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

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 $({\alpha }^{\prime },{\beta }_{+}^{\prime },{\beta }_{-}^{\prime })$ thereabout. Employing a canonical transformation, we isolate two decoupled gauge-invariant linearized variables (${\beta }_{-}^{\prime }$ and $Q_{+}^{\prime }=p_{+}{\alpha }^{\prime }+p_{\alpha }{\beta }_{+}^{\prime }$), 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 Schrödinger equations. For the case of $Q_{+}^{\prime }$, we are able to solve for the discrete solutions and the exact quantum squeezed states.

LaTeX, 6 pages, 1 figure

References in corpus (2)

• The Bianchi Ix (MIXMASTER) Cosmological Model is Not Integrable
• On Close Relationship between Classical Time-Dependent Harmonic Oscillator and Non-Relativistic Quantum Mechanics in One Dimension