Self-adjoint Wheeler-DeWitt operators, the problem of time, and the wave function of the Universe

Abstract

We discuss minisuperspace aspects a non empty Robertson-Walker universe containing scalar matter field. The requirement that the Wheeler-DeWitt (WDW) operator be self adjoint is a key ingredient in constructing the physical Hilbert space and has non-trivial cosmological implications since it is related with the problem of time in quantum cosmology. Namely, if time is parametrized by matter fields we find two types of domains for the self adjoint WDW operator: a non trivial domain is comprised of zero current (Hartle-Hawking type) wave functions and is parametrized by two new parameters, whereas the domain of a self adjoint WDW operator acting on tunneling (Vilenkin type) wave functions is a single ray. On the other hand, if time is parametrized by the scale factor both types of wave functions give rise to non trivial domains for the self adjoint WDW operators, and no new parameters appear in them.

Brane classical and quantum cosmology from an effective action

“…As argued in ref. [68], it should be addressed by the specification of boundary conditions on Θ, which is an issue that we will not consider here. Finally, one can verify using identity (26a) that…”

Dynamical Vacuum in Quantum Cosmology

“…It is well-known that in such circumstances one has to impose boundary conditions on the allowed wave functions otherwise the relevant differential operators will not be self-adjoint. The need to impose boundary conditions to ensure self-adjointness has been long recognized by practitioners of the Arnowitt-Deser-Misner (ADM) reduced phase space formalism as applied to quantum cosmology [8,[10][11][12], and very recently it has also been seen to have non-trivial cosmological implications in the Wheeler-DeWitt approach [13].…”

Wormholes, Classical Limit and Dynamical Vacuum in Quantum Cosmology

“…We shall take p = (γ − 1) ρ as equation of state for the fluid, where γ is a constant and ρ is the fluid's energy density (we shall eventually put γ = 1). Performing an ADM reduction described in detail in [8,9], and setting γ = 1 for dust, we can write the full action (S g + S m ) as [8][9][10]…”

“…Once again Schutz , s variables and a FRW metric are used. As before, the ADM reduction takes the total action to the form [5,10]…”