TL;DR
This paper introduces quantum geometry and loop quantum gravity, focusing on conceptual and mathematical foundations, including Hilbert space construction and geometric operators, for an audience of physicists and mathematicians.
Contribution
It provides an accessible overview emphasizing the mathematical structures and conceptual issues of quantum geometry and loop quantum gravity.
Findings
Construction of Hilbert spaces for quantum geometry
Definition and analysis of geometric operators like areas and volumes
Clarification of conceptual relationships with traditional approaches
Abstract
The purpose of this contribution is to give an introduction to quantum geometry and loop quantum gravity for a wide audience of both physicists and mathematicians. From a physical point of view the emphasis will be on conceptual issues concerning the relationship of the formalism with other more traditional approaches inspired in the treatment of the fundamental interactions in the standard model. Mathematically I will pay special attention to functional analytic issues, the construction of the relevant Hilbert spaces and the definition and properties of geometric operators: areas and volumes.
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