Astronomy and Astrophysics – Astrophysics – General Relativity and Quantum Cosmology
Scientific paper
1998-05-05
Class.Quant.Grav. 15 (1998) 3763-3775
Astronomy and Astrophysics
Astrophysics
General Relativity and Quantum Cosmology
LaTeX 2.09, 14 pages, no figures
Scientific paper
10.1088/0264-9381/15/12/007
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.
Garay Luis J.
Mena Marugan Guillermo A.
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