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Applying the Jacobi method of second variation to the Bianchi IX system inMisner variables $(\alpha, \beta_+, \beta_-)$, we specialize to the Taub spacebackground $(\beta_- = 0)$ and obtain the governing equations for linearizedhomogeneous perturbations $(\alpha', \beta_+', \beta_-')$ thereabout. Employinga canonical transformation, we isolate two decoupled gauge-invariant linearizedvariables ($\beta_-'$ and $Q_+' = p_+ \alpha' + p_\alpha \beta_+'$), togetherwith their conjugate momenta and linearized Hamiltonians. These two linearizedHamiltonians are of time-dependent harmonic oscillator form, and we quantizethem 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 squeezedstates.

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