Unitary evolution of the quantum Universe with a Brown–Kuchař dust

Abstract

We study the time evolution of a wave function for the spatially flat FriedmannLemaître-Robertson-Walker universe governed by the Wheeler-DeWitt equation in both analytical and numerical methods. We consider a Brown-Kuchař dust as a matter field in order to introduce a "clock" in quantum cosmology and adopt the Laplace-Beltrami operator-ordering. The Hamiltonian operator admits an infinite number of self-adjoint extensions corresponding to a one-parameter family of boundary conditions at the origin in the minisuperspace. For any value of the extension parameter in the boundary condition, the evolution of a wave function is unitary and the classical initial singularity is avoided and replaced by the big bounce in the quantum system. Exact wave functions show that the expectation value of the spatial volume of the universe obeys the classical time evolution in the late time but its variance diverges.

Singularity avoidance for collapsing quantum dust in the Lemaître-Tolman-Bondi model

“…The Hamiltonian (6) also matches the gravitational Hamiltonian (with its negative kinetic term) for a flat Friedmann model with vanishing cosmological constant when identifying the scale factor as a r o = R o , where r o is the parametric radius of the dust cloud [1]. When using Brown-Kuchař dust as matter and dust proper time as the time coordinate, the full Hamiltonian constraint for this Friedmann model reads H + P τ = 0, where P τ is the momentum conjugate to τ [39]. Quantizing this constraint gives exactly the same Schrödinger equation as discussed below.…”

“…It should be noted that in agreement with the Misner-Sharp mass, vanishes at the bounce. The expression (39) need not necessarily be negative. 2 The shells may get 'scrambled', that is, their order may (perhaps partially) be reversed.…”

A New Version of the Aharonov–Bohm Effect

“…Such choice corresponds, mathematically, to the choice of a self-adjoint extension of the initial magnetic Hamiltonian, since usually the preliminary physical information provides only a symmetric operator; the self-adjoint property is required (in fact equivalent) for a unitary time evolution (conservation of probability) in quantum dynamics. The topic is related to the existence of anomalies [13], the presence of anisotropic scale invariances [14], different surface spectra of Weyl semimetals [35], creation of a pointlike source in quantum field theory [39], studies of topological quantum phases [3] and models in quantum gravity whose time evolution depend on boundary conditions at the origin [28], to mention only a handful of examples that illustrate the well-known fact that different self-adjoint extensions correspond to different physics. An instructive discussion about self-adjoint extensions of the simple case of a particle in a potential well appears in [12].…”

Elimination of cosmological singularities in quantum cosmology by suitable operator orderings

“…In the Brown-Kuchař description [21,22], dust is described by a matter fluid determined by the rest mass density ρ and a 4-velocity field U µ which can be parametrized by certain (noncanonical) scalar fields. The classical action including gravity is 2…”