On the space of generalized fluxes for loop quantum gravity

Abstract

We show that the space of generalized fluxes - momentum space - for loop quantum gravity cannot be constructed by Fourier transforming the projective limit construction of the space of generalized connections - position space - due to the non-abelianess of the gauge group SU(2). From the abelianization of SU(2), U(1)^3, we learn that the space of generalized fluxes turns out to be an inductive limit, and we determine the consistency conditions the fluxes should satisfy under coarse-graining of the underlying graphs. We comment on the applications to loop quantum cosmology, in particular, how the characterization of the Bohr compactification of the real line as a projective limit opens the way for a similar analysis for LQC.

arXiv comment25 pages, 2 figures; matches published version

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• Relational evolution of effectively interacting GFT quantum gravity condensates
• Noncommutative fields and the short-scale structure of spacetime
• Projective Loop Quantum Gravity I. State Space
• A new realization of quantum geometry
• Towards a Hartle-Hawking state for loop quantum gravity