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Dimensional Phenomenology in Polymeric Quantization Framework

Kourosh Nozari, Hamed Ramezani

TL;DR

This work investigates statistical mechanics in the polymer quantization framework, motivated by a minimal length and a maximal momentum from quantum gravity. Using a noncanonical phase-space transformation, it derives a polymer-deformed density of states and a corresponding partition function for an $n$-dimensional harmonic oscillator, revealing a finite phase-space volume and bound momentum. In the high-temperature regime, the approach yields thermodynamic dimensional reduction, effectively reducing degrees of freedom from $n$ to $n/2$ and drastically constraining internal energy, specific heat, and entropy through deformed Bessel-function expressions. The results provide a phenomenological bridge between quantum gravity concepts and statistical mechanics, offering insights into spacetime microstructure and potential observable consequences at Planck-scale thermodynamics. In the low-energy limit, standard classical thermodynamics are recovered, validating the framework as a controlled semiclassical probe of minimal length effects.

Abstract

In this paper, we study the statistical mechanics within the polymer quantization framework in the semiclassical regime. We apply a non-canonical transformation to the phase space variables. Then, we use this non-canonical transformation to calculate the deformed density of states of the $2n$-dimensional phase space, which encompasses all polymer effects. In the next step, some thermodynamic features of a system of $n$-dimensional harmonic oscillators are studied by computing the deformed partition function. The results show that the number of microstates decreases because there is an upper bound on the momentum within the polymer framework. We found that in the high-temperature regime, when the thermal de Broglie wavelength is close to the Planck length, $n$ degrees of freedom of the system are frozen in this setup. In other words, there is an effective reduction in space dimensions from $n$ to $\frac{n}{2}$ in the polymeric framework, which also signals the fractional dimension for odd-dimensional oscillators.

Dimensional Phenomenology in Polymeric Quantization Framework

TL;DR

This work investigates statistical mechanics in the polymer quantization framework, motivated by a minimal length and a maximal momentum from quantum gravity. Using a noncanonical phase-space transformation, it derives a polymer-deformed density of states and a corresponding partition function for an -dimensional harmonic oscillator, revealing a finite phase-space volume and bound momentum. In the high-temperature regime, the approach yields thermodynamic dimensional reduction, effectively reducing degrees of freedom from to and drastically constraining internal energy, specific heat, and entropy through deformed Bessel-function expressions. The results provide a phenomenological bridge between quantum gravity concepts and statistical mechanics, offering insights into spacetime microstructure and potential observable consequences at Planck-scale thermodynamics. In the low-energy limit, standard classical thermodynamics are recovered, validating the framework as a controlled semiclassical probe of minimal length effects.

Abstract

In this paper, we study the statistical mechanics within the polymer quantization framework in the semiclassical regime. We apply a non-canonical transformation to the phase space variables. Then, we use this non-canonical transformation to calculate the deformed density of states of the -dimensional phase space, which encompasses all polymer effects. In the next step, some thermodynamic features of a system of -dimensional harmonic oscillators are studied by computing the deformed partition function. The results show that the number of microstates decreases because there is an upper bound on the momentum within the polymer framework. We found that in the high-temperature regime, when the thermal de Broglie wavelength is close to the Planck length, degrees of freedom of the system are frozen in this setup. In other words, there is an effective reduction in space dimensions from to in the polymeric framework, which also signals the fractional dimension for odd-dimensional oscillators.

Paper Structure

This paper contains 12 sections, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Polymeric statistical mechanics
  3. Polymerization
  4. Density of States
  5. Partition Function
  6. Thermodynamics
  7. Thermodynamical Dimensional Reduction of a System of n-Dimensional Harmonic Oscillators in Polymeric framework
  8. Internal Energy
  9. Specific Heat
  10. Entropy
  11. Summary and Conclusions
  12. Expansion of the Partition Function

Figures (4)

  • Figure 1: The internal energy in terms of temperature (left), and the internal energy in terms of thermal de Broglie wavelength (right). The figures are plotted in units of $\mathcal{N}=5, m=100, \omega=10, \beta = 0.004$.
  • Figure 2: The number of degrees of freedom in terms of temperature (left), and the number of degrees of freedom in terms of thermal de Broglie wavelengththe (right). The figures are plotted in units of $\mathcal{N}=5, m=100, \omega=10, \beta = 0.004$.
  • Figure 3: The specific heat in terms of temperature (left), and the specific heat in terms of thermal de Broglie wavelengththe (right). The figures are plotted in units of $\mathcal{N}=5, m=100, \omega=10, \beta = 0.004$.
  • Figure 4: The entropy in terms of temperature (left), and the entropy in terms of thermal de Broglie wavelengththe (right). The figures are plotted in units of $\mathcal{N}=5, m=100, \omega=10, \beta = 0.004$.