Hamiltonian quantization of General Relativity with the change of signature

TL;DR

This paper develops a modified Hamiltonian formalism for General Relativity to incorporate Euclidean solutions and analyze the impact of signature change on superspace and quantization.

Contribution

It introduces a formalism that includes Euclidean solutions and studies the effects of signature change on the superspace structure and quantization.

Findings

Signature of the supermetric changes from (-+++++) to (+-----).

Euclidean solutions can be incorporated into the Hamiltonian framework.

Different boundary conditions affect the quantization of Euclidean solutions.

Abstract

We show in this article how the usual hamiltonian formalism of General Relativity should be modified in order to allow the inclusion of the Euclidean classical solutions of Einstein's equations. We study the effect that the dynamical change of signature has on the superspace and we prove that it induces a passage of the signature of the supermetric from ($-+++++$) to ($+-----$). Next, all these features are more particularly studied on the example of minisuperspaces. Finally, we consider the problem of quantization of the Euclidean solutions. The consequences of different choices of boundary conditions are examined.