Kiefer C. Quantum gravity

70 COVARIANT APPROACHES TO QUANTUM GRAVITY
footing, with intriguing results, in spite of the formal non-renormalizability of
the perturbation theory.
2.3 Quantum supergravity
Supergravity (SUGRA) is a supersymmetric theory of gravity encompassing GR.
SUSY is a symmetry which mediates between bosons and fermions. It exhibits
interesting features; for example, the running coupling constants in the Standard
Model of particle physics can meet at an energy of around 10 16 GeV if SUSY
is 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 the
perturbative UV behaviour of quantum gravity discussed in the last section can
be improved by going over to SUGRA.
SUSY arose from the question whether the Poincaré group (and therefore
space–time symmetries) can be unified with an internal (compact) group such as
SO(3). A no-go theorem states that in a relativistic quantum field theory, given
‘natural’ assumptions of locality, causality, positive energy, and a finite number
of elementary particles, such an invariance group can only be the direct product
of the Poincaré group with a compact group, preventing a real unification. There
is, however, a loophole. A true unification is possible if anticommutators are used
instead of commutators in the formulation of a symmetry, leading to a ‘graded Lie
algebra’. 32 It was shown by Haag et al. (1975) that, with the above assumptions
of 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 of
SUSY generators, and all anticommutators among the Qs andthe ¯Qs themselves
vanish. 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–time
translations.) In addition, there are the remaining commutators of the Poincaré
group, (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, for
example, in Weinberg (2000). The SUSY algebra is compatible with relativistic
32 Anticommutators were, of course, used before the advent of SUSY, in order to describe
fermions, but not in the context of symmetries.