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

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.
Comment: 23 pages, 16 figures, source code as ancillary files