On the mass dependence of the modular operator for a double cone

TL;DR

This paper introduces a numerical method to approximate the modular operator in linear quantum fields, revealing its dependence on mass and angular momentum for double cones in Minkowski spacetime.

Contribution

It develops a discretization-based numerical scheme to analyze the modular operator at the one-particle level for local subalgebras in quantum field theory, extending understanding beyond wedge regions.

Findings

Component of the modular generator is close to a multiplication operator.

The modular operator component depends on mass and angular momentum.

Results suggest mass dependence in double cone regions.

Abstract

We present a numerical approximation scheme for the Tomita-Takesaki modular operator of local subalgebras in linear quantum fields, working at one-particle level. This is applied to the local subspaces for double cones in the vacuum sector of a massive scalar free field in (1 + 1)- and (3 + 1)-dimensional Minkowski spacetime, using a discretization of time-0 data in position space. In the case of a wedge region, one component of the modular generator is well-known to be a mass-independent multiplication operator; our results strongly suggest that for the double cone, the corresponding component is still at least close to a multiplication operator, but that it is dependent on mass and angular momentum.