Complex lapse, complex action and path integrals

TL;DR

This paper introduces complex lapse as a conceptually robust alternative to imaginary time in quantum gravity, unifying Lorentzian and Riemannian path integrals and providing insights into quantum tunnelling in cosmology.

Contribution

It proposes a complex lapse function in the gravitational action, enabling a unified path integral framework for Lorentzian and Riemannian signatures in quantum gravity.

Findings

Complex lapse yields a well-defined complex action for gravity.

The theory interpolates between Lorentzian and Riemannian actions.

It suggests the universe's lapse argument is small and negative, supporting quantum tunnelling scenarios.

Abstract

Imaginary time is often used in quantum tunnelling calculations. This article advocates a conceptually sounder alternative: complex lapse. In the ``3+1'' action for the Einstein gravitational field minimally coupled to a Klein-Gordon field, allowing the lapse function to be complex yields a complex action which generates both the usual Lorentzian theory and its Riemannian analogue, and in particular allows a change of signature between the two. The action and variational equations are manifestly well defined in the Hamiltonian representation, with the momentum fields consequently being complex. The complex action interpolates between the Lorentzian and Riemannian actions as they appear formally in the respective path integrals. Thus the complex-lapse theory provides a unified basis for a path-integral quantum theory of gravity involving both Lorentzian and Riemannian aspects. A major motivation is the quantum-tunnelling scenario for the origin of the universe. Taken as an explanation for the observed quantum tunnelling of particles, the complex-lapse theory determines that the argument of the lapse for the universe now is extremely small but negative.