Probability distribution for the quantum universe

Abstract

We determine the inner product on the Hilbert space of wavefunctions of the universe by imposing the Hermiticity of the quantum Hamiltonian in the context of the minisuperspace model. The corresponding quantum probability density reproduces successfully the classical probability distribution in the ħ → 0 limit, for closed universes filled with a perfect fluid of index w. When −1/3 < w ≤ 1, the wavefunction is normalizable and the quantum probability density becomes vanishingly small at the big bang/big crunch singularities, at least at the semiclassical level. Quantum expectation values of physical geometrical quantities, which diverge classically at the singularities, are shown to be finite.

WKB Approaches to Restore Time in Quantum Cosmology: Predictions and Shortcomings

“…In the canonical quantum picture [51], this is equivalent to a timeless Schrödinger equation with null eigenvalues describing a trivial evolution leading to the so-called problem of time [11,14,47,52,53]. This issue has been long discussed in the literature since the formulation of the DeWitt theory [54][55][56][57][58][59][60]: for example, in [56] some time choices (scalar field, cosmological constant conjugate, and proper time) models are discussed via a semiclassical expansion in h. Indeed, the time coordinate could in principle be regarded together with the gravitational degrees of freedom and integrated over [17], such that there is no clear choice for the definition of another time parameter; subsequently, the definition of a conserved and well-defined probability distribution is troublesome, unless one imposes further conditions, e.g., hermicity of the Hamiltonian [61] or finiteness of the probability density [62]. One of the most followed approaches is the definition of a relational time [5,[7][8][9]37,54,63,64] to recover a time parameter leading to a Schrödinger dynamics; such "emergence of time" has been discussed not only for quantum gravity but also in the context of non-relativistic quantum mechanics, for example in [65].…”

Quantum gravity corrections to the matter dynamics in the presence of a reference fluid

“…One of the most puzzling questions affecting the canonical quantization [1] of the gravitational field [2][3][4][5][6][7] is the socalled "frozen formalism," i.e., the absence of a time evolution of the Universe wave function [8][9][10][11]. This basic problem of the quantum gravitational field dynamics has been faced in the literature by many approaches, some dealing with the Schrödinger equation [12][13][14][15][16][17][18] and others facing related topics [19][20][21][22][23][24], although this problem still remains an open issue, especially for what concerns the definition of a causality relation.…”