Phonon Gravity, Non-equilibrium QFT, and the Tolman Thermal Equivalence Principle

We describe an extension of the Keldysh method for fermions from constant temperature to steady state case with spatially varying temperature field.This is done with the use on the imaginary section of the Keldysh path of a thermal Hamiltonian obtained from the zero temperature relativistic Hamiltonian by coupling its density multiplicatively to the temperature field. We show that the two Hamiltonians commute, provided appropriate boundary conditions are imposed. A microscopical equation on the temperature field and the corresponding microscopical Fourier law of heat transfer are derived for the relativistic and the non-relativistic cases. We discuss application of the proposed method to the thermoelectric effect and point out a remarkable correspondence between the non-equilibrium thermal Hamiltonian and the zero temperature fermionic Hamiltonian in general relativity for the metric obtained from the Minkowski metric by rescaling time with the inverse temperature.Our results suggest the existence of the correspondence principle between gravitating equilibrium and non-gravitating non-equilibrium quantum field theories, which we call the Tolman thermal equivalence principle in honor of his pioneering work.