Statistical Thermodynamics of Polymer Quantum Systems

Abstract

Abstract. Polymer quantum systems are mechanical models quantized similarly as loop quantum gravity. It is actually in quantizing gravity that the polymer term holds proper as the quantum geometry excitations yield a reminiscent of a polymer material. In such an approach both non-singular cosmological models and a microscopic basis for the entropy of some black holes have arisen. Also important physical questions for these systems involve thermodynamics. With this motivation, in this work, we study the statistical thermodynamics of two one dimensional polymer quantum systems: an ensemble of oscillators that describe a solid and a bunch of non-interacting particles in a box, which thus form an ideal gas. We first study the spectra of these polymer systems. It turns out useful for the analysis to consider the length scale required by the quantization and which we shall refer to as polymer length. The dynamics of the polymer oscillator can be given the form of that for the standard quantum pendulum. Depending on the dominance of the polymer length we can distinguish two regimes: vibrational and rotational. The first occur for small polymer length and here the standard oscillator in Schrödinger quantization is recovered at leading order. The second one, for large polymer length, features dominant polymer effects. In the case of the polymer particles in the box, a bounded and oscillating spectrum that presents a band structure and a Brillouin zone is found. The thermodynamical quantities calculated with these spectra have corrections with respect to standard ones and they depend on the polymer length. When the polymer length is small such corrections resemble those coming from the phenomenological generalized uncertainty relation approach based on the idea of the existence of a minimal length. For generic polymer length, thermodynamics of both systems present an anomalous peak in their heat capacity C V . In the case of the polymer oscillators this peak separates the vibrational and rotational regimes, while in the ideal polymer gas it reflects the band structure which allows the existence of negative temperatures.

Casimir effect in polymer scalar field theory

“…Using these results, the authors in Ref. [12] have studied the statistical thermodynamics of such systems.…”

“…PQ has been used to study quantum gravitational effects upon simple quantum systems, such as the harmonic oscillator [5], the particle in a box [6], the diffraction in time [7], the tunneling phenomena [8], the Coulomb potential [9,10], the quantum bouncer [11] and within statistical thermodynamics [12], just to name a few. It * Electronic address: alberto.martin@nucleares.unam.mx has also been applied to the scalar field theory [13][14][15][16] and the electromagnetic field [17].…”

Polymer quantization versus the Snyder noncommutative space

“…Thus, one may work within two equivalent pictures on the separable Hilbert space (22): i) Working with the standard canonical momentum p which satisfies the standard Heisenberg algebra (2) together with the polymermodified Hamiltonian operator (25). ii) Working with the polymeric momentum p µ which satisfies the polymermodified Heisenberg algebra (30) and the corresponding Hamiltonian operator (31) with standard functional form.…”

Strong equivalence principle in polymer quantum mechanics and deformed Heisenberg algebra

“…In this interpretation, various aspects of polymer Quantum Mechanics have been explored and contrasted with the usual Schrodinger Quantum Mechanics. There have been studies of polymer corrections to the dynamics [13][14][15][16][17] or thermodynamics [18][19][20][21] in different quantum systems as well comparison with regards to general features such as implementation of symmetries [22].…”