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.

**G. Tatara, H. Kohno, and J. Shibata**: *Microscopic approach to current-driven domain wall dynamics* (Phys. Rep. **468** no. 6 (2008) 213, arXiv:0807.2894).

- 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.

**T. Kita**: *Introduction to Non-equilibrium Statistical Mechanics with Quantum Field Theory* (Prog. Theor. Phys. **123** (2010) 581, arXiv:1005.03932).

- 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).

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

- 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.

Finally, this slide of mine also contains useful references, as does this indroductory presentation, both written by me.

This post imported from StackExchange Physics at 2014-08-07 15:35 (UCT), posted by SE-user Kouki Nakata