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I am looking to understand the impact of finite temperature quantum field theory. specifically low energy version applied to condensed matter. Certainly it helps to treat many-body systems a fundamentally as possible, however under finite temperature most of these quantum effects are negligible or can be treated without field theory.

I am looking the historical context of formalisms like that of Matsubara, Keldysh and KMS. I think these concepts has become much more settled now than a few decades ago. Here is a picture of a wall in the Warsaw University's Centre of New Technologies where the KMS condition is engraved as one of the main equations of science.

Were there historical problems in condensed matter that needed to be fully treated under a quantum field version? How these different versions developed? Where them inspired from relativistic quantum field theory first? Any general account of the matter would be helpful.

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An important early work is L. Dolan and R. Jackiw, Phys. Rev. D 9, 3320 (1974), which unfortunately is behind a paywall. In their introduction they refer to earlier work by Kirzhnits and Linde, Phys. Lett. 42B, 471 (1972), and parallel work by Weinberg, Phys. Rev. D 9, 3357 (1974). The paper by Dolan and Jackiw has more than 3000 citations and (together with the cited publications and a few others) marks the beginning of the "modern period" of QFT at finite temperature.

For work done before the 1970s, they refer to the book by Kadanoff and Baym, Quantum Statistical Mechanics (Benjamin, 1962) which can be read online here. A 2019 reprint is described by the publisher (CRC Press, now Taylor and Francis) as follows:

This book is a very early systematic treatment of the application of the field-theoretical methods developed after the Second World War to the quantum mechanical many-body problem at finite temperature. It describes various techniques that remain basic tools of modern condensed matter physicists.

Here's a screenshot from Kadanoff and Baym's Supplementary Reading (p.201):

Kadanoff and Baym, p.201

The parallel development in the Soviet Union was a recurrent theme. At the time of writing, the authors could not have been aware of the important contribution by L. Keldysh, Diagram technique for nonequilibrium processes, Sov. Phys. JETP. 20, 1018 (1965), regarding what is now called the Schwinger-Keldysh method.

Of course, all papers cited above contain further references.

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    $\begingroup$ Thanks for the references ! Just to be clear I was not necessarily looking for books and papers on the topic. I was looking for a historical perspective, the motivation and the interconnection of all these concepts, even if it is just a brief summary. Do you think you could expand more on that? $\endgroup$
    – Mauricio
    Apr 5, 2023 at 13:26
  • $\begingroup$ I'm afraid that is too tall an order for me at the moment. You'll get an idea from the introductions of the references above. The 1970s revival was based on phase transitions and symmetry restoration in gauge theories at finite temperature, relevant for the physics of the early universe. Kirzhnits and Linde employed an analogy with the Meissner effect of superconductivity which is kind of a paradigm for the physics involved. $\endgroup$
    – Tom Heinzl
    Apr 6, 2023 at 9:18

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