Thiemann transform for gravity with matter fields

TL;DR

This paper extends Thiemann's Wick transform to gravity coupled with matter fields, providing a geometric interpretation and linking it to an inverse Wick rotation and complex scaling, with implications for quantum gravity formulations.

Contribution

The paper generalizes Thiemann's transform to include matter fields in the Ashtekar formulation, establishing its equivalence to inverse Wick rotation and complex scaling, and clarifying its geometric meaning.

Findings

Thiemann transform is equivalent to inverse Wick rotation plus complex scaling.

The transform applies to functions of gravitational and matter variables, including shift and lapse.

The generator of the transform depends only on the spin of the fields.

Abstract

The generalised Wick transform discovered by Thiemann provides a well-established relation between the Euclidean and Lorentzian theories of general relativity. We extend this Thiemann transform to the Ashtekar formulation for gravity coupled with spin-1/2 fermions, a non-Abelian Yang-Mills field, and a scalar field. It is proved that, on functions of the gravitational and matter phase space variables, the Thiemann transform is equivalent to the composition of an inverse Wick rotation and a constant complex scale transformation of all fields. This result holds as well for functions that depend on the shift vector, the lapse function, and the Lagrange multipliers of the Yang-Mills and gravitational Gauss constraints, provided that the Wick rotation is implemented by means of an analytic continuation of the lapse. In this way, the Thiemann transform is furnished with a geometric interpretation. Finally, we confirm the expectation that the generator of the Thiemann transform can be determined just from the spin of the fields and give a simple explanation for this fact.