  In the theory of nets of observable algebras, the modular operators
associated with wedge regions are expected to have a natural geometric action,
a generalization of the Bisognano-Wichmann condition for nets associated with
Poincare-covariant fields. Here many possible such modular covariance
conditions are discussed (in spacetime of at least three dimensions), including
several conditions previously proposed and known to imply versions of the PCT
and spin-statistics theorems. The logical relations between these conditions
are explored: for example, it is shown that most of them are equivalent, and
that all of them follow from appropriate commutation relations for the modular
automorphisms alone. These results allow us to reduce the study of modular
covariance to the case of systems describing non-interacting particles. Given
finitely many Poincare-covariant non-interacting particles of any given mass,
it is shown that modular covariance and wedge duality must hold, and the
modular operators for wedge regions must have the Bisognano-Wichmann form, so
that the usual free fields are the only possibility. For models describing
interacting particles, it is shown that if they have a complete scattering
interpretation in terms of such non-interacting particles, then again modular
covariance and wedge duality must hold, and the modular operators for wedge
regions must have the Bisognano-Wichmann form, so that wedge duality and the
PCT and spin-statistics theorems must hold.
