Quantizing the Homogeneous Linear Perturbations about Taub using the Jacobi Method of Second Variation

TL;DR

This paper applies the Jacobi method to linear perturbations of the Taub space in Bianchi IX cosmology, deriving gauge-invariant variables, quantizing the system, and obtaining exact quantum states.

Contribution

It introduces a novel application of the Jacobi second variation method to linearize and quantize perturbations around the Taub background in Bianchi IX models.

Findings

Decoupled gauge-invariant variables identified

Quantization leads to time-dependent Schrödinger equations

Exact quantum squeezed states derived for one variable

Abstract

Applying the Jacobi method of second variation to the Bianchi IX system in Misner variables $(\alpha, \beta_+, \beta_-)$, we specialize to the Taub space background $(\beta_- = 0)$ and obtain the governing equations for linearized homogeneous perturbations $(\alpha', \beta_+', \beta_-')$ thereabout. Employing a canonical transformation, we isolate two decoupled gauge-invariant linearized variables ($\beta_-'$ and $Q_+' = p_+ \alpha' + p_\alpha \beta_+'$), together with their conjugate momenta and linearized Hamiltonians. These two linearized Hamiltonians are of time-dependent harmonic oscillator form, and we quantize them to get time-dependent Schr\"{o}dinger equations. For the case of $Q_+'$, we are able to solve for the discrete solutions and the exact quantum squeezed states.