Anisotropic models are unitary: A rejuvenation of standard quantum cosmology

Abstract

The present work proves that the folklore of the pathology of non-conservation of probability in quantum anisotropic models is wrong. It is shown in full generality that all operator ordering can lead to a Hamiltonian with a self-adjoint extension as long as it is constructed as a symmetric operator. It is indicated that the self-adjoint extension, however, is not unique and this non-uniqueness is suspected not to be a feature of anisotropic models only, in the sense that there exists operator orderings such that Hamiltonian for an isotropic homogeneous cosmological model does not have unique self-adjoint extension. For isotropic model, there is a special unique extension associated with quadratic form of Hamiltonian, i.e., a Friedrich’s extension. Details of calculations are carried out for a Bianchi III model as an example.