  Quantum weak energy inequalities have recently been extensively discussed as
a condition on the dynamical stability of quantum field states, particularly on
curved spacetimes. We formulate the notion of a quantum weak energy inequality
for general dynamical systems on static background spacetimes and establish a
connection between quantum weak energy inequalities and thermodynamic stability
in the general setting. We show that the free scalar field in representations
induced by quasifree Hadamard states provides an example system, and we
indicate that (1) the microlocal spectrum condition, (2) quantum weak energy
inequalities and (3) the existence of passive states (e.g., mixtures of ground-
and thermal equilibrium states) are essentially equivalent, which is
significant because each of these conditions becomes effective at a different
length scale. [See title page of paper for the full abstract.]
