A stochastic quasi-classical wavefunction of the Universe from the third quantization procedure

TL;DR

This paper demonstrates that solutions to the Wheeler-DeWitt equation for a closed universe with a cosmological constant exhibit quasi-classical behavior at large scale factors, linking quantum wavefunctions to classical cosmological states.

Contribution

It introduces a stochastic quasi-classical wavefunction approach derived from third quantization, connecting quantum solutions to classical cosmology and providing a potential boundary condition for the WdW equation.

Findings

Wavefunction behaves as a random quasi-classical field at large scale factors.

Statistical properties of the wavefunction relate to the Hartle-Hawking wavefunction.

Density matrix describes a mixed state with a probability distribution over field velocities.

Abstract

(abbreviated) We study quantized solutions of WdW equation describing a closed FRW universe with a $\Lambda $ term and a set of massless scalar fields. We show that when $\Lambda \ll 1$ in the natural units and the standard $in$-vacuum state is considered, either wavefunction of the universe, $\Psi$, or its derivative with respect to the scale factor, $a$, behave as random quasi-classical fields at sufficiently large values of $a$, when $1 \ll a \ll e^{{2\over 3\Lambda}}$ or $a \gg e^{{2\over 3\Lambda}}$, respectively. Statistical r.m.s value of the wavefunction is proportional to the Hartle-Hawking wavefunction for a closed universe with a $\Lambda $ term. Alternatively, the behaviour of our system at large values of $a$ can be described in terms of a density matrix corresponding to a mixed state, which is directly determined by statistical properties of $\Psi$. It gives a non-trivial probability distribution over field velocities. We suppose that a similar behaviour of $\Psi$ can be found in all models exhibiting copious production of excitations with respect to $out$-vacuum state associated with classical trajectories at large values of $a$. Thus, the third quantization procedure may provide a 'boundary condition' for classical solutions of WdW equation.