  For every local quantum field theory on a static, globally hyperbolic
spacetime of arbitrary dimension, assuming the Reeh-Schlieder property, local
preparability of states, and the existence of an energy density as
operator-valued distribution, we prove an approximate quantum energy inequality
for a dense set of vector states. The quantum field theory is given by a net of
von Neumann algebras of observables, and the energy density is assumed to
fulfill polynomial energy bounds and to locally generate the time translations.
While being approximate in the sense that it is controlled by a small parameter
that depends on the respective state vector, the derived lower bound on the
expectation value of the spacetime averaged energy density has a universal
structure. In particular, the bound is directly related to the Tomita-Takesaki
modular operators associated to the local von Neumann algebras. This reveals
general, model-independent features of quantum energy inequalities for a large
class of quantum field theories on static spacetimes.
