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General Relativity and Quantum Cosmology, gr-qc
Abstract:
A loop quantization of the diagonal class A Bianchi models starting from the
complex-valued self-dual connection variables is presented in this paper. The
basic operators in the quantum theory correspond to areas and generalized
holonomies of the Ashtekar connection and the reality conditions are
implemented via the choice of the inner product on the kinematical Hilbert
space. The action of the Hamiltonian constraint operator is given explicitly
for the case when the matter content is a massless scalar field (in which case
the scalar field can be used as a relational clock), and it is shown that the
big-bang and big-crunch singularities are resolved in the sense that singular
and non-singular states decouple under the action of the Hamiltonian constraint
operator.
