When Do Measures on the Space of Connections Support the Triad Operators of 
Loop Quantum Gravity?

Sahlmann, Hanno

doi:10.1063/1.3525706

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When Do Measures on the Space of Connections Support the Triad Operators of Loop Quantum Gravity?

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Sahlmann, Hanno
AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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gr-qc_0207112
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Citation

Sahlmann, H. (2011). When Do Measures on the Space of Connections Support the Triad Operators of Loop Quantum Gravity? Journal of Mathematical Physics, 52: 012503. doi:10.1063/1.3525706.

Cite as: https://hdl.handle.net/11858/00-001M-0000-000F-112F-1

Abstract

In this work we investigate the question, under what conditions Hilbert spaces that are induced by measures on the space of generalized connections carry a representation of certain non-Abelian analogues of the electric flux. We give the problem a precise mathematical formulation and start its investigation. For the technically simple case of U(1) as gauge group, we establish a number of "no-go theorems" asserting that for certain classes of measures, the flux operators can not be represented on the corresponding Hilbert spaces. The flux-observables we consider play an important role in loop quantum gravity since they can be defined without recourse to a background geometry, and they might also be of interest in the general context of quantization of non-Abelian gauge theories.