Complex actions and causality violations: applications to Lorentzian quantum cosmology

For the construction of the Lorentzian path integral for gravity one faces two main questions: firstly, what configurations to include, in particular whether to allow Lorentzian metrics that violate causality conditions. And secondly, how to evaluate a highly oscillatory path integral over unbounded domains. Relying on Picard–Lefschetz theory to address the second question for discrete Regge gravity, we will illustrate that it can also answer the first question. To this end we will define the Regge action for complexified variables and study its analytical continuation. Although there have been previously two different versions defined for the Lorentzian Regge action, we will show that the complex action is unique. More precisely, starting from the different definitions for the action one arrives at equivalent analytical extensions. The difference between the two Lorentzian versions is only realized along branch cuts which arise for a certain class of causality violating configurations. As an application we discuss the path integral describing a finite evolution step of the discretized de Sitter Universe. We will in particular consider an evolution from vanishing to finite scale factor, for which the path integral defines the no-boundary wave function.