• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,075 questions , 2,226 unanswered
5,348 answers , 22,757 comments
1,470 users with positive rep
818 active unimported users
More ...

  LSZ like formalism for (nonequilibrium) statistical mechanics?

+ 2 like - 0 dislike

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?

asked Aug 4, 2016 in Theoretical Physics by Dilaton (6,240 points) [ revision history ]
edited Aug 4, 2016 by Dilaton

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

1 Answer

+ 2 like - 0 dislike

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 Arnold Neumaier (15,787 points) [ revision history ]
edited Aug 9, 2016 by Arnold Neumaier

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...

Thanks in advance :-)

@Dilaton: I added two additional paragraphs to my answer.

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights