Uniqueness of diffeomorphism invariant states on holonomy-flux algebras

TL;DR

This paper proves that in loop quantum gravity, there is a unique diffeomorphism-invariant representation of the basic algebra of observables, which is crucial for the consistency of the theory and applies broadly to similar connection-based models.

Contribution

It establishes the uniqueness of the diffeomorphism-invariant cyclic representation of the holonomy-flux algebra in loop quantum gravity and related theories.

Findings

Only one cyclic representation invariant under spatial diffeomorphisms exists.

The result applies to any theory with connection variables and a compact structure group.

The algebra's definition is general for connection-based theories.

Abstract

Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms. While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract *-algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory.