A Numerical Experiment in DLCQ: Microcausality, Continuum Limit and all that

TL;DR

This paper investigates microcausality violations and the continuum limit in (1+1)D scalar field theory using DLCQ, demonstrating numerical convergence to continuum results and discussing implications for interacting theories.

Contribution

It provides numerical evidence that DLCQ reproduces continuum results for free theories and discusses the continuum limit in interacting cases, clarifying misconceptions about zero modes.

Findings

DLCQ results match continuum Pauli-Jordan and Feynman propagators.

Zero mode contributions are smaller than commonly believed.

Continuum limit can be properly taken in DLCQ for both free and interacting theories.

Abstract

Issues related with microcausality violation and continuum limit in the context of (1+1) dimensional scalar field theory in discretized light-cone quantization (DLCQ) are addressed in parallel with discretized equal time quantization (DETQ) and the fact that Lorentz invariance and microcausality are restored if one can take the continuum limit properly is emphasized. In the free case, it is shown with numerical evidence that the continuum results can be reproduced from DLCQ results for the Pauli-Jordan function and the real part of Feynman propagator. The contributions coming from $k^+$ near zero region in these cases are found to be very small in contrast to the common belief that $k^+=0$ is an accumulation point. In the interacting case, aspects related to the continuum limit of DLCQ results in perturbation theory are discussed.