Polymer-Fourier quantization of the scalar field revisited

Abstract

The Polymer Quantization of the Fourier modes of the real scalar field is studied within algebraic scheme. We replace the positive linear functional of the standard Poincaré invariant quantization by a singular one. This singular positive linear functional is constructed as mimicking the singular limit of the complex structure of the Poincaré invariant Fock quantization. The resulting symmetry group of such Polymer Quantization is the subgroup $\mbox{SDiff}(\mathbb{R}^4)$ which is a subgroup of $\mbox{Diff}(\mathbb{R}^4)$ formed by spatial volume preserving diffeomorphisms. In consequence, this yields an entirely different irreducible representation of the Canonical Commutation Relations, non-unitary equivalent to the standard Fock representation. We also compared the Poincaré invariant Fock vacuum with the Polymer Fourier vacuum.

29 pages

References in corpus (7)

• Polymer Quantum Mechanics and its Continuum Limit
• Loop Quantum Gravity: An Inside View
• The propagator in polymer quantum field theory
• Hamiltonian and physical Hilbert space in polymer quantum mechanics
• The Quantum Configuration Space of Loop Quantum Cosmology
• Exploring the diffeomorphism invariant Hilbert space of a scalar field
• Canonical Quantization of the Scalar Field: The Measure Theoretic Perspective

Cited by in corpus (2)

• Response of a rotating detector coupled to a polymer quantized field
• Casimir effect in polymer scalar field theory