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!

 

[Entropia]

Hello,

Integrable Quantum Field Theory (IQFT) is a branch of Quantum Field Theory that focuses on the study of integrable systems in the context of quantum field theory. It combines the principles of quantum mechanics and classical integrable systems to study the behavior of quantum fields in a way that is solvable and tractable. This approach has been successful in solving many complex problems in quantum field theory, such as the calculation of correlation functions, scattering amplitudes, and exact solutions to certain models.

The theoretical background of Integrable QFT lies in the concept of integrability, which is the ability of a system to be described by a set of conserved quantities that do not change over time. In the context of IQFT, these conserved quantities are related to symmetries of the system, such as the Yang-Baxter equation or the Bethe ansatz equations. These symmetries play a crucial role in understanding the behavior of the system and solving it analytically.

The relationship between QFT and IQFT is that IQFT is a subset of QFT. While QFT deals with a wide range of systems, IQFT focuses specifically on integrable systems within QFT. This means that IQFT provides a useful framework for understanding and solving certain problems within QFT, but not all problems can be solved using IQFT methods.

The most interesting aspect of IQFT, in my opinion, is its ability to solve complex problems in QFT using analytical techniques. This has led to significant advances in our understanding of quantum field theories and their applications in various fields, including high energy physics and condensed matter physics.

The Ising Model is a good example of an integrable system that can be studied using IQFT methods. It is a simple model that describes the behavior of interacting spins in a lattice, and it has been extensively studied in the context of statistical mechanics and condensed matter physics. IQFT has provided exact solutions for the Ising Model, which have helped us understand the properties of this system in great detail.

The work on IQFT is ongoing, and there is still much to be explored and improved upon. One direction of research is to extend IQFT methods to non-integrable systems, which would greatly broaden its applications. Another direction is to apply IQFT techniques to other areas of physics, such as quantum gravity and cosmology. Overall, IQFT continues to be a valuable tool for studying complex systems in quantum field theory, and its development and applications are