  In these notes I explain how to describe one-dimensional quantum systems that
are simultaneously near to, but not exactly at, a critical point, and in a
far-from-equilibrium steady state. This description uses a density matrix on
scattering states (of the type of Hershfield's density matrix), or equivalently
a Gibbs-like ensemble of scattering states. The steady state I am considering
is one where there is a steady flow of energy along the chain, coming from the
steady draining / filling of two far-away reservoirs put at different
temperatures. The context I am using is that of massive relativistic quantum
field theory, which is the framework for describing the region near quantum
critical points in any universality class with translation invariance and with
dynamical exponent z equal to 1. I show, in this completely general setup, that
a particular steady-state density matrix occurs naturally from the physically
motivated Keldysh formulation of the steady state. In this formulation the
steady state occurs as a result of a large-time evolution from an initial state
where two halves of the system are separately thermalized. I also show how this
suggests a particular dependence (a "factorization") of the average current on
the left and right temperatures. The idea of this density matrix was proposed
already in a recent publication with my collaborator Denis Bernard, where we
studied it in the context of conformal field theory.
