About the quantum plane.

Question

We know taht the quantum plane, denoted by $\mathbb F_q[x,y]$ or $\mathcal{O}_q(\mathbb F^2)$, is the $\mathbb F$-algebra generated by $x$ and $y$ subject to the relation $yx-qxy=0,$ where $q\in \mathbb F\backslash\{0\}:=\mathbb F^{*}$ is some scalar.

My question is that what does "quantum" refers to in quantum plane? Why it is called the "quantum" plane? Is it related to "q"? Is it related to quantum in the physics literature?

Many thanks!

Answer 1

In English, the term "quantum plane" appears to be due to Manin (1987), who relayed this notion as a way to naturally define the quantum group $\mathrm{GL}_q(2)$ as the automorphism group of some object $\mathbb{A}_q^{2|0}$.

In turn, the term "quantum group" originates (in English) with Drinfel'd (1986), based on analogy:

• In quantum mechanics, the observables are self-adjoint members of a non-commutative $C^*$-algebra and the pure states are extreme points of the set of normalized positive linear functionals on that algebra. When we "set $\hbar=0$", the observables comprise a commutative algebra that is a ring of functions on some Poisson manifold, and the states are points on that manfiold.
• In an algebraic group, the "observables" are the commutative ring of regular functions on the group and the "pure states" are points of that group. Heuristically, a quantum group is a "quantization" of some reductive algebraic group with noncommutative ring of regular functions that, when $q=1$, reduces to that of the "classical" group.

The use of $q$ in $q$-analogues is a much earlier invention (it dates back at least to 1842 when the first paper on basic hypergeometric series was presented).

Drinfel’d, V. G. 1986. “Quantum Groups.” In Proceedings of the International Congress of Mathematicians, 798–820. Berkeley, CA: American Mathematical Society.

Manin, Yu. I. 1987. “Some Remarks on Koszul Algebras and Quantum Groups.” Annales de l’Institut Fourier 37 (4): 191–205.

Answer 2

Maybe you already realize this by now, but maybe someone else finds it useful to note that the quantum plane can also be viewed as a deformation quantization of the "commutative plane".

In the context of deformation quantization, it is natural to look at quantizing the algebra of complex-valued functions on a real Poisson manifold, where here we should take $\mathbb R^2$ with the Poisson structure given by the bivector field $\pi = x y \frac{\partial}{\partial y} \wedge \frac{\partial}{\partial x}$ so that $\{ f, g \} = xy \bigl( \frac{\partial f}{\partial y} \frac{\partial g}{\partial x} - \frac{\partial f}{\partial x} \frac{\partial g}{\partial y} \bigr)$ for any smooth functions $f, g$ on $\mathbb R^2$. This Poisson structure is often called the log-canonical Poisson structure on $\mathbb R^2$ since $\log x$ and $\log y$ are "canonical coordinates" in the sense that $\{ \log y, \log x \} = 1$.

Setting $q = 1 + \mathrm i \hbar$, or even $q = 1 + \mathrm i \hbar + \dotsb$ for any formal power series in $\mathrm i \hbar$ starting with $1 + \mathrm i \hbar$, one has that $$ \mathbb C \langle x, y \rangle [[ \hbar ]] / (yx - q xy) $$ is a formal deformation quantization of $\pi$. One choice of star product is the one determined by $x \star y = xy$ and $y \star x = q xy$. If $q = 1 + \mathrm i \hbar$ or $q = \mathrm e^{\mathrm i \hbar}$, say, then $\hbar$ (corresponding to the reduced Planck constant) can be evaluated to any complex constant — this corresponds to taking $q \in \mathbb C$ or in $\mathbb C \setminus \{ 0 \}$.

The "classical limit" $\hbar = 0$ corresponds to $q = 1$, which recovers the algebra of "classical observables" $\mathbb C \langle x, y \rangle / (yx - xy) = \mathbb C [x, y]$.

In general, one might want to look not only at polynomial functions on $\mathbb R^2$ as above, but at bigger function spaces, but at least in some sense the "quantum plane" can be viewed as the algebra of quantum observables for the plane $\mathbb R^2$, and I would say that the case of general $\mathbb F$ is by analogy to the real/complex case in the physics literature.