Loop quantum cosmology with self-dual variables

Abstract

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.

15 pages, v2: Extended with a more complete treatment of parity transformations, other minor improvements, v3: Discussion significantly extended, typos corrected, references added, v4: Typos corrected

References in corpus (7)

• Quantum Nature of the Big Bang: Improved dynamics
• Loop quantum cosmology of k=1 FRW models
• Corrections to the Friedmann Equations from LQG for a Universe with a Free Scalar Field
• Closed FRW model in Loop Quantum Cosmology
• Hybrid quantization of an inflationary universe
• Loop Quantum Cosmology with Complex Ashtekar Variables
• Loop quantum gravity without the Hamiltonian constraint

Cited by in corpus (16)

• Cosmological Effective Hamiltonian from full Loop Quantum Gravity Dynamics
• Testing loop quantum cosmology
• Phenomenological investigation of a quantum gravity extension of inflation with the Starobinsky potential
• The Thiemann Complexifier and the CVH algebra for Classical and Quantum FLRW Cosmology
• Linking loop quantum gravity quantization ambiguities with phenomenology
• Spacetime and dark matter from spontaneous breaking of Lorentz symmetry
• Protected $SL(2,\mathbb{R})$ Symmetry in Quantum Cosmology
• Anisotropic loop quantum cosmology with self-dual variables
• Noncommutative gravity with self-dual variables
• Hamiltonian formulation of gravity as a spontaneously-broken gauge theory of the Lorentz group
• Super Cartan geometry and the super Ashtekar connection
• Supersymmetric minisuperspace models in self-dual loop quantum cosmology
• Towards self dual Loop Quantum Gravity
• Simplicity constraints: a 3d toy-model for Loop Quantum Gravity
• Spacetime Quanta? : The Discrete Spectrum of a Quantum Spacetime Four-Volume Operator in Unimodular Loop Quantum Cosmology
• Covariant single-field formulation of effective cosmological bounces