# What is the correlation between QFT and thermodynamics?

This may be a naive question. In physics many processes are symmetric, except a few involving entropy, or the arrow of time. Another one has to do with heat generation. We can generate heat, or energy in general, for example by making a fire, or through chemical and atomic processes, like nuclear fusion.

We cannot produce cold, though. A refrigerator works by moving heat outside, similarly an AC system. We can make hot, we cannot make cold, we can just displace the heat away.

I assume this has to do with the fact that making cold would mean making small amounts of energy into matter, and it would contradict the laws of thermodynamics as we could make the heat into matter, then transform it back into energy.

This question has an answer that talks about QFT and why energy cannot transform back into mass. So, can the laws of thermodynamics be explained in terms of QFT?

• Energy and mass are equivalent. I think what you are really asking is the relationship between radiation and matter, which is what QFT describes and where thermodynamics inevitably comes in. When we annihilate matter and antimatter or when the weak force leads to the decay of particles and nuclei, the entropy of the final state is higher than the entropy of the initial state (because more degrees of freedom get excited) and that is no different from heat only moving from warmer objects to colder objects. It's basically the second law of thermodynamics at work. It's a very good question, btw. Jun 1, 2016 at 9:11
• I am sorry but your question is not that clear. What do you mean by "explained by QFT"? In general if you go by the path integral formulation of QFT you can see a lot of similarity between Statistical Mechanics and field theory. Jun 1, 2016 at 11:27

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.