Mathematical introduction to loop quantum gravity
Seminar Room 1, Newton Institute
The aim of this talk is to introduce the basic definitions of Quantum Geometry and Loop Quantum Gravity. The starting point will be the Ashtekar-Isham holonomy C*-algebra and characterization of its Gel'fand spectrum - the space of quantum connections. The spectrum is endowed with a diffeomorphism invariant measure and plays the role of a (quantum) configuration space. Next, the holonomy-flux *-algebra is introduced. It should be thought of as an algebra of quantum position-momentum variables. There exists a unique diffeomorphism invariant positive functional on the algebra. The corresponding GNS representation is used to define operators of Quantum Geometry -- the kinematic quantum theory of initial data of gravitational field. The quantum Einstein vector constraints generate the group of diffeomomorphisms. The space of solutions is contained in the suitably defined dual vector space. The next step is introduction of the quantum scalar constraint defined by Thiemann. The constraint admits a large family of solutions. The derivations of all the quantum operators are free of infinities. Remaining ambiguities are reduced by various consistency conditions.