Restoring Time Dependence Into Quantum Cosmology

Abstract

Mini superspace cosmology treats the scale factor a(t), the lapse function n(t), and an optional dilation field φ(t) as canonical variables. While pre-fixing n(t) means losing the Hamiltonian constraint, pre-fixing a(t) is serendipitously harmless at this level. This suggests an alternative to the Hartle-Hawking approach, where the pre-fixed a(t) and its derivatives are treated as explicit functions of time, leaving n(t) and a now mandatory φ(t) to serve as canonical variables. The naive gauge pre-fix a(t) = const is clearly forbidden, causing evolution to freeze altogether, so pre-fixing the scale factor, say a(t) = t, necessarily introduces explicit time dependence into the Lagrangian. Invoking Dirac's prescription for dealing with constraints, we construct the corresponding mini superspace time dependent total Hamiltonian, and calculate the Dirac brackets, characterized by {n, φ} D = 0, which are promoted to commutation relations in the quantum theory.

Time-covariant Schrödinger equation and invariant decay probability: the $$\Lambda $$-Kantowski–Sachs universe

“…As far as ( 103) and ( 104) are concerned, which regard the quantization with respect to the boost symmetry of the problem, we do not see them being recovered with the previous method. However, the analysis performed has the big advantage of offering the possibility of a quantum description for the special solution of Bertotti and Kasner, for which we have no minisuperspace approximation through (84) since G i j = 0.…”

“…At this point, we need to mention of another interesting work, where a similar procedure with the one we follow here is employed. It regards the derivation of a Schrödinger equation and the introduction of time dependence in the wave function through the process of gauge fixing the scale factor while leaving the lapse as a degree of freedom, for more details see [84].…”

Is spacetime absolutely or just most probably Lorentzian?

“…Counter intuitively, such an unconventional pre-gauging [8] does constitute a physically viable option. Indeed, the reduced Euler-Lagrange equation ∂L 2 ∂n = 0, albeit an algebraic rather than a differential equation, gives rise to the exact solution of the full theory, free of any fake degree of freedom.…”

Schwarzschild mass uncertainty

“…For the hereby adopted S(r) = r gauge, it is the invariant spherical surface area A(r) = 4πr 2 which is T, R-independent as required. The procedure has been successfully implemented in the cosmological case [9], where an admissible gauge choice fixes not the lapse function, but rather the scale factor. The path to the quantum black hole is governed by the Hamiltonian formalism, and to be more specific, by Dirac's prescription [10] for dealing with constraint systems.…”