Propagator in polymer quantum field theory

We study free scalar field theory on flat spacetime using a background-independent (polymer) quantization procedure. Specifically we compute the propagator using a method that takes the energy spectrum and position matrix elements of the harmonic oscillator as inputs. We obtain closed form results in the infrared and ultraviolet regimes that give Lorentz invariance violating dispersion relations and show suppression of propagation at sufficiently high energy.

Figure 1

Energy levels of the polymer oscillator associated with $\pi$-periodic (red solid curves) and $\pi$-antiperiodic (blue dashed curves) eigenfunctions. The energy levels associated with these two sets are degenerate in the small and large $g$ limits, with the exception of the ground state energy (labeled $n=0$) which goes to zero for large $g$.