In the study of quantum field theory, one may encounter S-matrix a lot. Recently, in the study of integrability, I encountered R-matrix formulation which I am not familiar with.

First of all, the S-matrix is a scattering matrix which comes from scattering processes.

In the context of quantum mechanics and quantum field theory, we often compute the S-matrix, and we use well-known formulas, i.e, the LSZ reduction formula

On Wikipedia, they describe a R-matrix, as related with the Yang-Baxter equation (governing equation for integrability), and they add some comments that it is related with resonance.

My questions are, then:

  1. Can you give me a clear definition and a governing equation for a R-matrix?

  2. How are S-matrix and R-matrix related to each other?

  3. Why is the R-matrix important in integrability (Yang-Baxter equation)?


Mathematically, the R-matrix is an (invertible) element of a quasi-triangular Hopf algebra. The R-matrix there is what "controls" the failure of the cocommutativity of your Hopf-algebra, and the Yang-Baxter equation is a consequence of all that. You can interprete it as a "braid-like" equation. See the wiki-pages on these subjects for more precisions.

In physics, you usually have symmetry "groups" of your models. These symmetry "groups" appear as representation of abstract "groups". In some cases, it appears that the symmetry "groups" of your models are not groups, nor Lie groups/algebras, but (quasi-triangular) Hopf algebras. This is what happens for quantum spin-chains, say. See the wiki article on "quantum groups".

Now, this formalism is important for integrability because the Yang-Baxter equation ensures that your model (either a statistical mechanics one, a quantum one, or even a QFT one) has enough preserved quantities to be "solved". See for example the quantum inverse scattering methods for one application of this. Note that there are many different applications of this in physics, and as far as I know, it is not necessarily trivial to relate all of them to one abstract framework (surely it has been done).

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