Moment Problems and the Causal Set Approach to Quantum Gravity

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.

24 pages. To appear: Journal of Mathematical Physics

References in corpus (3)

• Discrete structures in gravity
• Algebraic description of spacetime foam
• Quantum Space-Time as a Quantum Causal Set

Cited by in corpus (11)

• Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School)
• Causal structures and causal boundaries
• The Random Walk in Generalized Quantum Theory
• Indications of de Sitter Spacetime from Classical Sequential Growth Dynamics of Causal Sets
• Stable Homology as an Indicator of Manifoldlikeness in Causal Set Theory
• Light, Links and Causal Sets
• Statistical minimax approach of the Hausdorff moment problem
• Universal homogeneous causal sets
• The structure of covtree: searching for manifestly covariant causal set dynamics
• Observables for cyclic causal set cosmologies
• Hopf algebras from poset growth models