Canonical analysis of the Bianchi models in an Ashtekar formulation

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Abstract
The authors discuss the reduction of the physical degrees of freedom in the Ashtekar formulation of general relativity which leads to the Bianchi models. They also perform a canonical analysis of the Bianchi models on compact spacelike surfaces. In the case of the class B Bianchi models the constraint algebra does not close because some of the constraints are second class. This property does not exclude a consistent dynamics, but due to purely geometrical reasons, the class B models cannot exist on compact spacelike surfaces. In the case of the class A models the constraints are all first class. However, the number of independent components of the vector constraint varies from 0 (type I) to 3 (type IX). This is an agreement with the results of a similar analysis recently performed by Ashtekar and Samuel (1991) in the metric formulation. The reduced phase space is analysed through a gauge fixing procedure in the case of the type I, II and IX Biachi models.

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