Triad representation of the Chern-Simons state in quantum gravity

TL;DR

This paper transforms the Chern-Simons state in quantum gravity into a real triad representation, revealing multiple gauge-invariant wavefunctionals and analyzing their semiclassical geometries, especially in homogeneous cosmological models.

Contribution

It introduces a generalized Fourier transform to express the Chern-Simons state in the triad variables, uncovering topologically distinct, gauge-invariant wavefunctionals and analyzing their properties.

Findings

Multiple linearly independent wavefunctionals from a single Chern-Simons state.

Explicit asymptotic expressions for wavefunctionals in various regimes.

The Chern-Simons state is non-normalizable under the chosen inner product.

Abstract

We investigate a triad representation of the Chern-Simons state of quantum gravity with a non-vanishing cosmological constant. It is shown that the Chern-Simons state, which is a well-known exact wavefunctional within the Ashtekar theory, can be transformed to the real triad representation by means of a suitably generalized Fourier transformation, yielding a complex integral representation for the corresponding state in the triad variables. It is found that topologically inequivalent choices for the complex integration contour give rise to linearly independent wavefunctionals in the triad representation, which all arise from the one Chern-Simons state in the Ashtekar variables. For a suitable choice of the normalization factor, these states turn out to be gauge-invariant under arbitrary, even topologically non-trivial gauge-transformations. Explicit analytical expressions for the wavefunctionals in the triad representation can be obtained in several interesting asymptotic parameter regimes, and the associated semiclassical 4-geometries are discussed. In restriction to Bianchi-type homogeneous 3-metrics, we compare our results with earlier discussions of homogeneous cosmological models. Moreover, we define an inner product on the Hilbert space of quantum gravity, and choose a natural gauge-condition fixing the time-gauge. With respect to this particular inner product, the Chern-Simons state of quantum gravity turns out to be a non-normalizable wavefunctional.