TITLE: Physical interpretation of $q$-deformation
QUESTION [6 upvotes]: I am currently reading the paper Quantum Group Particles and Non-Archimedean Geometry by Volovich and Aref'eva. Here they discuss the difference between $q$-deformation and $\hslash$-deformation. In a classical system, one has
$$
\begin{align}
x_ix_j&=x_jx_i\\
p_ip_j&=p_jp_i\\
p_ix_j&=x_jp_i,
\end{align}
$$
where $x_i$ and $p_i$ are position and momentum, respectively. Then an $\hslash$-deformation gives
$$
\begin{align}
[x_i,x_j]&=0\\
[p_i,p_j]&=0\\
[p_i,x_j]&=i\hslash \delta_{ij},
\end{align}
$$
where $\delta_{ij}$ is the Kronecker delta. This is a quantum system, and I am familiar with this. Then they say that a $q$-deformation gives
$$
\begin{align}
x_ix_j&=qx_jx_i, \quad i<j,\\
p_ip_j&=qp_jp_i\\
p_ix_j&=qx_jp_i + ...,
\end{align}
$$
which is different from an $\hslash$-deformation. This seems to be in connection with what is called a quantum line or quantum plane.
My questions are:
1) What is the physical interpretation of $q$-deformation? It doesn't look like it is the same as going from classical mechanics to quantum mechanics. I have seen $q$-deformation in the case of quantum groups, though nothing was said about the physical interpretation.
2) What is the physical interpretation of a quantum line or quantum plane?
Thanks in advance!

REPLY [3 votes]: 1) What is the physical interpretation of q-deformation? It doesn't look like it is the same as going from classical mechanics to quantum mechanics. I have seen q-deformation in the case of quantum groups, though nothing was said about the physical interpretation.

A relation between the two notions is 

If $[x,y]=h$  (a scalar or central element) then $e^x e^y = q e^y e^x$ with $q = e^h$  (assume the exponential series converge) 

This is consistent with commutative algebras being the case $\hbar = 0$ and $q=1$ respectively. 

2) What is the physical interpretation of a quantum line or quantum plane?

There is no standard physical interpretation at this time.  In some physical models, turning on a magnetic field is a form of noncommutative deformation, and there is braiding in some experimentally realizable 2-d systems, and probably other examples, but there is no physical system you can learn about and then use as an intuitive model for quantum groups.