A constraint on canonical quantum gravity?

Abstract

Gamma rays from the γ-ray burst (GRB) 021206 have been reported to be strongly linearly polarized¹, with the estimated degree of polarization (80 ± 20%) being close to the absolute maximum of 100% — affording us the opportunity to constrain models of quantum gravity, which has had 10¹⁰ years to act on the photons as they travelled towards us. Here I show that if the effects of quantum gravity are linearly proportional to the ratio of the photon energy to the characteristic scale energy of quantum gravity, then the polarization of photons with energies of about 0.1 MeV should be completely random, contrary to what is observed. I conclude that, should the polarization measurement be confirmed, quantum gravity effects act with a power that is greater than linearity, or that loop quantum gravity is not viable. Compared with previous methods and results (see ref. 2, for example), testing of the linear polarization of cosmic γ-ray bursts may substantially extend the observational window on the theory of quantum gravity.

Main

GRBs are characterized by a highly variable flux of high-energy photons that propagate over cosmological distances. It has been suggested³ that they are the best candles in cosmological space, allowing us either to study or to constrain the effects of quantum gravity. These effects are known^3,4 to be proportional to the ratio (E/E_QG)ⁿ of the photon energy, E, to the Planck energy, E_QG ≈ 10¹⁹ GeV, and to the distance, D, of the photon's propagation. The linear case (n = 1) is the best studied^3,4, but the quadratic case (n = 2) has also recently been considered (see preprint at http://xxx.lanl.gov/PS_cache/gr-qc/pdf/0305/0305057.pdf). For n = 1, the effect of the energy-dependent refraction of photons in quantum space-time should lead to a measurable difference in arrival time (of the order of milliseconds) for photons with different energies².

The linear polarization of γ-rays from GRB 021206 allows us to test another possible effect of quantum space-time, which is predicted for canonical quantum gravity in loop representation. In this case, space-time exhibits the property of birefringence⁴: two photons with opposite states of helicity, +1 and −1, have different group velocities

The factor χ is about 1 for loop representation of quantum gravity³. A linearly polarized electromagnetic wave may be represented as the superposition of two monochromatic waves with opposite circular polarizations. When a linearly polarized wave propagates inside the substance with birefringence, the plane of polarization rotates along the path because of the difference in group velocity between the two circular components.

For the linear case n = 1, the phase angle, ϕ₁, of a plane of linear polarization changes along a distance D (in light years) as

This angle depends on the photon energy as E². Linear polarization measured within a broad energy range should vanish, provided that the difference in accumulated angles is large for photons with different energies. Two photons with energies of around 0.1 MeV and with a difference of energy of about 0.01% will therefore accumulate a difference of δϕ ≈ χ in polarization phase angle after a year of propagation in space with birefringence (see equation (2)).

For cosmological GRBs, which have a travel distance of D ≈ 10¹⁰ light years, the planes of linear polarization of photons with different energies should be totally randomized. The bulk linear polarization of photons with energies greater than 0.01 eV over a broad energy range must become zero even if they were all originally 100% polarized in a single plane.

In the quadratic case of quantum space-time birefringence with n = 2, the rotation of a plane of linear polarization is rather small for photons with energy of around 0.1 MeV

However, the distance D ≈ 10¹⁰ light years is so large that even the quadratic case of birefrigence could be tested by polarization measurements of photons with energies greater than 100 MeV. The detection⁵ of a high-energy component of GRB941017 (energies up to 200 MeV), which dominates the total fluence of the event, suggests that quadratic space-time birefringence could be tested experimentally in the future by polarimetry of such GRBs.

I therefore conclude that either the birefringence of quantum space-time with n = 1 should be below the level of χ > 10⁻¹⁴, or it should be quadratic (n = 2), assuming that the strong linear polarization of GRBs is confirmed by a second measurement.