An approximate local modular quantum energy inequality in general quantum field theory

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.