# Is there any published physics article where $q$-mathematics is applied?

Excuse me for the concern, but I want to ask you a question.

In 2002 Professor John Baez had published a few articles on his page regarding the possibility of applying $q$-mathematics in the science of physics (see [1]). The scripts were interesting but so far I could not find any article where $q$-mathematics was applied. Is there any published article where $q$-mathematics is applied?

There has been quite a lot of literature on the applications of $q$-numbers, $q$-derivatives, $q$-deformations, etc, of various algebraic models of physics. Such applications range from $q$-deformations of simple harmonic oscillator(s) and angular momentum algebras to the development of quantum groups and their applications in nuclear physics, particle physics and field theories. They can be -roughly- divided in two broad categories (although experts might argue that such classifications can be made much more fine):

A significant amount of related literature can be found at mathematical physics journals such as the Journal of Mathematical Physics, Journal of Physics A: Mathematical and General, Communications of Mathematical Physics, SIGMA etc.

• Just out of curiosity, is there any example of the "phenomenological nature" that is widely accepted by the physics community as the way to go? – Jules Lamers Apr 25 '18 at 15:22

As another example of the second category in Kostantinos Kanakoglou's answer I think it is fair to mention quantum-integrable systems: this topic in physics was pivotal in the historical development of the notion of quantum groups by the Leningrad group (Faddeev et al), and the Japanese group (Jimbo and Miwa et al).

In particular, $U_q(\widehat{\mathfrak{sl}_2})$, the quantum-affine version of $\mathfrak{sl}_2$, naturally arises in the (algebraic Bethe-ansatz) analysis of the partially isotropic (or "XXZ") Heisenberg quantum spin chain, and the closely related six-vertex model in statistical physics. Here the deformation parameter $q$ characterizes the spin chain's partial anisotropy as $\Delta = (q+q^{-1})/2$, with $\Delta=q=1$ the completely isotropic ($\mathfrak{sl}_2$-invariant) point.

More, including many references to published articles, can e.g. be found in the books

Finally, although I am by no means an expert on the topic, let me point out that the second half of the book by Gómez et al is devoted to quantum groups in conformal field theory, see also

• I should probably also mention applications of $q$-mathematics in integrable quantum many-body models (Calogero--Sutherland, Ruijsenaars--Schneiders), where Hecke algebras, Macdonald polynomials, etc play an important role in the model's analysis. – Jules Lamers Aug 22 '17 at 14:55
• you are right. The quantum inverse scattering method, its application(s) in quantum integrable systems (among which the Heisenberg quantum spin chain has a special historical importance) and the consequent developments in $q$-deformations of Lie algebras, quantum groups, etc have gone hand-by-hand with the development and the applications of $q$-mathematics. These examples certainly deserve to be mentioned on their own. – Konstantinos Kanakoglou Aug 23 '17 at 17:18