Abstract
The system under study is the Lambda -Kantowski–Sachs universe. Its canonical quantization is provided based on a recently developed method: the singular minisuperspace Lagrangian describing the system, is reduced to a regular (by inserting into the dynamical equations the lapse dictated by the quadratic constraint) possessing an explicit (though arbitrary) time dependence; thus a time-covariant Schrödinger equation arises. Additionally, an invariant (under transformations t=f({tilde{t}})) decay probability is defined and thus “observers” which correspond to different gauge choices obtain, by default, the same results. The time of decay for a Gaussian wave packet localized around the point a=0 (where a the radial scale factor) is calculated to be of the order sim 10^{-42}{-}10^{-41}~text {s}. The acquired value is near the end of the Planck era (when comparing to a FLRW universe), during which the quantum effects are most prominent. Some of the results are compared to those obtained by following the well known canonical quantization of cosmological systems, i.e. the solutions of the Wheeler–DeWitt equation.
Highlights
Relativity with Quantum Mechanics regarding the notions of space and time
It is quite difficult to cite them all, so we just mention a handful of them: The quantization of the FLRW geometry in the presence of a scalar field has been studied extensively from various perspectives [37,38,39,40,41,42,43,44]
The authors of [46,47,48] exploit the notion of conditional symmetries to provide a quantum description of some Bianchi types and of the Reissner–Nordström black hole
Summary
Relativity with Quantum Mechanics (and even more Quantum Field Theory) regarding the notions of space and time. This can be extended by considering the true degrees of freedom as functions of the spatial coordinates on the hypersurfaces, leading eventually to the so called “multi-finger” Schrödinger equation [29,72,73] Note that this generalization is possible only in the case of field theory and not for the minisuperspace models. We present the de facto covariant method of the typical Wheeler–DeWitt quantization of the given model, in order to study where the two approaches meet and what differences there exist At this point, we need to mention of another interesting work, where a similar procedure with the one we follow here is employed. For simplicity and in order to avoid more numeric factors we just use M for the constant of integration
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