion), and directiondependence (anisotropy). Certain coefficients for Lorentz violation can bedetected at leading order by studying propagation in the vacuum, whileothers require nonvacuum boundary conditions. The details of these effectsdepend on features of the specific radiation being considered, such as itsfrequency, polarization, and direction of travel. Surprisingly, this plethoraof new effects is almost unexplored in relativity tests. No dedicated lab-oratory experiments have searched for these behaviors, and the existingastrophysical tests are limited to a few comparatively simple cases.For coefficients governing leading-order birefringence in the vacuum, themost sensitive tests involve polarimetry of astrophysical sources. Birefrin-gent effects are controlled by the ratio of the wavelength to the sourcedistance, so the sharpest tests involve polarimetry of high-frequency radia-tion propagating over cosmological distances. Although still in its infancy,the polarimetry of gamma-ray bursts has already led, for example, to con-straints of order 10 − GeV − on certain operators at d = 5.For vacuum-nonbirefringent operators causing dispersion, interestingtests can be performed by studying the separation of a propagating pulse.The sensitivity to the corresponding coefficients is controlled by the ratio of the pulse separation to the source distance. For cosmological sources, thisdispersion-based sensitivity is typically many orders of magnitude weakerthan polarimetric measurements, but nonetheless provides the best accessto vacuum-nonbirefringent operators.Finally, for the vast numbers of ‘vacuum-orthogonal’ operators that pro-duce no leading-order effects on photon propagation in the vacuum, the bestoption is investigation via laboratory tests. Typical experiments with res-onant cavities and interferometers produce sensitivities given by the ratioof the frequency shift to the frequency. Along with studies of astrophysicalbirefringence and dispersion, the investigations of these Lorentz-violatingeffects on light present an open experimental challenge, with a real potentialfor discovery in an area that is almost unexplored to date.
3. Gravity
The key feature of Special Relativity is the isotropy of spacetime. An observ-able background Lorentz vector or tensor implies a spacetime anisotropy ofthe vacuum and hence Lorentz violation. Similarly, a key component of GRis the local isotropy of spacetime. Lorentz violation in this context can beunderstood as the presence of an observable background vector or tensorin a local Lorentz frame.A local Lorentz frame at a given point is a tangent spacetime to thespacetime manifold. Since local Lorentz violation is a property of the tan-gent spacetime rather than the manifold, the ‘vierbein formalism’ is appro-priate for studies of local Lorentz violation and gravity. In this approach,the vierbein e aµ implements the conversion from local Lorentz coordinates a , b , . . . to spacetime manifold coordinates µ , ν , . . . . ‘No-go’ result for explicit Lorentz violation. The ramifications of thesesimple observations are surprisingly broad. One powerful result is that ex-plicit Lorentz violation is incompatible with generic Riemann geometriesand therefore with GR. The basic point is that explicit Lorentz violationoccurs when the background tensors are externally prescribed, but this isinconsistent with the Bianchi identities for general Riemann spacetimes.To illustrate this no-go result, suppose explicit Lorentz violation appearsin the matter sector. The energy-momentum tensor is then nonconservedin most spacetimes and the equations of motion are inconsistent with theBianchi identities,0 ≡ D µ G µν = 8 πG N D µ T µν = 0 (explicit breaking) . (1)In contrast, in spontaneous Lorentz violation the background tensors are dynamically determined along with the metric and are therefore compatiblewith the spacetime geometry,0 ≡ D µ G µν = 8 πG N D µ T µν = 0 (spontaneous breaking) . (2)The no-go result holds also for explicit Lorentz violation in the grav-ity sector and for Riemann-Cartan spacetimes. In the general case withexplicit Lorentz breaking, imposing consistency with the Bianchi identitiesenforces an additional nondynamical constraint on the spacetimes solvingthe theory. The constraint often forbids any solution, but in any case itrepresents at best a post hoc assumption slicing the solution spacetimes ofthe theory. The no-go result also presents an obstruction to reproducingGR from a theory with explicit Lorentz violation, including theories suchas ‘Lifschitz gravity’ that attempt to generate GR through the running ofexplicit Lorentz-violating couplings. Gravity theories in which the gravitonarises from spontaneous Lorentz violation avoid the no-go result. Gravitational signals from spontaneous Lorentz violation.
Lorentz vio-lation can occur in the pure-gravity and matter-gravity sectors. The no-goresult shows it must be spontaneous, so the coefficients for Lorentz viola-tion must originate as dynamical fields. Each coefficient field can thereforebe written as the sum of the vacuum coefficient for Lorentz violation anda fluctuation. Since the breaking is spontaneous, the fluctuation includesmassless NG modes and so can affect the dynamics even at low energies.The problem of solving for these modes and eliminating them to recover aneffective post-newtonian gravitational theory is challenging but has beensolved in both the pure-gravity and the matter-gravity sectors.Observable effects arise from Lorentz violation in the gravitational fieldof the source and in the trajectory of a test body. As an example, the localgravitational acceleration experienced by a test body near the surface of theEarth acquires sidereal and annual variations that can depend on the com-position of the test body and the Earth. In general, signals can appear ingravimeters (free fall and force comparison), tests of the weak equivalenceprinciple (free fall, force comparison, and space based), exotic matter (an-tihydrogen, higher-generation particles, etc.), solar-system measurements(lunar laser ranging, perihelion shift, gyroscopes, etc.), binary pulsars, andvarious photon tests (Shapiro delay, Doppler shift, gravitational redshift,null redshift, etc.). Also, a nonzero background torsion can be understoodin terms of certain coefficients for Lorentz violation, so sensitive constraintson torsion can be obtained. The overall prospects for new and improvedsearches for gravitational Lorentz violation are excellent.
Acknowledgments
This work was supported in part by U.S. D.o.E. grant DE-FG02-91ER40661and by the Indiana University Center for Spacetime Symmetries.
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