Wick Rotation, Regularization of Propagators by a Complex Metric and Multidimensional Cosmology

TL;DR

This paper explores Wick rotation in quantum field theory and multidimensional cosmology, demonstrating how analytical continuation in complex metrics and dimensions can regularize propagators and facilitate cosmological models.

Contribution

It introduces a method of Wick rotation via analytical continuation in dimensions, extending previous regularization techniques to multidimensional cosmological settings.

Findings

Wick rotation can be performed by continuation in space dimensions.

Regularization by complex metrics preserves convergence and regularity.

Application to multidimensional cosmology models.

Abstract

The Wick rotation in quantum field theory is considered in terms of analytical continuation in the signature matrix parameter w = eta_00. Regularization of propagators by a complex metric parameter in most cases preserves (i) the convergence of Feynmann integrals (understood as Lebesgue integrals) if the corresponding integrals of Euclidean theory are convergent; (ii) the regularity of propagators in the coordinate representation if there is regularity in the Euclidean case. The well-known covariant regularization by a complex mass does not in general satisfy these conditions. Theories with a large family of propagators regularized by complex metric were previously considered by the author, and analogues of the Bogoliubov-Parasiuk-Hepp-Zimmermann theorems were proved. [V.D.Ivashchuk, Izv. Akad. Nauk Mold. SSR, Ser. Fiz.-Tekhn. i Math. Nauk, 3 (1987), 8; 1 (1988), 10]. This paper shows that in the case of multidimensional cosmology describing the evolution of n spaces M_i, i = 1, ..., n, the Wick rotation in the minisuperspace may be performed by analytical continuation in the dimensions N_i = dim M_i or in the dimension of the time submanifold M_0.