Can We Obtain Conserved Currents in Supersymmetric Quantum Cosmology?

In this paper we report whether conserved currents can be sensibly defined in supersym-metric minisuperspaces. Our analysis deals with a k = +1 FRW model. Supermatter in the form of scalar supermultiplets is included. We show that conserved currents cannot be adequately established except for some very simple scenarios. More details can be found in ref. 3,5. N=1 supergravity 1 constitutes a " square-root " 2 of gravity: in finding a physical state Ψ, it is sufficient to solve the Lorentz and supersymmetry constraints; Ψ will then consequently obey the Hamiltonian constraints a. Supersymmetry thus induces advantageous features: in many cases we have to solve simple first-order differential equations in the bosonic variables 3. This contrasts with the situation without supersymmetry: a second-order (Wheeler-DeWitt) equation has to be solved, employing boundary conditions 4. Therefore, it is quite tempting to address from a supersymmetric point of view the issue of probability densities for a quantum state Ψ and conservation equations of the type ∇ a J a = 0 (see also ref. 5). The approach that we employ here is based on a differential operator representation for the fermionic variables. This constitutes the correct procedure: it is totally consistent with the existence of second–class constraints and subsequent Dirac brackets in supergravity theories. These then imply that fermionic variables and their Hermitian conjugates are intertwined within a canonical coordinate– momentum relation. It should also be pointed out that other authors have persued objectives similar to ours but with different methods 6,7,8. In particular, rigid su-persymmetry was used 6 , other approach 7 was not supersymmetric. Furthermore, a wave function arranged as a vector was used 8 where the Lorentz constraints were severely restricted, but an improved approach seems to have been found 9. Let us consider the action of the more general theory of N=1 supergravity in the presence of gauged supermatter 1. Our physical variables include the tetrad e AA ′ µ and the gravitinos which are represented by ψ A µ , ¯ ψ A ′ µ. The " overline " denotes Hermitian conjugation. The tetrad for a k = +1 FRW model can be be written as e aµ = diag [N (τ), aEâi ], wherê a and i run from 1 to 3 and E ˆ ai is a basis of left-invariant 1-forms on the unit S 3 with volume σ 2 = 2π 2. We take ψ …