Dimensional Phenomenology in Polymeric Quantization Framework

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.