Quantum field theory on noncommutative spaces

Lorentz violation at high energy: Concepts, phenomena, and astrophysical constraints

The behavior of photons in the presence of Lorentz and CPT violation is studied. Allowing for operators of arbitrary mass dimension, we classify all gauge-invariant Lorentz- and CPT-violating terms in the quadratic Lagrange density associated with the effective photon propagator. The covariant dispersion relation is obtained, and conditions for birefringence are discussed. We provide a complete characterization of the coefficients for Lorentz violation for all mass dimensions via a decomposition using spin-weighted spherical harmonics. The resulting nine independent sets of spherical coefficients control birefringence, dispersion, and anisotropy in the photon propagator. We discuss the restriction of the general theory to various special models, including among others the minimal standard-model extension, the isotropic limit, the case of vacuum propagation, the nonbirefringent limit, and the vacuum-orthogonal model. The transformation of the spherical coefficients for Lorentz violation between the laboratory frame and the standard Sun-centered frame is provided. We apply the results to various astrophysical observations and laboratory experiments. Astrophysical searches of relevance include studies of birefringence and of dispersion. We use polarimetric and dispersive data from gamma-ray bursts to set constraints on coefficients for Lorentz violation involving operators of dimensions four through nine, and we describe the mixing of polarizations induced by Lorentz and CPT violation in the cosmic-microwave background. Laboratory searches of interest include cavity experiments. We present the general theory for searches with cavities, derive the experiment-dependent factors for coefficients in the vacuum-orthogonal model, and predict the corresponding frequency shift for a circular-cylindrical cavity.

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Electrodynamics with Lorentz-violating operators of arbitrary dimension

We consider the standard model on a non-commutative space and expand the action in the non-commutativity parameter $\theta^{\mu \nu}$
 . No new particles are introduced; the structure group is $SU(3)\times SU(2)\times U(1)$
 . We derive the leading order action. At zeroth order the action coincides with the ordinary standard model. At leading order in $\theta^{\mu\nu}$
 we find new vertices which are absent in the standard model on commutative space-time. The most striking features are couplings between quarks, gluons and electroweak bosons and many new vertices in the charged and neutral currents. We find that parity is violated in non-commutative QCD. The Higgs mechanism can be applied. QED is not deformed in the minimal version of the NCSM to the order considered.

The standard model on non-commutative space-time

The theoretical description of fermions in the presence of Lorentz and $CPT$ violation is developed. We classify all Lorentz- and $CPT$-violating and invariant terms in the quadratic Lagrange density for a Dirac fermion, including operators of arbitrary mass dimension. The exact dispersion relation is obtained in closed and compact form, and projection operators for the spinors are derived. The Pauli Hamiltonians for particles and antiparticles are extracted, and observable combinations of operators are identified. We characterize and enumerate the coefficients for Lorentz violation for any operator mass dimension via a decomposition using spin-weighted spherical harmonics. The restriction of the general theory to various special cases is presented, including isotropic models, the nonrelativistic and ultrarelativistic limits, and the minimal Standard-Model Extension. Expressions are derived in several limits for the fermion dispersion relation, the associated fermion group velocity, and the fermion spin-precession frequency. We connect the analysis to some other formalisms and use the results to extract constraints from astrophysical observations on isotropic ultrarelativistic spherical coefficients for Lorentz violation.

M. Hayakawa

Papers

Perturbative analysis on infrared aspects of noncommutative QED on R4

Mass hierarchy from compositeness hierarchy in supersymmetric gauge theory