Separable Hilbert space for loop quantization

Abstract

We discuss, within the simplified context provided by the polymeric harmonic oscillator, a construction leading to a separable Hilbert space that preserves some of the most important features of the spectrum of the Hamiltonian operator. This construction may be applied to other polymer quantum mechanical systems, including those of loop quantum cosmology, and is likely generalizable to certain formulations of full loop quantum gravity. It is helpful to sidestep some of the physically relevant issues that appear in that context, in particular those related to superselection and the definition of suitable ensembles for the statistical mechanics of these types of systems.

RevTex4-1, BibTex, 6 pages, replaced with revised version to appear in Phys.Rev.D

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Cited by in corpus (13)

• Loop quantum cosmology of a radiation-dominated flat FLRW universe
• Semi-classical and quantum analysis of the isotropic Universe in the polymer paradigm
• Observations on interfacing loop quantum gravity with cosmology
• Polymer dynamics of isotropic universe in Ashtekar and in volume variables
• Projective Limits of State Spaces: Quantum Field Theory without a Vacuum
• The symplectic group in Polymer Quantum Mechanics
• Instanton solutions on the polymer harmonic oscillator
• Polymer quantization, stability and higher-order time derivative terms
• Geometric phases and cyclic isotropic cosmologies
• Critical Insight into the Cosmological Sector of Loop Quantum Gravity
• The relation between the symplectic group $Sp(4,\mathbb{R})$ and its Lie algebra: its application in polymer quantum mechanics
• Coherent states for fractional powers of the harmonic oscillator Hamiltonian
• Squeeze operator: a classical view