Polymer quantization and the saddle point approximation of partition functions

Abstract

The saddle point approximation of the path integral partition functions is an important way of deriving the thermodynamical properties of black holes. However, there are certain black hole models and some mathematically analog mechanical models for which this method cannot be applied directly. This is due to the fact that their action evaluated on a classical solution is not finite and its first variation does not vanish for all consistent boundary conditions. These problems can be dealt with by adding a counterterm to the classical action, which is a solution of the corresponding Hamilton-Jacobi equation.In this work we study the effects of polymer quantization on a mechanical model presenting the aforementioned difficulties and contrast it with the above counterterm method. This type of quantization for mechanical models is motivated by the loop quantization of gravity which is known to play a role in the thermodynamics of black hole systems.The model we consider is a nonrelativistic particle in an inverse square potential, and analyze two polarizations of the polymer quantization in which either the position or the momentum is discrete. In the former case, Thiemann's regularization is applied to represent the inverse power potential but we still need to incorporate the Hamilton-Jacobi counterterm which is now modified by polymer corrections. In the latter, momentum discrete case however, such regularization could not be implemented. Yet, remarkably, owing to the fact that the position is bounded, we do not need a Hamilton-Jacobi counterterm in order to have a well-defined saddle point approximation. Further developments and extensions are commented upon in the discussion.

Path integral polymer propagator of relativistic and nonrelativistic particles

“…More precisely, once one fixes a certain λ as a free parameter of the theory, and starts from an initial state |q 0 , it is seen from (2.13) (and also (2.12)) that the wave functions q|Ψ are restricted to the lattice points {q n |q n = q 0 + nλ , n ∈ Z} and the eigenvalues of the operatorq are discrete. On the contrary, values of p corresponding to the basis |p are not discrete but take values on a circle, i.e., − πh λ ≤ p < πh λ (see for example appendix A of [28]). The discreteness in q is thus inherent in polymer type of theories.…”

“…Interestingly, a parallelism between the two point functions for cosmology and the relativistic particle has shed some light on the timeless and deparametrized frameworks [27]. For other systems some mechanical models have been considered that resemble the problematics in analyzing the semiclassical approximation of certain black holes [28]. Also polymer field theories have been dealt with by adopting this approach [29,30].…”

Instanton solutions on the polymer harmonic oscillator

“…Nevertheless, it also renders periodic the semiclassical description, via path integral formulation [28][29][30] which is, in this sense, a desired result. This feature results from the fact that (1) is a particular case of an almost periodic Schrödinger operator [31,32] i.e., operators formed by a finite sum of holonomies.…”

Bounds on the polymer scale from gamma ray bursts

“…This procedure is somehow technical, and it is thus described in Appendix A (see also Refs. [23,27,[29][30][31][32][33][34]). It turns out that such an effective Hamiltonian can be obtained by replacing1 For the polymer harmonic oscillator, the dynamics superselects equidistant graphs with polymer scale µ [23].…”

“…arXiv:1704.08750v4 [gr-qc] 30 Nov 2017 which has the form of a harmonic oscillator for each mode k and each polarization r. We now implement the polymer representation on this classical theory in the spirit of Ref. 7.…”