Specialization of Ashtekar's Formalism to Bianchi Cosmology

Recently Ashtekar found a new canonical formalism of general relativity in whichi the constraints and the dynamical equations are all written formally as polynomials of canonical variables. In this paper we apply his formalism to the Bianchi cosmology and write down the constraints and the dynamical equations explicitly in terms of Ashtekar-type canonical variables which depend only on time. In particular we prove that for the vacuum Bianchi IX model the gauge freedom is decoupled from the dynamics and the canonical variables reduce to three diagonal components of the metric and their Ashtekar-type conjugate momentum, preserving the polynomial nature of the equations. Further we discuss the quantization of this model and show that the quantized hamiltonian constraint takes a quite simple and beautiful form. To illustrate the tractability of this equation, we present an explicit non-trivial solution to it.