I am looking for a reference where the Feynman-Vernon influence functional was defined and used in the context of relativistic quantum field theory. This functional is one method to describe non-equilibrium dynamics for open systems (e.g. coupled to noise) which seems (naively, as an outsider) to be particularly well-suited for field theories where path integral methods are more intuitive.

As a consolation prize, I'd be also interested in applications to other areas of physics (such as dissipative quantum systems), or for more effective or popular methods to describe non-equilibrium dynamics (of open or closed systems) in the context of relativistic quantum field theory (preferably in the path integral language).

  • $\begingroup$ One of the standard framworks to define and develop non-equilibirum QFT is to use Keldysh Green functions defined on a countour going from $t=-\infty$ to now and back to $t=+\infty$. Is your search related to that? $\endgroup$
    – Slaviks
    Sep 21, 2011 at 18:38
  • $\begingroup$ I am familiar with the close path contours in the context of real time correlators in thermal equilibrium. I am more interested in systems far from equilibrium and open systems, and how you deal with them using QFT. This is kinda vague, I know... $\endgroup$
    – user566
    Sep 21, 2011 at 18:48
  • $\begingroup$ The point of Keldysh is to be able to use arbitrary density matrix as the asysmptotic initial condition, not neccessarily equilibrium. The whole thing was designed to define non-equilibrium rigorously. That's one of the directions to check out for answers. $\endgroup$
    – Slaviks
    Sep 21, 2011 at 18:54
  • $\begingroup$ More on Keldysh formalism: physics.stackexchange.com/q/13997/2451 $\endgroup$
    – Qmechanic
    Jun 15, 2013 at 22:37

2 Answers 2


The book Quantum dissipative systems by Weiss dedicates a subsection to the Feynman Vernon method, see also the original reference. See also this article and chapter 18.8 of the book by Kleinert.

It's applied to the Caldeira-Leggett model, which is a toy model for a particle in contact with a heat bath. There are a number of mesoscopic systems out there in which a Feynman-Vernon functional of similar type pops up. I don't have any references, but tunneling junctions in fractional quantum Hall edges, impurities in Luttinger liquids and SQUID devices form three examples. I'm sure the book by Weiss has some references as well.

The Keldysh-Schwinger or real-time formalism is required to treat systems out of equilibrium. For a list of references see this thread here. But this formalism by itself is not enough. You need to make some assumptions regarding the degrees of freedom of the heat bath, the coupling between the subsystem and the external heat bath and also the initial (untangled or not) state of the system as a whole.

The idea as follows: you model the system under consideration in contact with a heat bath. In the Caldeira-Leggett model the heat bath is a macroscopic number of harmonic oscillators, each of which is in contact with the degrees of freedom of the system under consideration. The Feynman-Vernon functional is obtained by integrating out the degrees of freedom associated with the heat bath, all by using a path integral formalism. We can think of this functional as describing the time evolution of the reduced density matrix.


One of the avenues to search for an answer is the so-called Keldysh formalism which is used extensively in condensed matter, in particular in mescopic physics, to define and study steady-state and time-dependent quantum phenomena in systems with infinitely many degrees of freedom. A recent comprehensive review is given by Kamenev and Levchenko, arXiv:0901.3586.

The general idea is as follows: time evolution is defined an a real-time contour going from $t=-\infty$ to $t=+\infty$ and then back, to avoid reference to an unknown final state. The two-time Green functions $G^{ab}(t',t)$ acquire indices $a,b=\pm$ denoting the forward- ($+$) or backward- ($-$) propagating branches of the contour. This gives extra matrix structure to correlators, but many QFT techniques can be adopted to handle this generalization.

I'm not aware of relativistic applications but almost sure it has been done somewhere.


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