  We investigate the relation between the actions of Tomita-Takesaki modular
operators for local von Neumann algebras in the vacuum for free massive and
massless bosons in four dimensional Minkowskian spacetime. In particular, we
prove a long-standing conjecture that says that the generators of the mentioned
actions differ by a pseudo-differential operator of order zero. To get that,
one needs a careful analysis of the interplay of the theories in the bulk and
at the boundary of double cones (a.k.a. diamonds). After introducing some
technicalities, we prove the crucial result that the vacuum state for massive
bosons in the bulk of a double cone restricts to a KMS state at its boundary,
and that the restriction of the algebra at the boundary does not depend anymore
on the mass. The origin of such result lies in a careful treatment of classical
Cauchy and Goursat problems for the Klein-Gordon equation as well as the
application of known general mathematical techniques, concerning the interplay
of algebraic structures related with the bulk and algebraic structures related
with the boundary of the double cone, arising from quantum field theories in
curved spacetime. Our procedure gives explicit formulas for the modular group
and its generator in terms of integral operators acting on symplectic space of
solutions of massive Klein-Gordon Cauchy problem.
