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

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.