  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 inner product for 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 of which some are well-defined;
these operators in turn are used in the definition of the Hamiltonian
constraint operator where the scalar field can be used as a relational clock.
The resulting quantum theory is closely related, 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.
