Predictions for possible Lorentz violations are a key area of interesting phenomenology in quantum gravity. Within loop quantum gravity, it has not been possible so far to reliably extract predictions for Lorentz violations. Existing claims were based on simplified toy models inspired by LQG, but not implied. This situation hasn’t changed so far, but there is an interesting development which hints at which order we might expect such Lorentz violations to be found.

In recent years, investigations in the anomaly freedom of effective constraints for cosmological perturbations and spherical symmetry have brought forward evidence for a smooth signature change at densities close to the Planck density. This effect shows up in a deformation of the hypersurface deformation algebra, where the commutator

$$ \{ H[M], H[N] \} = H_a[\beta ~ q^{ab}(M \partial_b N - N \partial_b M)]$$

between two Hamiltonian constraints is modified by a function $\beta$. In the simplest case of holonomy corrections, we have $\beta = 1 - 2 \rho / \rho_\text{crit}$, where $\rho$ is the background energy density and $\rho_\text{crit} \sim 1 / G^2 \hbar$ is the critical energy density at which the quantum bounce substituting the big bang occurs. Since the hypersurface deformation algebra of Euclidean gravity has the opposite sign in the previous Poisson bracket, we can interpret this as a smooth signature change from Lorentzian to Euclidean.

As a physical effect, one finds that also the propagation equation for matter fields gets modified as

$$ \frac {\partial^2 \phi^2}{\partial t^2} - \frac{\beta}{a^2} \Delta \phi = S[\phi] $$

where $S[\phi]$ contains source and lower derivative terms. It follows that $\beta \neq 1$ implies a change in the speed of light.

So far, $\beta$ only depends on the background energy density, meaning that no particle-energy dependent speed of light can be deduced. However, one should also take into account backreaction of a particle on the background. This has not been done rigorously so far, but we can try a heuristic estimate of the effect to be expected.

Consider a photon. Its energy density is roughly given by

In recent years, investigations in the anomaly freedom of effective constraints for cosmological perturbations and spherical symmetry have brought forward evidence for a smooth signature change at densities close to the Planck density. This effect shows up in a deformation of the hypersurface deformation algebra, where the commutator

$$ \{ H[M], H[N] \} = H_a[\beta ~ q^{ab}(M \partial_b N - N \partial_b M)]$$

between two Hamiltonian constraints is modified by a function $\beta$. In the simplest case of holonomy corrections, we have $\beta = 1 - 2 \rho / \rho_\text{crit}$, where $\rho$ is the background energy density and $\rho_\text{crit} \sim 1 / G^2 \hbar$ is the critical energy density at which the quantum bounce substituting the big bang occurs. Since the hypersurface deformation algebra of Euclidean gravity has the opposite sign in the previous Poisson bracket, we can interpret this as a smooth signature change from Lorentzian to Euclidean.

As a physical effect, one finds that also the propagation equation for matter fields gets modified as

$$ \frac {\partial^2 \phi^2}{\partial t^2} - \frac{\beta}{a^2} \Delta \phi = S[\phi] $$

where $S[\phi]$ contains source and lower derivative terms. It follows that $\beta \neq 1$ implies a change in the speed of light.

So far, $\beta$ only depends on the background energy density, meaning that no particle-energy dependent speed of light can be deduced. However, one should also take into account backreaction of a particle on the background. This has not been done rigorously so far, but we can try a heuristic estimate of the effect to be expected.

Consider a photon. Its energy density is roughly given by

$$\rho_\text{Photon} \sim E_\text{Photon} / \lambda_\text{Photon}^3 = \hbar \omega^4 / (2 \pi)^3 \text{.}$$

The difference in the speed of two photons thus depends on the difference of their energy densities that we input in $\beta$. We estimate

$$ 1-\beta \sim \frac{\rho_\text{Photon}}{\rho_\text{crit}} \sim \hbar^2 G^2 \omega^4 \sim \frac{E_\text{Photon}^4}{E_\text{Planck}^4} \text{.} $$

We conclude that based on our argument, we would expect an energy dependence of the speed of light which deviates from the standard speed of light by a term of order $4$ in $\frac{E_\text{Photon}}{E_\text{Planck}}$. Given current experimental data, there seems to be no contradiction with experiment.

It has to be stressed that this argument was very qualitative and has several weak points. While signature change has been observed by several authors in different contexts, it is not established yet in full loop quantum gravity. Also, our treatment of backreaction was very naive and a more detailed study should be undertaken. Still, it seems that the possible deduction of Lorentz violations from the deformed algebra approach is an interesting subject of study, and that the appearance of the effect only at order 4 goes against common expectations.

This argument has been written up here in other words. I am currently aware of some other works trying to find modified dispersion relations from the deformed algebra approach, but I could not find the argument made here in there. In any case, I would be very happy about comments on this topic.

$$ 1-\beta \sim \frac{\rho_\text{Photon}}{\rho_\text{crit}} \sim \hbar^2 G^2 \omega^4 \sim \frac{E_\text{Photon}^4}{E_\text{Planck}^4} \text{.} $$

We conclude that based on our argument, we would expect an energy dependence of the speed of light which deviates from the standard speed of light by a term of order $4$ in $\frac{E_\text{Photon}}{E_\text{Planck}}$. Given current experimental data, there seems to be no contradiction with experiment.

It has to be stressed that this argument was very qualitative and has several weak points. While signature change has been observed by several authors in different contexts, it is not established yet in full loop quantum gravity. Also, our treatment of backreaction was very naive and a more detailed study should be undertaken. Still, it seems that the possible deduction of Lorentz violations from the deformed algebra approach is an interesting subject of study, and that the appearance of the effect only at order 4 goes against common expectations.

This argument has been written up here in other words. I am currently aware of some other works trying to find modified dispersion relations from the deformed algebra approach, but I could not find the argument made here in there. In any case, I would be very happy about comments on this topic.

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