The relations between thermodynamics and quantum field theories is treated in books and papers about non-equilibrium statistical mechanics. Here macroscopic many-particle systems are considered, and the focus is either on equilibrium, or on a dynamical description at finite times. For a readable introduction see, e.g.,
J. Berges, Introduction to Nonequilibrium Quantum Field Theory, AIP Conf. Proc. 739 (2004), 3–62. (preprint version: hep-ph/0409233)
What is computable in finite-time QFT? The central objects in finite-time QFT are the expectations of products of one or more field operators at different spacetime positions in a given Heisenberg (space-time independent) state. The
state characterizes the system under consideration; the expectations are collectively referred to as N-point functions; for N = 2 as correlations functions. If the system is of macroscopic, human-scale size we have
a piece of solid or fluid matter.
To find out what the N-point functions mean, we turn to the macroscopic limit of quantum field theories. These are treated in a more or less approximate way in typical books on nonequilibrium statistical mechanics, at least for the case of ideal gases. The end result is always a classical description, usually hydromechanic or kinetic. From these limiting descriptions one can infer that the 1-point functions are just the quantum analogues of the classical fields in 4-dimensional space-time, whereas the Wigner transforms of the 2-point functions are the quantum analogues of the classical fields on a 7-dimensional mass shell of an 8-dimensional phase space. Thus not the field operators themselves but the N-point functions with N = 1 or N = 2 contain the classically observable information. N-point functions with larger N no longer have a direct classical meaning but appear in the BBGKY-like truncation schemes for constructing classical dynamical equations from the quantum description.
Note that the nonlinearity of the quantum field equations directly translate into strong nonlinearities of the macroscopic effective equations (Navier–Stokes, Boltzmann, quantum BBGKY). This shows that quantum field theory does not only predict probabilities but predicts the whole of macroscopic classical mechanics (though proofs are available only in special cases). Quantum field theory predicts – under the usual assumptions of statistical
mechanics, which include local equilibrium – hydrodynamics and elasticity theory, and hence everything computable from it. Of course it predicts only the general theoretical structure, since all the detail depends on the initial conditions. But it predicts in principle all material properties, and quantum chemists are doing precisely that when they use the Dirac–Fock–Bogoliubov approximation of QED.
All items mentioned are essentially exact predictions of QFT, with errors dominated by the computational techniques available rather than the uncertainty due to the averaging. Together with prepared or observed initial conditions it predicts the values of the macroscopic observables at later times. For example, computational fluid dynamics is an essential tool for the optimization of modern aircrafts. Local equilibrium itself is usually justified in an ad hoc way assuming fast relaxation scales. These can probably be derived, too, though I haven’t seen a derivation. But one knows when this condition is not satisfied in practice – namely if the mean free path length is too long. This happens for very dilute gases, where the Boltzmann equation must be used instead of hydrodynamic equations (and can be derived from QFT).
Note that the standard properties of expectation values imply intrinsic uncertainty estimates for the accuracy of the observable fields (N-point functions). For application to actual macroscopic measurements we do not need definite values but only values accurate enough to match experimental practice. This is a much less severe condition. We all know from classical non-equilibrium thermodynamics that the macroscopic local observables
are a small set of fields (in the simplest case just internal energy density and mass density). We also know from statistical mechanics in the grand canonical ensemble that these are given microscopically not by eigenvalues but by certain well-defined expectations. Under the assumption of local equilibrium, the fluctuations of the corresponding averaged quantum fields around the expectations are negligible.
Thus the values of the macroscopic effective fields (obtained by corresponding small-scale averaging in the statistical coarse-graining procedure) are sharp for all practical purposes. Mathematically, this becomes exact only in the thermodynamic limit. But for observable systems, which have finite extent, one can estimate the uncertainties through the standard fluctuation formulas of statistical mechanics. One finds that for macroscopic observations at the human length and time scale, we typically get engineering accuracy. This is the reason why engineering was already successful long before the advent of quantum mechanics. Thus quantum field theory provides a satisfying description of isolated quantum systems ranging from the microscopic scattering system to the macroscopic fluid and solid systems of everyday life.
