Massless Scalar Field Propagator in a Quantized Space-Time

TL;DR

This paper analyzes the behavior of the massless scalar field propagator in Snyder's discretized non-commuting spacetime, revealing how it behaves across different energy scales and its implications for multiparticle states.

Contribution

It provides a detailed analytic study of the scalar propagator in a non-commutative discretized spacetime, highlighting its real and imaginary parts and their physical implications.

Findings

Propagator is real and behaves like 1/p^2 near zero momentum.

Real part smoothly vanishes at the discretization scale and remains zero beyond.

Imaginary part indicates multiparticle production and saturates beyond the discretization scale.

Abstract

We consider in detail the analytic behaviour of the non-interacting massless scalar field two-point function in H.S. Snyder's discretized non-commuting spacetime. The propagator we find is purely real on the Euclidean side of the complex $p^2$ plane and goes like $1/p^2$ as $p^2\to 0$ from either the Euclidean or Minkowski side. The real part of the propagator goes smoothly to zero as $p^2$ increases to the discretization scale $1/a^2$ and remains zero for $p^2>1/a^2$. This behaviour is consistent with the termination of single-particle propagation on the ultraviolet side of the discretization scale. The imaginary part of the propagator, consistent with a multiparticle-production branch discontinuity, is finite and continuous on the Minkowski side, slowly falling to zero when $1/a^2<p^2<\infty$. Finally, we argue that the spectral function for the multiparticle states appears to saturate as $p^2$ probes just beyond the $1/a^2$ discretization scale. We speculate on the cosmological consequences of such a spectral function.