Time-space duality in 2D quantum gravity

TL;DR

This paper explores the inherent quantum uncertainties in spacetime causal relations within 2D quantum gravity, revealing a symmetry that causes equal probabilities for timelike and spacelike separations, affecting causal structure understanding.

Contribution

It identifies a global $Z_2$ symmetry in 2D quantum gravity and a local symmetry in simplicial models, demonstrating the fundamental nature of causal uncertainties.

Findings

Equal probability for timelike and spacelike separations in the path integral

Causal uncertainties are generically present even with boundary conditions

Symmetries imply maximal causal uncertainty possible

Abstract

An important task faced by all approaches of quantum gravity is to incorporate superpositions and quantify quantum uncertainties of spacetime causal relations. We address this task in 2D. By identifying a global $Z_2$ symmetry of 1+1D quantum gravity, we show that gravitational path integral configurations come in equal amplitude pairs with timelike and spacelike relations exchanged. As a consequence, any two points are equally probable to be timelike and spacelike separated in a universe without boundary conditions. In the context of simplicial quantum gravity we identify a local symmetry of the action which shows that even with boundary conditions causal uncertainties are generically present. Depending on the boundary conditions, causal uncertainties can still be large and even maximal.