Good reading on the Keldysh formalism

Question

I'd like some suggestions for good reading materials on the Keldysh formalism in a condmat context. I'm familiar with the imaginary time, coherent state, path integral formalism, but lately I've been seeing Keldysh more and more in papers. My understanding is that it is superior to the imaginary time formalism it at least one way: one can evaluate non-equilibrium expectations.

Thanks.

Answer 1

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 - Quantum Field Theory of Non-Equilibrium States. 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 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- Quantum Statistical Mechanics: Green's Function Methods in Equilibrium and Nonequilibrium Problems. A classic.

Kubo, Toda and Hashitsume - 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:

Rammer and Smith Review article. Solid.

Kamenev and Levchenko is very advanced, but treats some important details.

Answer 2

[For researchers who study condensed matter physics (i.e. low-energy physics)]

It might be helpful to read following 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 P.35-69 (section 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.

G. Tatara, H. Kohno, and J. Shibata: Microscopic approach to current-driven domain wall dynamics, Phys. Rep. 468 (2008) 213/arXiv:0807.2894.[link]

• We can learn the essence of Keldysh formalism by reading P.289-295 of Phys. Rep.(Appendix B. Brief introduction to non-equilibrium Green function);　it also explains Langreth method in P.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.

T. Kita:Introduction to Nonequilibrium Statistical Mechanics with Quantum Field Theory, Prog. Theor. Phys. 123 (2010) 581, arXiv:1005.0393.[link]

• We can learn the (minimal) essence of Keldysh formalism by reading P.5-20 (section 2-3) of arXiv:1005.0393.　In particular, this article closely explains Feynman rules (Feynman diagram) from the viewpoint of practical use.　On top of this, we can review the point of the second quantization method and Matsubara formalism (i.e. nonrelativistic quantum field theory) by P. 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 Matsubara formalsim (i.e. imaginary-time formalsim) as well as Keldysh formalism (i.e. real-time formalism) and hence, we can learn with comparing each other. In particular, it will be helpful to read section 4-5 (P.79-149).

D. A. Ryndyk, R. Gutierrez, B. Song, and G. Cuniberti: Energy Transfer Dynamics in Biomaterial Systems, (Springer,Heidelberg,2009)/arXiv:0805.0628.

• I happened to find this article, which is also self-contained; we can learn the essence of Keldysh formalism by reading P.47-77 (section 3; Nonequilibrium Green function theory of transport) of arXiv:0805.0628.

The above articles will be reliable and readable. On top of them, we 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, et al.

• 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.

I hope my contribution helps someone to learn Keldysh formalism.

[Comments]

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, Schwinger-Keldysh formalism (i.e. closed time path formalism or the real-time formalism) is not based on the assumption called Gell-Mann and Low theorem (i.e. 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 also can 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.

Answer 3

Here are a few more references that may be useful:

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