[Paper Review] Geometric interpretation of Thiemann's generalized Wick transform

Guillermo A. Mena Marugán|ArXiv.org|May 13, 1997

Advanced Measurement and Metrology Techniques3 citations

TL;DR

This paper provides a geometric interpretation of Thiemann's generalized Wick transform in canonical quantum gravity by showing it arises from an inverse Wick rotation combined with an imaginary conformal transformation. For cosmological models like the Gowdy universe and general vacuum gravity, the transform is rigorously shown to correspond to analytic continuation in time, mapping Euclidean to Lorentzian metrics via an imaginary factor, differing from standard four-metric rescaling.

ABSTRACT

In the Ashtekar and geometrodynamic formulations of vacuum general relativity, the Euclidean and Lorentzian sectors can be related by means of the generalized Wick transform discovered by Thiemann. For some vacuum gravitational systems in which there exists an intrinsic time variable which is not invariant under constant rescalings of the metric, we show that, after such a choice of time gauge and with a certain identification of parameters, the generalized Wick transform can be understood as an analytic continuation in the explicit time dependence. This result is rigorously proved for the Gowdy model with the topology of a three-torus and for a whole class of cosmological models that describe expanding universes. In these gravitational systems, the analytic continuation that reproduces the generalized Wick transform after gauge fixing turns out to map the Euclidean line element to the Lorentzian one multiplied by an imaginary factor; this transformation rule differs from that expected for an inverse Wick rotation in a complex rescaling of the four-metric. We then prove that this transformation rule for the line element continues to be valid in the most general case of vacuum gravity with no model reduction nor gauge fixing. In this general case, it is further shown that the action of the generalized Wick transform on any function of the gravitational phase space variables, the shift vector, and the lapse function can in fact be interpreted as the result of an inverse Wick rotation and a constant, imaginary conformal transformation.

Motivation & Objective

• To clarify the geometric meaning of Thiemann's generalized Wick transform in canonical quantum gravity.
• To investigate how the transform relates to time gauge choices and metric rescalings in cosmological models.
• To establish a rigorous connection between the generalized Wick transform and analytic continuation in explicitly time-dependent systems.
• To extend the interpretation beyond model reductions to full vacuum gravity without gauge fixing.
• To demonstrate that the transform acts as an inverse Wick rotation combined with an imaginary conformal transformation on phase space variables.

Proposed method

• The analysis is conducted in the context of Ashtekar and geometrodynamic formulations of vacuum general relativity.
• The study focuses on systems with an intrinsic time variable not invariant under metric rescalings, particularly the Gowdy model on a three-torus.
• The generalized Wick transform is interpreted as an analytic continuation in the explicit time dependence after gauge fixing.
• The transformation rule for the line element is derived, showing it maps the Euclidean metric to the Lorentzian metric multiplied by an imaginary factor.
• The method is extended to general vacuum gravity, proving the transform acts as an inverse Wick rotation and a constant imaginary conformal transformation on all phase space variables.
• The proof relies on consistent identification of parameters and gauge-invariant structure in the phase space of gravity.

Experimental results

Research questions

• RQ1 How does Thiemann's generalized Wick transform relate to time-dependent gauge choices in cosmological models?
• RQ2 Can the generalized Wick transform be interpreted as an analytic continuation in time for specific gravitational systems?
• RQ3 Does the transform preserve its structure when extended beyond model-reduced systems to full vacuum gravity?
• RQ4 How does the transform differ from standard inverse Wick rotations in four-metric rescaling?
• RQ5 What is the geometric action of the transform on the gravitational phase space, including the lapse and shift functions?

Key findings

• In the Gowdy model with three-torus topology, the generalized Wick transform is rigorously shown to correspond to an analytic continuation in time, mapping the Euclidean line element to the Lorentzian one multiplied by an imaginary factor.
• The transformation rule for the line element differs from that expected in a complex rescaling of the four-metric, indicating a distinct geometric mechanism.
• The same transformation rule is proven to hold in the most general case of vacuum gravity without model reduction or gauge fixing.
• The action of the generalized Wick transform on any function of the gravitational phase space variables, shift vector, and lapse function is equivalent to an inverse Wick rotation combined with a constant imaginary conformal transformation.
• The result establishes a geometric and covariant interpretation of Thiemann's transform, linking it to time evolution and conformal structure in canonical quantum gravity.
• The findings provide a deeper understanding of the role of analytic continuation in quantum gravity and the physical meaning of the generalized Wick transform.