Quantum Geometry from the Formalism of Loop Quantum Gravity

Introducing Quantum Geometry as a consequence of the quantisation procedure of loop quantum gravity. By recasting general relativity in terms of 1 2 -flat connections, specified by the Holst’s Modification to the Palatini action, we can recast general relativity as a gauge theory. By preforming a 3+1 split of space-time a Legendre transformation can be preformed to give an expression of the Hamiltonian, in terms of the configuration variables, and the Vector, Gauss and Hamiltonian constraints on it. The system is then quantised by refined algebraic quantisation; a suitable choice of polarisation is chosen and by smearing the phase space variables a suitable representation of the Poisson algebra is constructed. Lattice gauge theory is applied on graphs and the quantum configuration space and the Hilbert space, which admits a spin decomposition, is constructed. The conjugate momentum defines an operator which, its representation, defines the Kinematical Hilbert space. The constraint operators, that generate symmetries and diffeomorphism, are constructed on the kinematical Hilbert space. The geometrical operators are constructed and their corresponding eigenvalues approximate to the continuum theory.