Institute for Mathematical Physics Quantization of Diieomorphism-invariant Theories with Fermions Quantization of Diieomorphism-invariant Theories with Fermions

We extend ideas developed for the loop representation of quantum gravity to diieomorphism-invariant gauge theories coupled to fermions. Let P ! be a principal G-bundle over space and let F be a vector bundle associated to P whose ber is a sum of continuous unitary irreducible representations of the compact connected gauge group G, each representation appearing together with its dual. We consider theories whose classical connguration space is A F, where A is the space of connections on P and F is the space of sections of F , regarded as a collection of Grassmann-valued fermionic elds. We construct thèquantum connguration space A F as a completion of A F. Using this we construct a Hilbert space L 2 (A F) for the quantum theory on which all automorphisms of P act as unitary operators, and determine an explicit`spin network ba-sis' of the subspace L 2 ((A F)=G) consisting of gauge-invariant states. We represent observables constructed from holonomies of the connection along paths together with fermionic elds and their conjugate momenta as operators on L 2 ((A F)=G). We also construct a Hilbert space Hdii of diieomorphism-invariant states using the group averaging procedure of Ashtekar, Lewandowski, Marolf, Mour~ ao and Thiemann.