Some results concerning the representation theory of the algebra underlying loop quantum gravity
Hanno Sahlmann
TL;DR
Addresses the problem of classifying representations of the loop quantum gravity holonomy–flux algebra and shows that under mild domain assumptions the representation data can be encoded by a set of measures on the space of generalized connections and corresponding functional data; flux operators act as a sum of a derivative and a multiplication operator, enabling a concrete parametrization of representations. The Ashtekar–Lewandowski representation emerges as the simplest case with vanishing data, while nontrivial representations with physically motivated data remain a challenge. The work links to broader QFT notions about inequivalent representations and diffeomorphism-invariant cyclic representations, and provides a mathematical platform for future physics-driven selection of representations.
Abstract
Important characteristics of the loop approach to quantum gravity are a specific choice of the algebra A of observables and of a representation of A on a measure space over the space of generalized connections. This representation is singled out by its elegance and diffeomorphism covariance. Recently, in the context of the quest for semiclassical states, states of the theory in which the quantum gravitational field is close to some classical geometry, it was realized that it might also be worthwhile to study different representations of the algebra A of observables. The content of the present note is the observation that under some mild assumptions, the mathematical structure of representations of A can be analyzed rather effortlessly, to a certain extent: Each representation can be labeled by sets of functions and measures on the space of (generalized) connections that fulfill certain conditions. These considerations are however mostly of mathematical nature. Their physical content remains to be clarified, and physically interesting examples are yet to be constructed.