Specialization of Ashtekar's Formalism to Bianchi Cosmology

Abstract

Recently Ashtekar found a new canonical formalism of general relativity in which 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. § 1. Introduction Recently Ashtekar proposed a new canonical formalism of general relativity.1),2) Iri his formalism the constraint equations and the dynamical equations become polynomials when they are expressed in terms of the new complex canonical vari­ ables, the Ashtekar variables_ Hence the severe nonlinearity, which has been thought to be an inevitable nature of general relativity and has bothered us so far, disappears. In spite of this fascinating aspect, Ashtekar's formalism has various problems yet to be solved. First the role of the newly introduced gauge freedom which corre­ sponds to rotation of triad is not clear. Second the complex nature of the canonical variables brings about new problems when one tries to quantize the theory, such as the structure of state space, interpretation and the correspondence with the Wheeler-DeWitt theory based on the conventional ADM formalism. In order to solve these problems and also to check the power of the formalism, it is desirable to see how the Ashtekar formalism works in simple systems. From this point of view, in the present paper, we examine how the canonical dynamics of spatially homogeneous spacetime is described in terms of the Ashtekar variables. In particular, as an application we study the vacuum Bianchi IX model in detail. We show that in this model the gauge freedom is decoupled from the dynamics and the canonical variables reduce to the three diagonal components of the spatial metric and their Ashtekar-type conjugate momentum, preserving the polynomial nature of the equations. This implies that there is a direct correspond­ ence between the Ashtekar formalism and the ADM formalism in this case. Further we show that when the theory is quantized it leads to an equation for the wavefunc­ tion which is much more tractable than the Wheeler-DeWitt equation in the conven­ tional ADM formalism. In order to illustrate the tractability of the equation, we give an example of non-trivial exact solutions, though it is unphysical. The relation