  Using the "Liouville space'' (the space of operators) of the massive Ising
model of quantum field theory, there is a natural definition of form factors in
any mixed state. These generalize the usual form factors, and are building
blocks for mixed-state correlation functions. We study the cases of mixed
states that are diagonal in the asymptotic particle basis, and obtain exact
expressions for all mixed-state form factors of order and disorder fields. We
use novel techniques based on deriving and solving a system of nonlinear
functional differential equations. We then write down the full form factor
expansion for mixed-state correlation functions of these fields. Under weak
analytic conditions on the eigenvalues of the density matrix, this is an exact
large-distance expansion. The form factors agree with the known
finite-temperature form factors when the mixed state is specialized to a
thermal Gibbs ensemble. Our results can be used to analyze correlation
functions in generalized Gibbs ensembles (which occur after quantum quenches).
Applying this to the density matrix for non-equilibrium steady states with
energy flows, we observe that non-equilibrium form factors have branch cuts in
rapidity space. We verify that this is in agreement with a non-equilibrium
generalization of the KMS relations, and we conjecture that the leading
large-distance behavior of order and disorder non-equilibrium correlation
functions contains oscillations in the log of the distance between the fields.
