Perturbative polymer quantum field theory and high energy scalar propagator

Abstract

We advance a proposal of perturbative polymer field theory motivated by the study of the high energy regime of scalar propagator in flat spacetime. To do so we revisit the model by Hossain, Husain and Seahra (HHS) with the field modes taken as polymer harmonic oscillators. In momentum basis, the mode Hamiltonian is a pendulum formed by a rotor plus a trigonometric potential which contains a discrete polymer scale. Mode frequencies much greater than the inverse squared of the polymer scale allow to treat the potential term as a perturbation. Using a Hamiltonian scheme we show the field two-point function acquires corrections over the mechanical rotors both through energy spectrum as well as wave functions. Then we implement an alternative, yet equivalent, path integral scheme for the same field quantity expressed in terms of the mechanical rotor propagator and its corresponding two-point function which also get perturbative corrections. Our approaches can be performed systematically to arbitrary order; previous results are regained at lowest non trivial (second) order. In this way our work provides a different interpretation of HHS model which was developed on the basis of Mathieu functions. Moreover, they pave the way to extensions beyond quadratic operators that can allow the study of interacting fields. These require further work not done here however. Finally we summarize our results.