Ashtekar Variables: Structures in Bundles

Abstract

Canonical gravity can be formulated by means of a densitized dreibein field together with an SU(2) connection. These so-called Ashtekar variables are the fundamental quantities, loop quantum gravity is resting on. In this paper, we review these variables from the perspective of fibre bundles. This is straightforward for the dreibein field as this is simply a frame field. The Ashtekar connection, however, is more complicated. It turns out that, at the level of the tangent bundle, it is given by the Levi-Civita connection plus a multiple of the Weingarten mapping, whose action on vector fields is induced from the vector product on \(\mathbb {R}^{3}\). Lifted to the spin bundle, one regains the well-known SU(2) Ashtekar connection. At the end, we apply our results to Friedmann–Robertson–Walker spacetimes.