ausblenden:
Schlagwörter:
General Relativity and Quantum Cosmology, gr-qc
Zusammenfassung:
Using the complex-valued self-dual connection variables, the loop quantum
cosmology of a closed Friedmann universe coupled to a massless scalar field is
studied. It is shown how the reality conditions can be imposed in the quantum
theory by choosing a particular measure for the inner product in the
kinematical Hilbert space. While holonomies of the self-dual Ashtekar
connection are not well-defined in the kinematical Hilbert space, it is
possible to introduce a family of generalized holonomy-like operators, some of
which are well-defined; these operators in turn are used in the definition of a
Hamiltonian constraint operator where the scalar field can be used as a
relational clock. The resulting quantum dynamics are similar, although not
identical, to standard loop quantum cosmology constructed from the
Ashtekar-Barbero variables with a real Immirzi parameter. Effective Friedmann
equations are derived, which provide a good approximation to the full quantum
dynamics for sharply-peaked states whose volume remains much larger than the
Planck volume, and they show that for these states quantum gravity effects
resolve the big-bang and big-crunch singularities and replace them by a
non-singular bounce. Finally, the loop quantization in self-dual variables of a
flat Friedmann space-time is recovered in the limit of zero spatial curvature
and is identical to the standard loop quantization in terms of the real-valued
Ashtekar-Barbero variables.
