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Automorphism covariant representations of the holonomy-flux *-algebra

Andrzej Okolow·Jerzy Lewandowski
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Abstract

We continue an analysis of representations of cylindrical functions and fluxes which are commonly used as elementary variables of Loop Quantum Gravity. We consider an arbitrary principal bundle of a compact connected structure group and following Sahlmann's ideas define a holonomy-flux *-algebra whose elements correspond to the elementary variables. There exists a natural action of automorphisms of the bundle on the algebra; the action generalizes the action of analytic diffeomorphisms and gauge transformations on the algebra considered in earlier works. We define the automorphism covariance of a *-representation of the algebra on a Hilbert space and prove that the only Hilbert space admitting such a representation is a direct sum of spaces L^2 given by a unique measure on the space of generalized connections. This result is a generalization of our previous work (Class. Quantum. Grav. 20 (2003) 3543-3567, gr-qc/0302059) where we assumed that the principal bundle is trivial, and its base manifold is R^d.

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Andrzej Okolow, Jerzy Lewandowski. 2005-02-11. Automorphism covariant representations of the holonomy-flux *-algebra. https://doi.org/10.1088/0264-9381%2F22%2F4%2F002

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