Polymer quantization and Symmetries

Abstract

Polymer quantization was discovered during the construction of Loop Quantum Cosmology. For the simplest quantum theory of one degree of freedom, the implications for dynamics were studied for the harmonic oscillator as well as some other potentials. For more degrees of freedom, the possibility of continuous, kinematic symmetries arises. While these are realised on the Hilbert space of polymer quantum mechanics, their infinitesimal versions are not supported. For an invariant Hamiltonian, these symmetry realizations imply infinite degeneracy suggesting that the symmetry should be spontaneously or explicitly broken. The estimation of symmetry violations in some cases have been analysed before. Here we explore the alternative of shifting the arena to the distributional states. We discuss both the polymer quantum mechanics case as well as polymer quantized scalar field.

20 pages, version 2, added further explanations and references. Results and conclusions unchanged. Version to appear in CQG

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Cited by in corpus (8)

• Protected $SL(2,\mathbb{R})$ Symmetry in Quantum Cosmology
• Path Integral representation for Polymer Quantized Scalar Fields
• Quantum cosmology of a Bianchi III LRS geometry coupled to a source free electromagnetic field
• The Polymer Bouncer
• The symplectic group in Polymer Quantum Mechanics
• Relativistic Planck-scale polymer
• Quantum Groups and Polymer Quantum Mechanics
• Strong Equivalence Principle in Polymer Quantum Mechanics and deformed Heisenberg Algebra