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## Main Question or Discussion Point

Is there a

*simple*derivation of Ginzburg-Landau theory of superconductivity, with emphasis on**simple**, from the BCS theory?- A
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Is there a *simple* derivation of Ginzburg-Landau theory of superconductivity, with emphasis on **simple**, from the BCS theory?

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I think he wants it

Still, that's no consolation since I do not think there is also a "simple" way to do the derivation.

Zz.

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Quantum field theory, or second quantization, is simple in my view. But the way how it is used, for instance, in the book by Abrikosov, Gorkov and Dzyaloshinskii is not simple. On the other hand, the book by Mattuck is much simpler. An even simpler book on QFT is the one by Lancaster and Blundell (QFT for the Gifted Amateur), which in fact does give a rough simple idea of how to get GL from BCS and something more complete on that level would be very desirable. I hope it helps to get a picture of what do I mean by "simple".I think that you are asking for the impossible unless you consider quantum field theory simple.

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dRic2

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(about BCS theory)

then some pages after, this one:

This is above my current understanding, so I don't really know if that's what you are looking for. Anyway the book is very good and it cost only 20€ (Europe).

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That's the original Gorkov's derivation, which is anything but simple.This is above my current understanding, so I don't really know if that's what you are looking for.

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MathematicalPhysicist

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And the equations just keep on coming...

Edit: There's a saying that in maths and physics you don't understand the theories you just get accustomed to them. I believe this is attributed to John Von-Neuman but I say this from memory.

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The simplest explanation of BBGKY I am aware is presented in Tong's lectures:I find BBGKY difficult, tried to understand it from Kardar's textbook; for the life of me this is difficult.

http://www.damtp.cam.ac.uk/user/tong/kinetic.html

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MathematicalPhysicist

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Since I see some texts call it GL and others LG. Obviously there's a competition between the two who is the first to be attributed this theory...

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A. Schmitt, Introduction to Superfluidity, Springer 2015

https://doi.org/10.1007/978-3-319-07947-9

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dRic2

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It appears to be available on arxiv.A. Schmitt, Introduction to Superfluidity, Springer 2015

I know it is a bit off topic, but it is something I wanted to ask for some time now. I'll put it in between spoilers tag.

1) you want to calculate the green function for a (ground) state of interacting particles

2) you switch to the interaction picture

3) you use Gell-Mann and Low's theorem along with Wick's theorem to get a perturbation expansions in which each term is an integral of greens function for the non-interacting system

4) the end.

My question is the following: every time I hear talking about field-theory the are some Lagrangians involved, but in the book I mentioned there are no Lagrangians. Also in the book by Schmitt it seems that this Lagrangian approach is present. Is there a book that explain how these two methods are related to each other ?

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https://arxiv.org/abs/1404.1284

I think Fetter-Walecka is among the best books about non-relativistic QFT, because it emphasizes the equivalence between the "1st and 2nd quantization formulation" of many-body systems in the special case that you deal with situations of conserved particle numbers. It becomes clear rather quickly that also in this case where both formulations are equivalent the field-theory formulation is simpler, because it takes automatically care of the symmetrization (boson) and antisymmetrization (fermion) of the multi-particle wave functions, which is a complicated issue in the 1st-quantization formalism.

The 2nd quantization formalism goes beyond this when it comes to the quasiparticle approach, where the quasiparticle numbers are not conserved, and the quasiparticle approach is obviously very important in many-body physics.

Now it turns out that you get directly to the field-theoretical formulation by starting with a classical field theory, described in terms of the least-action principle, i.e., with a Lagrangian and using the canonical quantization approach to the fields. That's the same approach as Dirac's formulation of the 1st-quantization formalism using the least-action principle for classical point-particle systems.

From a more formal systematic point of view the great advantage of the least-action principle approach to physics is that it lets you formulate symmetry principles (aka Noether's theorems) in a simple way, and that is the key for a systematic understanding, where the entire formalism of QT comes from.

Another very important aspect is that the action principles opens the door to the formulation of QT and particularly also QFT in terms of the Feynman path-integral approach and other related functional approaches. Particularly for gauge theories this is a great simplification compared to the pure operator formalism.

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dRic2

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I'm not so familiar with non-relativistic many-body textbooks. One I know and like is

A. Altland, B. Simons, Condensed Matter Field Theory, Cambridge University press 2010

It contains both 2nd quantization in the operator and the path-integral formalism.

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