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Journal Article

Loop quantum cosmology with self-dual variables


Wilson-Ewing,  Edward
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Wilson-Ewing, E. (2015). Loop quantum cosmology with self-dual variables. Physical Review D, 92(12): 123536. doi:10.1103/PhysRevD.92.123536.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0026-A16A-B
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.