New method for constructing valid Hamiltonians for class-B spacetimes and the quantization of the general type-V cosmology

In this work we propose a new method for constructing valid Hamiltonians for homogeneous spacetimes. Through a careful examination of the role played by spatial diffeomorphisms, we identify the "correct" momentum constraints in phase space; we also search for a supermetric ${\Theta }_{\alpha \beta \gamma \delta }$, which is the analogue of the Wheeler-DeWitt supermetric $G_{\mathrm{ijkl}}$ in the case of the finite system under consideration. This way, a valid Hamiltonian (which, of course, need not be the only possible one) is constructed for the general type-V spacetime and the resulting system is quantized.