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Numerical Evaluation of the Causal Set Propagator in 2D Anti-de Sitter Spacetime

Arsim Kastrati, Haye Hinrichsen

Abstract

We numerically investigate the application of the path-sum-based causal set scalar propagator construction to (1+1)-dimensional Anti-de Sitter (AdS) spacetime. Building upon a generalization of Johnston's path sum approach, we simulate Poisson-sprinkled causal sets in AdS$_{1+1}$ and numerically evaluate the retarded scalar propagator, comparing it to the known continuum result. Our results confirm that even in curved spacetimes with constant negative curvature, the discrete causal set path sum reproduces the continuum propagator without modification of the flat-spacetime jump amplitudes, thereby providing further numerical support for former analytical results and the applicability of the path sum formalism to curved Lorentzian manifolds.

Numerical Evaluation of the Causal Set Propagator in 2D Anti-de Sitter Spacetime

Abstract

We numerically investigate the application of the path-sum-based causal set scalar propagator construction to (1+1)-dimensional Anti-de Sitter (AdS) spacetime. Building upon a generalization of Johnston's path sum approach, we simulate Poisson-sprinkled causal sets in AdS and numerically evaluate the retarded scalar propagator, comparing it to the known continuum result. Our results confirm that even in curved spacetimes with constant negative curvature, the discrete causal set path sum reproduces the continuum propagator without modification of the flat-spacetime jump amplitudes, thereby providing further numerical support for former analytical results and the applicability of the path sum formalism to curved Lorentzian manifolds.

Paper Structure

This paper contains 17 sections, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Theoretical framework
  3. Brief Introduction to Causal Set Theory
  4. 2D Anti de-Sitter Spacetime
  5. Continuum Scalar Field Theory
  6. Retarded Propagators in Minkowski and Anti de-Sitter Spacetime
  7. Expansion in the eigenmodes of the Laplace-Beltrami operator
  8. Construction of Discrete Propagators
  9. Discrete Propagators and Jump Amplitudes
  10. Sprinkling Averages
  11. Determining the Jump Amplitude for a given Propagator
  12. Numerical Evaluation and Results
  13. Sprinkling into AdS$_{1+1}$
  14. Jump Amplitudes in AdS$_{1+1}$
  15. Numerical Results
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Illustration of the path sum approach. Every trajectory from $x$ to $y$ is either a direct transition to $y$, or it involves an intermediate event $z$, followed by a path from $z$ to $y$. This is captured in the composition rule in Eq. \ref{['cstpropagator']}.
  • Figure 2: Sprinkled points in AdS$_{1+1}$ with density $\rho=1000$ and cutoff $\epsilon=0.1$. Left: Simple method which generates points in the entire strip $(-\tfrac{\pi}{2},\tfrac{\pi}{2})\times (-\tfrac{\pi}{2}+\epsilon,\tfrac{\pi}{2}-\epsilon)$. Right: Improved method which generates points exclusively inside a rhombus (red line). The cutoff is indicated as a dashed line.
  • Figure 3: Numerical evaluation of the retarded causal set propagator in Eq. \ref{['cstprop']} for $m=10$, obtained from a single sprinkling of $n=18000$ points. Panels (a)–(d) show results for $\mathrm{AdS}_{1+1}$ with different curvature radii $L$. In each case, the causal set propagators are presented as functions of the rescaled manifold proper time and compared with the corresponding continuum retarded propagators. In panels (c) and (d), the flat-space continuum propagator with effective mass $m_{\text{eff}} = mL$ is additionally included (dotted curve) for comparison.
  • Figure 4: Numerical results for the retarded causal set propagator (\ref{['cstprop']}) with $m=10$ for flat 2D Minkowski (a) and 2D anti de-Sitter spacetime with curvature radius $L=1$ from a single sprinkling with $n=18000$, together with the corresponding continuum solutions, all plotted over the (rescaled) manifold proper time.