TITLE: Is there any published physics article where $q$-mathematics is applied?
QUESTION [12 upvotes]: 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?
[1] http://math.ucr.edu/home/baez/week183.html

REPLY [11 votes]: 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 

Jimbo and Miwa, Algebraic Analysis of Solvable Lattice Models, esp Chapters 0--3;   
Gómez, Ruiz-Altaba and Sierra, Quantum Groups in Two-Dimensional Physics, esp Section 2.3 and its references;
Chari and Pressley, A Guide to Quantum Groups, esp Section 7.5 and its bibliographical notes. 

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 

Fuchs, Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory.