Moment Problems and the Causal Set Approach to Quantum Gravity

TL;DR

This paper explores a mathematical framework connecting discrete Markov chains in quantum gravity models to moment problems, providing a complete characterization of the underlying coupling constants through probability distributions.

Contribution

It introduces a novel application of the classical moment problem to describe causal set dynamics in quantum gravity, establishing a unique correspondence with probability distributions.

Findings

Complete description of coupling constants via moment sequences

Representation theorem linking quantum gravity models to probability distributions

Mathematical characterization of causal set dynamics

Abstract

We study a collection of discrete Markov chains related to the causal set approach to modeling discrete theories of quantum gravity. The transition probabilities of these chains satisfy a general covariance principle, a causality principle, and a renormalizability condition. The corresponding dynamics are completely determined by a sequence of nonnegative real coupling constants. Using techniques related to the classical moment problem, we give a complete description of any such sequence of coupling constants. We prove a representation theorem: every discrete theory of quantum gravity arising from causal set dynamics satisfying covariance, causality and renormalizability corresponds to a unique probability distribution function on the nonnegative real numbers, with the coupling constants defining the theory given by the moments of the distribution.