We present a Dirac-like framework for quantum cosmology. This is achieved by representing the gravitino field components, analogously to the Dirac equation, as matrices. In particular, we work out the Bianchi type IX diagonal model. The Lorentz constraints $J^{\alpha \beta }$ can then be applied explicitly on the wave function, which results in a 64-component vector, without any particular assumption on its expansion in fermionic (and bosonic) variables. The resulting wave function has only two nonvanishing components. Hence, the Dirac-like structure of the $S_A$ constraints manifests itself in a system of equations for these two functions, where the “potential” plays the role of the charge in Dirac’s theory. We exhibit two solutions whose explicit form is determined by the chosen factor ordering. The proposed construction can be systematically extended to other cosmologies and the inclusion of matter.

• Received 21 October 1996

DOI:https://doi.org/10.1103/PhysRevD.57.1015