Quantum Gravity

70 COVARIANT APPROACHES TO QUANTUM GRAVITYfooting, with intriguing results, in spite of the formal non-renormalizability ofthe perturbation theory.2.3 Quantum supergravitySupergravity (SUGRA) is a supersymmetric theory of gravity encompassing GR.SUSY is a symmetry which mediates between bosons and fermions. It exhibitsinteresting features; for example, the running coupling constants in the StandardModel of particle physics can meet at an energy of around 10 16 GeV if SUSYis added. SUGRA is a theory in its own right; see, for example, van Nieuwenhuizen(1981) for a review. The main question of concern here is whether theperturbative UV behaviour of quantum gravity discussed in the last section canbe improved by going over to SUGRA.SUSY arose from the question whether the Poincaré group (and thereforespace–time symmetries) can be unified with an internal (compact) group such asSO(3). A no-go theorem states that in a relativistic quantum field theory, given‘natural’ assumptions of locality, causality, positive energy, and a finite numberof elementary particles, such an invariance group can only be the direct productof the Poincaré group with a compact group, preventing a real unification. Thereis, however, a loophole. A true unification is possible if anticommutators are usedinstead of commutators in the formulation of a symmetry, leading to a ‘graded Liealgebra’. 32 It was shown by Haag et al. (1975) that, with the above assumptionsof locality etc., the algebraic structure is essentially unique.The SUSY algebra is given by the anticommutator[Q i α, ¯Q j β ] + =2δ ij (γ n ) αβ P n ,i,j=1,...,N , (2.138)where Q i α denotes the corresponding generators, also called spinorial charges,¯Q i α = Qi α γ0 with γ 0 being one of Dirac’s gamma matrices, N is the number ofSUSY generators, and all anticommutators among the Qs andthe ¯Qs themselvesvanish. There are also the commutators[P n ,Q i α]=0, [P m ,P n ]=0. (2.139)(P n denotes the energy–momentum four vector, the generator of space–timetranslations.) In addition, there are the remaining commutators of the Poincarégroup, (2.34)–(2.36), as well as[Q i α,J mn ]=(σ mn ) β αQ i β , (2.140)where here σ mn =i[γ m ,γ n ]; cf. also Section 1.1. More details can be found, forexample, in Weinberg (2000). The SUSY algebra is compatible with relativistic32 Anticommutators were, of course, used before the advent of SUSY, in order to describefermions, but not in the context of symmetries.