# LSZ like formalism for (nonequilibrium) statistical mechanics?

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In QFT, the LSZ formula can be used to calculate S-matrix elements from time ordered products of fields (say $\phi)$), which gives the probability for a system to evolve from an initial state at asymptotic time consisiting of free particles to another asymptotic state consisting of other free particles. The only thing that is needed for it to work is that the fields $\phi$ create free states at asymptotic states.The whole interacting physics is encoded in the S-matrix.

In (nonequilibrium) statistical mechanics, the evolution of a system can for example be calculated in certain cases from the master equation or from the Boltzmann equation. However, the master equation does only work for Markov processes (that do not depend on the past or have no memory) and the Boltzmann equation is only valid for dilute systems as it includes only binary collisions/interactions. So compared to the S-matrix formalism in QFT, these approaches to calculate the evolution of a system in nonequilibrium statistical mechanics seem to be rather limitied and incomplete.

Does there exist some kind of S-Matrix formalism or even something like an LSZ formula too, for example to calculate the transition of a system between two different long-lived (metastable) states that includes all interactions/correlations at least in principle?

edited Aug 4, 2016

The previous last paragraph has been moved to a new question...

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The S-matrix formalism can be applied to effective field theories involving unstable particles (simply by giving the masses a small imaginary part encoding their half-life - no other change to the usual setting).

Apart from that I do not know anything resembling an asymptotic theory for dissipative systems.

There is however, an interesting connection between S-matrices of QFT and master equations, explained in some detail in Weinberg's Quantum Field Theory book, Vol I, Section 3.6. Indeed, the H-theorem for the Pauli master equation is shown there to be intimately related to the unitarity of the S-matrix, via the Low equations and the resulting optical theorem. (If you are interested in that I could expand a bit on this.)

OK, here are some more details about the latter connection: Weinberg relates the S-matrix elements to transition rates (rather than transition probabilities) in Section 3.4, and uses this relation to define a corresponding master equation; the rate of change of the probability density $P_\alpha$ (where $\alpha$ indexes the possible scattering eigenstates) is obtained by the usual balance equation matching what goes in and what goes out (Weinberg, eq. (3.6.19)). The H-theorem for this equation can then be proved assuming that the transition rates satisfy detailed balance. This detailed balance condition is a consequence of time reversal invariance together with the Low equations. The latter are equivalent to the unitarity of the S-matrix.

The Master equation is meaningful of course only in a context where many collisions take place everywhere, thus in a chemical or nuclear reaction context. Time reversal invariance also requires the absence of external magnetic fields.

answered Aug 7, 2016 by (15,468 points)
edited Aug 9, 2016

Hi Arnold,

Yes I would like to read some more comments about the relation between the S-matrix and the master equation, as I do not (yet) have the Weinberg book...

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