Uniqueness of Diffeomorphism Invariant States on Holonomy Flux Algebras
Abstract
Loop quantum gravity is an approach to quantum gravity that starts from the Hamiltonian formulation in terms of a connection and its canonical conjugate. Quantization proceeds in the spirit of Dirac: First one defines an algebra of basic kinematical observables and represents it through operators on a suitable Hilbert space. In a second step, one implements the constraints. The main result of the paper concerns the representation theory of the kinematical algebra: We show that there is only one cyclic representation invariant under spatial diffeomorphisms. While this result is particularly important for loop quantum gravity, we are rather general: The precise definition of the abstract *algebra of the basic kinematical observables we give could be used for any theory in which the configuration variable is a connection with a compact structure group. The variables are constructed from the holonomy map and from the fluxes of the momentum conjugate to the connection. The uniqueness result is relevant for any such theory invariant under spatial diffeomorphisms or being a part of a diffeomorphism invariant theory.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 November 2006
 DOI:
 10.1007/s0022000601007
 arXiv:
 arXiv:grqc/0504147
 Bibcode:
 2006CMaPh.267..703L
 Keywords:

 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Mathematical Physics
 EPrint:
 38 pages, one figure. v2: Minor changes, final version, as published in CMP