Within loop quantum gravity we construct a coarse-grained approximation for the Einstein-Maxwell theory that yields effective Maxwell equations in flat spacetime comprising Planck scale corrections. The corresponding Hamiltonian is defined as the expectation value of the electromagnetic term in the Einstein-Maxwell Hamiltonian constraint, regularized in the manner of Thiemann, with respect to a would-be semiclassical state. The resulting energy dispersion relations entail Planck scale corrections to those in flat spacetime. Both the helicity dependent contribution of Gambini and Pullin and, for a value of a parameter of our approximation, that of Ellis and co-workers are recovered. The electric-magnetic asymmetry in the regularization procedure yields nonlinearities only in the magnetic sector which are briefly discussed. Observations of cosmological gamma ray bursts might eventually lead to the needed accuracy to study some of these quantum gravity effects.

Quantum gravity phenomenology opens up the possibility of probing Planck scale physics. Thus, by exploiting the generic properties that a semiclassical state of the compound system fermions plus gravity should have, an effective dynamics of spin-1/2 particles is obtained within the framework of loop quantum gravity. Namely, at length scales much larger than Planck length ${\mathcal{l}}_P\sim {10}^{-33}\mathrm{cm}$ and below the wavelength of the fermion, the spin-1/2 dynamics in flat spacetime includes Planck scale corrections. In particular we obtain modified dispersion relations in vacuo for fermions. These corrections yield a time of arrival delay of the spin-1/2 particles with respect to a light signal and, in the case of neutrinos, a novel flavor oscillation. To detect these effects the corresponding particles must be highly energetic and should travel long distances. Hence neutrino bursts accompanying gamma ray bursts or ultrahigh energy cosmic rays could be considered. Remarkably, future neutrino telescopes may be capable of testing such effects. This paper provides a detailed account of the calculations and elaborates on results previously reported in a Letter. These are further amended by introducing a real parameter ${\rm Y}$ aimed at encoding our lack of knowledge of scaling properties of the gravitational degrees of freedom.

The most important problem of fundamental physics is the quantization of the gravitational field. A main difficulty is the lack of available experimental tests that discriminate among the theories proposed to quantize gravity. Recently, Lorentz invariance violation by quantum gravity (QG) has been the source of growing interest. However, the predictions depend on an ad hoc hypothesis and too many arbitrary parameters. Here we show that the standard model itself contains tiny Lorentz invariance violation terms coming from QG. All terms depend on one arbitrary parameter $\alpha$ that sets the scale of QG effects. This parameter can be estimated using data from the ultrahigh energy cosmic ray spectrum to be $|\alpha |<\sim {10}^{-22}-{10}^{-23}$.

We study non-Abelian fields in the context of very special relativity (VSR). For this, we define the covariant derivative and the gauge field gauge transformations, both of them involving a fixed null vector $n_{\mu }$, related to the VSR breaking of the Lorentz group to the Hom(2) or Sim(2) subgroups. As in the Abelian case, the gauge field becomes massive. Moreover, we show that the VSR gauge transformations form a closed algebra. We then write actions coupling the gauge field to various matter fields (bosonic and fermionic). We mention how we can use the spontaneous symmetry-breaking mechanism to give a flavor-dependent VSR mass to the gauge bosons. Finally, we quantize the model using the Becchi-Rouet-Stora-Tyutin formalism to fix the gauge. The model is renormalizable and unitary and for non-Abelian groups, asymptotically free.

The very special relativity electroweak Standard Model (VSR EW SM) is a theory with $\mathrm{SU}(2{)}_L\times \mathrm{U}(1{)}_R$ symmetry, with the same number of leptons and gauge fields as in the usual Weinberg-Salam model. No new particles are introduced. The model is renormalizable and unitarity is preserved. However, photons obtain mass and the massive bosons obtain different masses for different polarizations. Besides, neutrino masses are generated. A VSR-invariant term will produce neutrino oscillations and new processes are allowed. In particular, we compute the rate of the decays $\mu \to e+\gamma$. All these processes, which are forbidden in the electroweak Standard Model, put stringent bounds on the parameters of our model and measure the violation of Lorentz invariance. We investigate the canonical quantization of this nonlocal model. Second quantization is carried out, and we obtain a well-defined particle content. Additionally, we do a counting of the degrees of freedom associated with the gauge bosons involved in this work, after spontaneous symmetry breaking has been realized. Violations of Lorentz invariance have been predicted by several theories of quantum gravity [J. Alfaro, H. Morales-Tecotl, and L. F. Urrutia, Phys. Rev. Lett. 84, 2318 (2000); Phys. Rev. D 65, 103509 (2002)]. It is a remarkable possibility that the low-energy effects of Lorentz violation induced by quantum gravity could be contained in the nonlocal terms of the VSR EW SM.

We compute the Itzykson-Zuber (IZ) integral for the superunitary group U(m|n). As a consequence, we are able to find the non-zero correlations of superunitary matricesComment: Latex, 16 page