Approaches to Quantum Gravity

Lorentz invariance violation & its role in Quantum Gravity phenomenology 529QG effects on the particular particle type. Note that ξ could depend on the particlespecies and its polarization. The dispersion relation can be written in a covariantfashion:P μ P μ = m 2 +ξ (P μ W μ ) 3 , (27.2)M Plwhere P μ is the particle’s 4-momentum, and W μ is the 4-velocity of the preferredframe. Amelino-Camelia et al. [7] noted that photons (m = 0) with different energieswould then travel with different velocities. For a gamma ray burst originatingat a distance D from us, the difference in time of arrival of different energy componentswould be t = ξ DE/M Pl . If the parameter ξ were of order 1 andD ∼ 100 Mpc, then for E ∼ 100 MeV, we would have t ∼ 10 −2 s, making itclose to measurable in gamma ray bursts.A second possible modification is that the parameter normally called the speedof light, c, is different for different kinds of particle. This is implemented by anon-universal particle-dependent coefficient of P 2 in Eq. (27.1). The differences inthe maximum speeds of propagation also gives sensitive tests: vacuum Cerenkovradiation etc. [19].There are in fact two lines of inquiry associated with modified dispersionrelations. One is the initial approach, where the equivalence of all referenceframes fails, essentially with the existence of a preferred frame. A second popularapproach preserves the postulate of the equivalence of all frames, but triesto find modifications of the standard Lorentz or Poincaré symmetries. The mostpopular version, with the name of doubly special relativity (DSR), replaces thestandard Poincaré algebra by a non-linear structure [6; 52; 48; 51]. Another line ofargument examines a deformed algebra formed by combining the Poincaré algebrawith coordinate operators one [71; 17; 16]. Related to these are field theorieson non-commutative space-time [15; 9; 24; 69]; they give a particular kind ofLIV at short distances that fits into the general field theoretic framework we willdiscuss.In this chapter we will concentrate on the first issue, actual violations of LI.Regarding DSR and its relatives, we refer the reader to the other contributions inthis volume and to critiques by Schützhold & Unruh [62; 63], by Rembieliński &Smoliński [59], and by Sudarsky [67]. A problem that concerns us is that the proposedsymmetry algebras all contain as a subalgebra the standard Poincaré algebra,and thus they contain operators for 4-momentum that obey the standard properties.The DSR approach uses a modified 4-momentum that has non-linear functions ofwhat we regard as the standard momentum operators. This of course raises the issueof which are the operators directly related to observations. In the discussion section27.9, we will summarize a proposal by Liberati, Sonego and Visser [49] who