Breakdown of Lorentz Invariance: The Order of Quantum Gravity Phenomenology. Thought of the Day 132.0

The purpose of quantum gravity phenomenology is to analyze the physical consequences arising from various models of quantum gravity. One hope for obtaining an experimental grasp on quantum gravity is the generic prediction arising in many (but not all) quantum gravity models that ultraviolet physics at or near the Planck scale, M_Planck = 1.2 × 10¹⁹ GeV/c², (or in some models the string scale), typically induces violations of Lorentz invariance at lower scales. Interestingly most investigations, even if they arise from quite different fundamental physics, seem to converge on the prediction that the breakdown of Lorentz invariance can generically become manifest in the form of modified dispersion relations

ω² = ω₀² + (1 + η₂) c²k² + η₄(ħ/M_{Lorentz violation})² + k⁴ + ….

where the coefficients η_n are dimensionless (and possibly dependent on the particle species under consideration). The particular inertial frame for these dispersion relations is generally specified to be the frame set by cosmological microwave background, and M_{Lorentz violation} is the scale of Lorentz symmetry breaking which furthermore is generally assumed to be of the order of M_Planck.

Although several alternative scenarios have been considered to justify the modified kinematics,the most commonly explored avenue is an effective field theory (EFT) approach. Here, the focus is explicitly on the class of non-renormalizable EFTs with Lorentz violations associated to dispersion relations. Even if this framework as a “test theory” is successful, it is interesting to note that there are still significant open issues concerning its theoretical foundations. Perhaps the most pressing one is the so called naturalness problem which can be expressed in the following way: The lowest-order correction, proportional to η₂, is not explicitly Planck suppressed. This implies that such a term would always be dominant with respect to the higher-order ones and grossly incompatible with observations (given that we have very good constraints on the universality of the speed of light for different elementary particles). If one were to take cues from observational leads, it is assumed either that some symmetry (other than Lorentz invariance) enforces the η₂ coefficients to be exactly zero, or that the presence of some other characteristic EFT mass scale μ ≪ M_Planck (e.g., some particle physics mass scale) associated with the Lorentz symmetry breaking might enter in the lowest order dimensionless coefficient η₂, which will be then generically suppressed by appropriate ratios of this characteristic mass to the Planck mass: η₂ ∝ (μ/M_Planck)^σ where σ ≥ 1 is some positive power (often taken as one or two). If this is the case then one has two distinct regimes: For low momenta p/(M_Planckc) ≪ (μ/M_Planck)^σ the lower-order (quadratic in the momentum) deviations will dominate over the higher-order ones, while at high energies p/(M_Planckc) ≫ (μ/M_Planck)^σ the higher order terms will be dominant.

The naturalness problem arises because such a scenario is not well justified within an EFT framework; in other words there is no natural suppression of the low-order modifications. EFT cannot justify why only the dimensionless coefficients of the n ≤ 2 terms should be suppressed by powers of the small ratio μ/M_Planck. Even worse, renormalization group arguments seem to imply that a similar mass ratio, μ/M_Planck would implicitly be present also in all the dimensionless n > 2 coefficients, hence suppressing them even further, to the point of complete undetectability. Furthermore, without some protecting symmetry, it is generic that radiative corrections due to particle interactions in an EFT with only Lorentz violations of order n > 2 for the free particles, will generate n = 2 Lorentz violating terms in the dispersion relation, which will then be dominant. Naturalness in EFT would then imply that the higher order terms are at least as suppressed as this, and hence beyond observational reach.

A second issue is that of universality, which is not so much a problem, as an issue of debate as to the best strategy to adopt. In dealing with situations with multiple particles one has to choose between the case of universal (particle-independent) Lorentz violating coefficients η_n, or instead go for a more general ansatz and allow for particle-dependent coefficients; hence allowing different magnitudes of Lorentz symmetry violation for different particles even when considering the same order terms (same n) in regards to momentum. Any violation of Lorentz invariance should be due to the microscopic structure of the effective space-time. This implies that one has to tune the system in order to cancel exactly all those violations of Lorentz invariance which are solely due to mode-mixing interactions in the hydrodynamic limit.

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