I'd like some suggestions for good reading materials on the Keldysh formalism in the condensed matter physics community. I'm familiar with the imaginary time, coherent state, and path integral formalisms, but lately, I've been seeing Keldysh more and more in papers. My understanding is that it is superior to the imaginary time formalism at least in that one can evaluate non-equilibrium expectations.
I am somewhat biased towards condensed matter physics, even though the subject extends also to fields such as cosmology and QCD.
In the context of condensed matter physics I recommend the following books (even though various techniques also apply outside this regime):
Rammer's Quantum Field Theory of Non-Equilibrium States. This was my first read on it, and I was quite content with it. If you are familiar with the idea of using periodic imaginary time to simulate a temperature then this book will explain the small additional step you need to take to grasp the basics of the Keldysh formalism. Unfortunately it's just formalism for the first 7 (!) chapters and sometimes the pace is a bit slow.
Kadanoff and Baym's Quantum Statistical Mechanics: Green's Function Methods in Equilibrium and Nonequilibrium Problems. A classic.
Kubo, Toda and Hashitsume's Statistical Physics II: Nonequilibrium Statistical Mechanics. Has some elements of classical statistical physics as well. The authors are very insightful.
There's also chapter 18 in Kleinert which I find a nice read. This book is huge though and treats a lot of other topics. Still, if you go through Rammer then this chapter by Kleinert nicely summarizes everything without dropping too many details. The newest edition of Altland and Simons has two chapters on classical and quantum systems out of equilibrium, but I was fairly disappointed with their treatment considering the rest of the book is fantastic.
As for quantum transport, where this formalism is frequently employed, I can recommend Di Ventra as an undergrad-level introductory book and this book by Datta for some other interesting topics. Weiss is excellent for dissipative (open) systems, although this field opens up a whole new can of worms so you might want to avoid at first.
Other sources not in book form:
Kamenev and Levchenko's article "Keldysh technique and non-linear σ-model: basic principles and applications" (Adv. Phys. 58 no. 3 (2009), pp. 197-319, arXiv:0901.3586) is very advanced, but it treats some important details.
For researchers who study condensed matter physics (i.e. low-energy physics), it might be helpful to read following books and articles.
H. Haug and A. P. Jauho: Quantum Kinetics in Transport and Optics of Semiconductors (Springer, New York, 2007).
- We can learn the (minimal) essence of Keldysh formalism by reading pp. 35-69 (sections 3 and 4). This article carefully explains Langreth method (theorem) in p. 66, which will be one of the most important properties of Keldysh formalism.
- They cover the essence of the Keldysh formalism on pp. 289-295 (Appendix B. Brief introduction to non-equilibrium Green function); they also explain the Langreth method in pp. 292-295 (Appendix B.2. Langreth method). This article will be instructive on the point that it contains many concrete examples of calculations in detail.
- One can learn the (minimal) essence of Keldysh formalism by reading pp. 5-20 (section 2-3). In particular, this article closely explains the Feynman rules (Feynman diagram) from the viewpoint of practical use. On top of this, one can review the point of the second quantization method and the Matsubara formalism (i.e. non-relativistic quantum field theory) on pp. 56-76 (Appendix A-D).
J. Rammer, Quantum Field Theory of Non-equilibrium States, (Cambridge University Press, 2011).
- Of course I have noted that there is a similar article written by the same author (J. Rammer and H. Smith, Rev. Mod. Phys. 58 (1986) 323.), but I would like to recommend this textbook because it is self-contained; it covers the Matsubara formalism (i.e. imaginary-time formalism) as well as the Keldysh formalism (i.e. real-time formalism) and hence, we can learn with comparing each other. In particular, it will be helpful to read sections 4-5 (pp. 79-149).
- I happened to find this article, which is also self-contained; one can learn the essence of Keldysh formalism by reading pp. 47-77 (section 3; Non-equilibrium Green function theory of transport).
The above articles will be reliable and readable. On top of them, one can learn important details from the sophisticated manuscripts by Alex Kamenev:
A. Kamenev: Field Theory of Non-Equilibrium Systems, (Cambridge University Press, 2011, arXiv:0412296).
- I should polish my understanding to comment on it. This article always helps me.
Although (as far as I know) I have listed the relevant articles, I guess I have missed a lot of other important papers. Please forgive me if I have. I hope my contribution helps someone to learn Keldysh formalism.
Last, let me remark the points of Keldysh formalism which I have learned by the above articles; thanks to the Schwinger-Keldysh closed time path, the Schwinger-Keldysh formalism (i.e. closed time path formalism or the real-time formalism) is not based on the assumption usually called the Gell-Mann and Low theorem (i.e. the adiabatic theorem).
Therefore, within the perturbative theory via Schwinger-Keldysh (or contour-ordered) Green's functions, the formalism can deal with an arbitrary time-dependent Hamiltonian and treat the system out of the equilibrium. On top of this, this formalism is applicable to systems at finite temperature; the well-known Matsubara formalism (i.e. the imaginary-time formalism), which can also deal with thermodynamic average values, can be regarded as a simple corollary of the Schwinger-Keldysh formalism.
That is, the Schwinger-Keldysh formalism includes the Matsubara formalism and information about finite temperature is contained in the greater and lesser Green's functions. Consequently we can treat non-equilibrium phenomena at finite temperature thanks to the Schwinger-Keldysh formalism. This will be the strong point of the formalism.
This answer contains some additional resources that may be useful. Please note that answers which simply list resources but provide no details are strongly discouraged by the site's policy on resource recommendation questions. This answer is left here to contain additional links that do not yet have commentary.
J. Berges, Introduction to Nonequilibrium Quantum Field Theory, AIP Conf. Proc. 739 (2004), 3--62. hep-ph/0409233
W. Botermans and R. Malfliet, Quantum transport theory of nuclear matter, Physics Reports 198 (1990), 115--194.
E. Calzetta and B.L. Hu, Nonequilibrium quantum fields: Closed-time-path effective action, Wigner function, and Boltzmann equation, Phys. Dev. D 37 (1988), 2878--2900.
K. Chou, Z. Su, B. Hao, and L. Yu, Equilibrium and nonequilibrium formalisms made unified, Physics Reports 118 (1985), 1--131.
Yu. B. Ivanov, J. Knoll, and D. N. Voskresensky, Self-Consistent Approximations to Non-Equilibrium Many-Body Theory, Nucl. Phys. A 657 (1999), 413--445. hep-ph/9807351
Yu. B. Ivanov, J. Knoll, and D. N. Voskresensky, Resonance Transport and Kinetic Entropy, Nucl. Phys. A 672 (2000), 313--356. nucl-th/9905028
If it is still of interest, next to all the excellent suggestions above there is a book from 2013 which I found rather helpful as it makes some neat observations I could not find in other texts:
Nonequilibrium Many-Body Theory of Quantum Systems - Stefanucci and Leeuwen
Below is my own list, in the context of studying transport in semiconductor nanostructures. Most of these have been already mentioned, but my ordering of the items and comments might be helpful.
- In principle, AGD combined with the Keldysh's original paper are quite sufficient for serious work (surprizingly not mentioned before).
- The introduction of the review article by Rammer & Smith is quite good as a short reference.
- Haug and Jauho's book is quite popular, however it mainly reproduces the results of the series of (very readable) papers by Meir, Wingreen and Lee, Meir and Wingreen, and Meir, Wingreen and Jauho.
- This review by Kamenev is a good entry point into Keldysh formalism in terms of the path integral formulation.
- Finally, the book by Kadanoff & Baym is a classic, that is worth mentioning, because they developed formalism simultaneously with Keldysh.