Should I study relativistic QFT to get non-relativistic QFT?

[pines-demon]

First time in PF, I am sorry if I did not choose the right category.

I have been doing theory in condensed matter (mostly numerics) as a PhD but I never got to learn proper quantum field theory (QFT). Aside from a few introductory courses at university, I never learned what is a many-body Green's function (MBGF) properly. I feel bad for not knowing this even if I do not use it. I want to properly learn MBGF for condensed matter, I know already very well second quantization, field operators and so on. However everytime I get one of these condensed matter quantum field theory (QFT) books I feel that they skip most of the understanding of what the formulas mean (I have tried Simons, Bruus, Mahan and others). I can follow everything until I get to MBGF where I find that there is a lot of assumptions, which are to be understood as "natural" or "shut-up and calculate using this" arguments.

I was wondering if it would be of any worth trying to understand relativistic QFT first (I have a good grasp of relativistic quantum mechanics). Taking something like the Peskin and working the way through to understand the philosophy of QFT before going back to condensed matter. My choice for doing that is that there are infinitely more resources for relativistic QFT than in non-relativistic MBGF.

Do you think that it is helpful to take this path? If you have limited time do you find that it is still worth it? Or the tools of relativistic QFT are so different that it is better to struggle with the books and keep it non-relativistic?

 

[anuttarasammyak]

QFT is relativistic in its nature as well as classical electromagnetism is relativitic. I am not sure of benefit of learning non relativistic QFT.

 

[pines-demon]

QFT is relativistic in its nature as well as classical electromagnetism is relativitic. I am not sure of benefit of learning non relativistic QFT.

Of course EM is relativistic. But when handling field theories in condensed matter the symmetries are not necessary the symmetries of free space. The difference is that the standard QFT books focus too much on elementary particles, while condensed QFT is about a whole array of systems (magnets, crystal vibrations, electrons in a metal and so on). People do study and research non-relativistic QFT.

 

[Frabjous]

Maybe @vanhees71 or @Demystifier might have some thoughts.

 

[haushofer]

The book "QFT for the gifted amateur" is written by condensed matter physicists.

I'm inclined to advice you to indeed study relativistic QFT for the reasons you give.

 

[Demystifier]

If the goal is to learn non-relativistic QFT in many-body theory, learning relativistic QFT will probably not be of much help. For learning many-body Green functions, I would suggest Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem.

 

[WernerQH]

I was wondering if it would be of any worth trying to understand relativistic QFT first (I have a good grasp of relativistic quantum mechanics).

I don't think so. For some time I had thought that application of QFT to problems in condensed matter physics was not quite legitimate, at least not what QFT was created for. But I'm now convinced that it's the other way round, that statistical field theory offers a better perspective of what QFT is about: a machinery for calculating correlation functions. You would be starting from the wrong end -- only after Wilson's work do we really have a better understanding of QFT!

I quite like the book "Quantum and Statistical Field Theory" by Michel Le Bellac (Oxford University Press):

This textbook emphasizes the underlying unity of the concepts and methods used in [phase transitions and elementary particle physics], and presents in clear language topics such as perturbative expansion, Feynman diagrams, renormalization, and the renormalization group. It contains detailed applications to condensed matter physics, such as the calculation of critical exponents and a discussion of the XY model. Applications to particle physics include quantum electrodynamics and chromodynamics, electroweak interactions, and lattice gauge theories.

(from the back cover)

 

[pines-demon]

I quite like the book "Quantum and Statistical Field Theory" by Michel Le Bellac (Oxford University Press):

Interesting how Le Bellac starts from condensed matter and then builds up to QED. I did not know about this book I'll give it a try. Thanks

 

[vanhees71]

If you want to do non-relativistic QFT, I don't see any reason, why you should learn relativistic QFT. Non-relativistic QFT is in some respects somewhat simpler than relativistic QFT, because there are no infinities in higher orders perturbation theory. It's just the most efficient formulation for many-body systems, because it builds in the symmetrization/antisymmetrization of states for bosons/fermions automatically, and you don't need to bother with it explicitly. On top you have the very useful tools of Green's function methods in the QFT formulation. The best introduction I can think of is still

L. Kadanoff and G. Baym, Quantum Statistical Mechanics,
The Benjamin/Cummings Publishing Company, New York
(1961).

Another standard book is

A. L. Fetter and J. D. Walecka, Quantum Theory of
Many-Particle Systems, McGraw-Hill Book Company, New
York (1971).

A more modern one is

A. Altland and B. Simons, Condensed Matter Field Theory,
Cambridge University Press, Cambridge, New York,
Melbourne, Madrid, Cape Town, 2 edn. (2010).

 

[pines-demon]

A. L. Fetter and J. D. Walecka, Quantum Theory of
Many-Particle Systems, McGraw-Hill Book Company, New
York (1971).

A more modern one is

A. Altland and B. Simons, Condensed Matter Field Theory,
Cambridge University Press, Cambridge, New York,
Melbourne, Madrid, Cape Town, 2 edn. (2010).

I have managed to follow Fetter & Walecka until 3.7 but then it becomes amazingly definitional or at least it becomes more obscure than in the previous chapters. I was looking for a complement book.

Atland & Simons is not my taste, it is clearly written as lectures notes for people that already know the subject as it prefers to skip many mathematical steps in order to provide an "intuition" of what it means. Also it seems unable to keep the same physical system for more than two pages.

Another problem I am finding when comparing sources is that there are two (maybe three) paths:

1. Going for MBGFs directly
2. Going to path integrals

And for (1) one has some times a whole discussion of Schrödinger's equation and Dyson equations before introducing MBGFs and sometimes this is left for later. And if that was not confusing, some books prefer to focus on bosons while other focus on Fermi systems (Fetter&Walecka).

 

[pines-demon]

So I have been advancing on this. Fetter&Walecka is still the preferred choice. I get more or less what the technique and rules to make Feynman diagrams are. However it still it feels very definitional and unmotivated.

I tried Mattuck's which is good for the motivation part but that book reads like a path integral! All the times you have to jump between the main chapters and the appendices in non-trivial ways. Seriously as suggested by the book the order is something like: 1,2,A,3,I,4,7, first half of 9, B,D,E,F,G,5,C,6,8,F,J,9 and so on.

I still think that relativistic QFT has more resources. Even Fetter-Walecka has a few comments here and there on how some of this stuff is more symmetric or generalizable from relativistic scenarios.