Diffeomorphism covariant representations of the holonomy-flux star-algebra

Details Diffeomorphism covariant representations of the holonomy-flux star-algebra Diffeomorphism covariant representations of the holonomy-flux star-algebra

2003-02-14

arxiv.org/abs/gr-qc/0302059v3

Class.Quant.Grav. 20 (2003) 3543-3568

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

37 pages, no figures, LaTeX2e, to be published in Class. Quant. Grav; typos corrected, minor clarifying remarks

Scientific paper

10.1088/0264-9381/20/16/302

Recently, Sahlmann proposed a new, algebraic point of view on the loop quantization. He brought up the issue of a star-algebra underlying that framework, studied the algebra consisting of the fluxes and holonomies and characterized its representations. We define the diffeomorphism covariance of a representation of the Sahlmann algebra and study the diffeomorphism covariant representations. We prove they are all given by Sahlmann's decomposition into the cyclic representations of the sub-algebra of the holonomies by using a single state only. The state corresponds to the natural measure defined on the space of the generalized connections. This result is a generalization of Sahlmann's result concerning the U(1) case.

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