Wave Packet in Quantum Cosmology and Definition of Semiclassical Time

TL;DR

This paper investigates the challenge of defining a real semiclassical time in quantum cosmology using wave packets, and proposes a phase-based real time variable that facilitates an approximate Schrödinger equation for matter fields.

Contribution

It introduces a method to define a real semiclassical time from wave packet phases in quantum cosmology, addressing the difficulty of real time definition without complex values.

Findings

A real semiclassical time variable is constructed from the wave packet phase.

An approximate Schrödinger equation for matter fields is derived in the large universe size and small momentum regime.

The formalism highlights the complexity of WKB time without complex extensions.

Abstract

We consider a quantum cosmology with a massless background scalar field $\pb$ and adopt a wave packet as the wave function. This wave packet is a superposition of the WKB form wave functions, each of which has a definite momentum of the scalar field $\pb$. In this model it is shown that to trace the formalism of the WKB time is seriously difficult without introducing a complex value for a time. We define a semiclassical real time variable $\tp$ from the phase of the wave packet and calculate it explicitly. We find that, when a quantum matter field $\pq$ is coupled to the system, an approximate Schr\"odinger equation for $\pq$ holds with respect to $\tp$ in a region where the size $a$ of the universe is large and $|\pb|$ is small.