Approaches to Quantum Gravity

530 J. Collins, A. Perez and D. Sudarskypropose that it is the measurement process that picks out the modified 4-momentumoperators as the measurable quantities.We will also touch on an aspect with important connections to the general fieldof QG: the problem of a physical regularization and construction of Quantum FieldTheories (QFT).27.2 Phenomenological modelsMethodical phenomenological explorations can best be quantified relative to adefinite theoretical context. In our case, of Lorentz invariance violation (LIV) ataccessible energies, the context should minimally incorporate known microscopicphysics, including Quantum Mechanics and Special Relativity (in order to considersmall deviations therefrom). This leads to the use of a conventional interactingQuantum Field Theory but with the inclusion of Lorentz violating terms in theLagrangian.One proposal is the Standard Model Extension (SME) of Colladay & Kostelecký[20] and Coleman & Glashow [19]. This incorporates within the Standard Modelof particle physics all the possible renormalizable Lorentz violating terms, whilepreserving SU(3) × SU(2) × U(1) gauge symmetry and the standard field content.For example, the terms in the free part of the Lagrangian density for a free fermionfield ψ are:L free = i ¯ψ(γ μ + c μν γ ν + d μν γ 5 γ ν + e μ + if μ γ 5 + 1 2 g μνρσ νρ )∂ μ ψ− ¯ψ(m + a ν γ ν + b ν γ 5 γ ν + 1 2 H νρσ νρ )ψ. (27.3)Here the quantities a μ , b μ , c μν , d μν , e μ , f μ , g μνρ and H μν are numerical quantitiescovariantly characterizing LIV, and can be thought of as arising from the VEV ofotherwise dynamical gravitational fields. The interacting theory is then obtainedin the same way as usual, with SU(3) × SU(2) × U(1) gauge fields and a Higgsfield. The expected renormalizability was shown by Kostelecký and Mewes [46]and Kostelecký et al. [47].A second approach, as used by Myers and Pospelov [54] is to take the LIV termsas higher dimension non-renormalizable operators. This is a natural proposal if onesupposes that LIV is produced at the Planck scale with power suppressed effectsat low energy; it gives modified dispersion relations at tree approximation. Forexample, there are dimension-5 terms with 1/M Pl suppression in the free part ofthe Lagrangian, such as1M PlW μ W ν W ρ ¯ψ(ξ f + ξ 5 f γ 5 )γ μ ∂ ν ∂ ρ ψ, (27.4)