Time-covariant Schrödinger equation and invariant decay probability: The $Λ$-Kantowski-Sachs universe

Abstract

The system under study is the Λ-Kantowski-Sachs universe. Its canonical quantization is provided based on a recently developed method: the singular minusuperspace Lagrangians describing the system, are reduced to regular (by inserting into the dynamical equations the lapse dictated by the quadratic constraint) plus explicit (though arbitrary) "time" dependence; thus a "time"-covariant Schrödinger equation arises. Additionally, an invariant (under transformations t = f ( t)) Decay Probability is defined and thus "observers" which correspond to different gauge choices will, by default, obtain the same results. The time of decay for a Gaussian wavepacket localized around the point a = 0 (where a the radial scale factor) is calculated to be of the order ∼ 10 −34 . This corresponds to the inflationary epoch, thus hinting to a connection between inflation and quantum gravity effects. Some of the results are compared with those obtained by following the well known quantization of constrained systems (Wheeler-DeWitt equation).