Scalar Product for a Version of Minisuperspace Model with the Grassmann Variables

Abstract

The Grassmann variables are used to transform a system with constraints into an unconstraint system. As a result, the Schrödinger equation arises instead of the Wheeler-DeWitt one. Formally, the Schrödinger equation describes a system’s evolution, but a definition of the scalar product is needed to calculate the mean values of the operators. We suggest an explicit formula for the scalar product. The calculation of the mean values is compared with the etalon method, in which a redundant degree of freedom is excluded. Nevertheless, we could note that a complete correspondence with the etalon picture is not found. Apparently, the picture with Grassmann variables requires further search for underlying Hilbert space.