Bosonic Physical States in N = 1 Supergravity?

It is argued that states in N = 1 supergravity that solve all of the constraint equations cannot be bosonic in the sense of being independent of the fermionic degrees of freedom. (Based on a talk given by Miguel Ortiz at the 7th Marcel Grossmann Meeting.) The canonical quantization of supergravity involves the solution of a number of constraint equations restricting the form of physical wave functionals, each of which corresponds to a symmetry of the theory. The constraint equations for N = 1 were discussed by D’Eath in 1984, where it was shown that in principle it is sufficient to solve the two supersymmetry constraints in order to obtain completely gauge invariant wave functionals (this assumes the absence of anomalies in the operator algebra). The reason for this is that the bracket of the two supersymmetry constraints yields the familiar Hamiltonian and momentum constraints that are present because of the diffeomorphism invariance of the theory. In a subsequent paper which was published this year, D’Eath used this simplifying feature to argue that explicit solutions of the quantum constraints can be found, and that these have the special property that they are bosonic. From this result, it has been argued by D’Eath that supergravity is a finite theory. However, this latter claim is dependent upon the existence of purely bosonic solutions. In response to Ref. 2, we demonstrated that states solving the supergravity constraints cannot be independent of the fermionic variables, and must almost certainly consist of an infinite product of Grassman valued fields. In this short paper, we review one argument presented there which shows that a bosonic state of the kind discussed by D’Eath cannot solve the supergravity constraints, and we shall attempt to update some of the arguments to clarify what is meant in our work by a bosonic state. The Lagrangian for N = 1 supergravity is L = 1 8κ2 εεabcdE a μE b νR cd ρσ − 1 2 ε ( ψ̄μE a ν σ̄aDρψσ −Dρψ̄μE a ν σ̄aψσ ) , (1) in terms of Weyl spinor gravitino fields ψAμ and ψ̄A′μ and a vierbein field E a μ. The conventions we use are the same as in Ref. 4. To work in the canonical formalism, we must choose a polarisation, and here we follow Ref. 1 in working in the holomorphic representation, which uses state functionals depending on ei and ψAi. As mentioned above, it is in principle sufficient to look at the supersymmetry constraints, provided that we ensure that any state functional is a Lorentz invariant combination of its arguments. These constraints take the relatively simple form: S̄ ′ F = [ −εeiσ̄ A′A a (DjψAk) + h̄κ 2 σ̄ ′ψAi δ δei ]