Note on the quantum plane

 Let $k$ be a field  of characteristic $0$ and $F(x,y)$ be the free $k$-algebra generated by $x$ and $y$. Then elements of $F(x,y)$ are finite sums of expressions of the form $cx^{n_1}y^{n_2}....x^{n_w}$ where $c\in k$, $n_i \geq 0$ where we consider that only $n_1$ ad $n_w$ can be zero. If $cm$ is such an expression then we consider $m$ to be the non-commutative monomial determined by $s$ and can also consider the 1-dimensional $k$-linear subspace  generated by $m$ (whose elements are of the form $cm$ for $c \in k$).

Now consider $M(x,y) = F(x,y)/ (xy -yx - h)$ for some non-zero $h \in k$. 

Notice that given a non-commutative monomial $m$ then $m$ determines a unique polynomial in $x$ and $y$ (with monomials all ordered in the form $x^ay^b$) with coeficients in $\mathbb{Z}[h]$ and such that the leading coeficient is $1$.  

Then we obtain a map 

$ Q_s:  Mon \rightarrow \mathbb{Z}[h][x,y] \rightarrow k[x,y]$

where $Mon$ are the non-commutative monomials in $M(x,y)$ determined by some $s$. We note that the coeficients in $Q_s(m)$ are uniquely determined by $s$.

Example: $yx^2y = yxxy = (xy - h)xy = xyxy - hxy = x(xy - h)y - hxy = x^2y^2 - hxy - hxy = x^2y^2 -2hxy$. 

This can be  extended to a $k$-linear map $Q$ on $M(x,y)$ but it is not clear which is the best way or what the range should be. Perhaps we define $Q : M(x,y) \rightarrow k[x,y]$ via the condition that its restriction to $Mon$ admits a factorization through $\mathbb{Z}[h][x,y]$ as for $Q_s$ above.

We can think of the $\mathbb{Z}[h]$ coeficients as generalizing energy eigenvalues, particle numbers, or spin and $M(x,y)$ as a kind of Fock space. The field $k$ itself is the "vacuum" and $h = xy - yx$ might be given a process interpretation (but this is not clear at the moment).

The prototype for $M(x,y)$ is the operator algebra generated by the operators $x$ and $\frac{d}{dx}$ on some suitable function  (or distribution) space. Since these are the operators that classical position and momentum are "promoted" to, it is natural to view $M(x,\frac{d}{dx})$ as a deformation of the Euclidean plane (phase-space) $(x,p)$ corresponding to the coordinate ring $k[x,p]$. 

How do we interpret our map $Q$? It suggests $M(x,y)$ is in a sense a more fundamental structure and that its projection (or emanation) onto classical space determines an intrinsic set of "quantum" integers (a kind of symbol).