Background independent quantizations: the scalar field I

Abstract

We are concerned with the issue of quantization of a scalar field in a diffeomorphism invariant manner. We apply the method used in Loop Quantum Gravity. It relies on the specific choice of scalar field variables referred to as the polymer variables. The quantization, in our formulation, amounts to introducing the `quantum' polymer *-star algebra and looking for positive linear functionals, called states. The assumed in our paper homeomorphism invariance allows to determine a complete class of the states. Except one, all of them are new. In this letter we outline the main steps and conclusions, and present the results: the GNS representations, characterization of those states which lead to essentially self adjoint momentum operators (unbounded), identification of the equivalence classes of the representations as well as of the irreducible ones. The algebra and topology of the problem, the derivation, all the technical details and more are contained in the paper-part II.

13 pages, minor corrections were made in the revised version

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• Polymer quantization of the free scalar field and its classical limit
• The Hole Argument for Covariant Theories
• Effective dynamics of the hybrid quantization of the Gowdy T^3 universe
• Background independent quantizations: the scalar field II
• Towards Loop Quantum Supergravity (LQSG) II. p-Form Sector
• Inflation from non-minimally coupled scalar field in loop quantum cosmology
• Quantum Hamiltonian for gravitational collapse
• Holonomy and inverse-triad corrections in spherical models coupled to matter
• Band structure in the polymer quantization of the harmonic oscillator
• The Immirzi parameter from an external scalar field
• Singularity resolution from polymer quantum matter
• Exploring the diffeomorphism invariant Hilbert space of a scalar field
• Towards new background independent representations for Loop Quantum Gravity
• Quantum reduced loop gravity: extension to scalar field
• Fermions in Loop Quantum Gravity and Resolution of Doubling Problem
• Bounds on the Polymer Scale from Gamma Ray Bursts
• Shape Dynamical Loop Gravity from a Conformal Immirzi Parameter
• Quantization of spherically symmetric loop quantum gravity coupled to a scalar field and a clock: the asymptotic limit
• Regularization of the cosmological sector of loop quantum gravity with bosonic matter and the related problems with general covariance of quantum corrections
• The Proca-field in Loop Quantum Gravity
• Abelian 2+1D Loop Quantum Gravity Coupled to a Scalar Field