Integrable Quantum Field Theory

[wam_mi]

Hi there,

I am currently studying Quantum Field Theory (well, for about 3 weeks isch), and it's really good fun! I would like to know how QFT relates Integrable QFT... I don't really know what it is. Can anyone tell me

(i) The theoretical background of Integrable QFT
(ii) The relationship between these two field theories
(iii) What's the most interesting part of Integrable QFT (in your opinion)
(iv) The Ising Model (I have read some papers about it, but I don't really understand what's going on there!)
(v) Is the work on Integrable QFT complete now, or can we still improve it? What direction though?

Thank you!

 

[Lonewolf]

You're asking a hell of a question. Are you familiar with classical integrable systems? The question is so wide that only a short reply is appropriate. There are a variety of approaches, mainly Bethe Anzatz, and quantum inverse scattering are used.

i) The main point of integrable QFT is we need an infinite number of conserved quantities in a field theory. There are an important class of field theories where this is automatically true, and these are conformal field theories, and are useful to statistical mechanics, amongst a variety of other areas of physics. Learn standard QFT first, it's much more relevant to reality.

ii) Hardly any quantum field theories are integrable.

iii) It can describe some materials that actually exist pretty well. This is a miracle...

iv) Don't read Onsager's original paper, it's a nightmare. Also, some care is needed. Ising model is a model of statistical mechanics, where 'integrable' is usually called solvable. Subtle, but imporant differences exist.

v) It's not complete. Calculating beyond the first few conservation laws is hard!