Transition probability spaces in loop quantum gravity

TL;DR

This paper explores the structure of transition probability spaces in loop quantum gravity, demonstrating their role in reconstructing quantum states and logical structures, and connecting to higher categorical frameworks and foundational proposals.

Contribution

It establishes that loop quantum gravity admits transition probability space structures and links them to spin foam models and higher categorical frameworks.

Findings

Transition probability spaces can be constructed in loop quantum gravity.

Spin foam models form 2-categories within this framework.

The approach offers new insights into quantum gravity foundations.

Abstract

We study the (generalized) transition probability spaces, in the sense of Mielnik and Cantoni, for spacetime quantum states in loop quantum gravity. First, we show that loop quantum gravity admits the structures of transition probability spaces. This is exemplified by first checking such structures in covariant quantum mechanics, and then identify the transition probability spaces in spin foam models via a simplified discrete version of general boundary formulation. The transition probability space thus defined gives a simple way to reconstruct the discrete analog of the Hilbert space of the canonical theory and the relevant quantum logical structures. Second, we show that the transition probability space and in particular the spin foam model are 2-categories. Then we discuss how to realize in spin foam models two proposals by Crane about the mathematical structures of quantum gravity, namely the quantum topos and causal sites. We conclude that transition probability spaces provide us with an alternative framework to understand various foundational questions of loop quantum gravity.