# H-Theorem in QFT?

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In nonequilibrium statistical mechanics, the H-Theorem says that some kind of entropy increases for a system evolving in accordance with the Boltzmann equation as expected by the second law of thermodynamics.

This can (for example for a classical system) be derived by inserting the one-particle distribution function $f_1$ and its evolution in accordance with the Boltzmann equation into the change of entropy

$\frac{dS}{dt} = -\frac{df_1}{dt}(1+\ln f_1) = \cdots \ge 0$

as for example nicely explained here.

As in QFT, the S matrix generally contains all interactions/correlations, I would not expect an H-Theorem to put restrictions on the scattering processes that are possible.

However, are there any regimes/situations that can be described by something like a Boltzmann or Master equation in QFT too, in which case something like a H-Theorem would become relevant to describe irreversible processes happening?

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An H-theorem can be expected only in dissipative situations, where some notion of entropy increases. In QFT proper. everything is reversible and the entropy $S=-$tr$\rho \log\rho$ is conserved, $dS/dt=0$, which is the degenerate form of an H-theorem.

However, once high energy degrees of freedoms are approximated (e.g. through an exact renormalization group equation), entropy is created in the resulting effective description since a little information is lost to the discarded degrees of freedom. Thus effective theories are strictly speaking dissipative and should have an H-theorem. In quantum optics, this is generally done and leads in a Markovian approximation to dissipative systems governed by a Lindblad equation (or quantum master equation). This equation describes the dynamics of the density matrix of the reduced system itself, taking into account weak coupling to the environment in an effective fashion.

However, for effective quantum field theories, it is general practice to discard the small dissipative contributions (in a similar spirit as one considers the motion of the solar system to be reversible) and then obtains approximate conservative effective theories.

When going to a coarsened semiclassical description dissipation is more relevant. This is why kinetic equations (Boltzmann and Kadanoff-Baym) and hydrodynamic equations (Navier-Stokes and reaction-diffusion) are dissipative and have an H-theorem.

In each case, the increase of entropy is due to the inability to recover information from neglected fast-changing degree of freedom.

The process of going from a conservative description ot a dissipative description is usually analyzed by the Zwanzig projection operator formalism. It is nicely and thoroughly described in a book by Grabert, "Projection operator techniques in nonequilibrium statistical mechanics", with a number of different typical situations distinguished.

In relativistic QFT, the dissipative semiclassical formulation gives the Kadanoff-Baym equations, derived using the closed time path (CTP) formalism and an exact RGE; see, e.g., the introductory paper by Berges.

answered Aug 7, 2016 by (15,608 points)

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