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We describe an extension of the Keldysh method for fermions from constanttemperature to steady state case with spatially varying temperature field.Thisis done with the use on the imaginary section of the Keldysh path of a thermalHamiltonian obtained from the zero temperature relativistic Hamiltonian bycoupling its density multiplicatively to the temperature field. We show thatthe two Hamiltonians commute, provided appropriate boundary conditions areimposed. A microscopical equation on the temperature field and thecorresponding microscopical Fourier law of heat transfer are derived for therelativistic and the non-relativistic cases. We discuss application of theproposed method to the thermoelectric effect and point out a remarkablecorrespondence between the non-equilibrium thermal Hamiltonian and the zerotemperature fermionic Hamiltonian in general relativity for the metric obtainedfrom the Minkowski metric by rescaling time with the inverse temperature.Ourresults suggest the existence of the correspondence principle betweengravitating equilibrium and non-gravitating non-equilibrium quantum fieldtheories, which we call the Tolman thermal equivalence principle in honor ofhis pioneering work.

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