Professor Achim Kempf in his lecture note mentioned that non-commutativity of quantum observables in the associated Poisson algebra to the system, impose CCR
It was Dirac who first realized that all of the Poisson algebra structure that we defined above can be kept (and therefore the ability to derive the equations of motion), while changing one little thing: allowing the symbols $ \hat{x}_{i}^{(r)}(t) $ and $ \hat{p}_{i}^{(r)}(t) $ to be noncommutative, though only in a particular way. Consistency with the Poisson algebra structure imposes strict conditions on the form that this noncommutativity can take. Namely, following Dirac, let us consider the Poisson bracket$$\left\{\hat{u}_{1} \hat{u}_{2}, \hat{v}_{1} \hat{v}_{2}\right\}\tag{3.3}$$where $ \hat{u}_{1}, \hat{u}_{2}, \hat{v}_{1}, \hat{v}_{2} $ are arbitrary polynomials in the variables $ \hat{x}_{i}^{(r)} $ and $ \hat{p}_{j}^{(s)} $. Expression Eq.3.3 can be evaluated in two ways and, of course, any noncommutativity of the $ \hat{x}_{i}^{(r)} $ and $ \hat{p}_{j}^{(s)} $ has to be such that both ways yield the same outcome:$$\begin{aligned}[t]\left\{\hat{u}_{1} \hat{u}_{2}, \hat{v}_{1} \hat{v}_{2}\right\} &=\hat{u}_{1}\left\{\hat{u}_{2}, \hat{v}_{1} \hat{v}_{2}\right\}+\left\{\hat{u}_{1}, \hat{v}_{1} \hat{v}_{2}\right\} \hat{u}_{2} \\&=\hat{u}_{1}\left(\hat{v}_{1}\left\{\hat{u}_{2}, \hat{v}_{2}\right\}+\left\{\hat{u}_{2}, \hat{v}_{1}\right\} \hat{v}_{2}\right)+\left(\hat{v}_{1}\left\{\hat{u}_{1}, \hat{v}_{2}\right\}+\left\{\hat{u}_{1}, \hat{v}_{1}\right\} \hat{v}_{2}\right) \hat{u}_{2}\end{aligned}\tag{3.4}$$This must agree with:$$\begin{aligned}[t]\left\{\hat{u}_{1} \hat{u}_{2}, \hat{v}_{1} \hat{v}_{2}\right\} &=\hat{v}_{1}\left\{\hat{u}_{1} \hat{u}_{2}, \hat{v}_{2}\right\}+\left\{\hat{u}_{1} \hat{u}_{2}, \hat{v}_{1}\right\} \hat{v}_{2} \\&=\hat{v}_{1}\left(\hat{u}_{1}\left\{\hat{u}_{2}, \hat{v}_{2}\right\}+\left\{\hat{u}_{1}, \hat{v}_{2}\right\} \hat{u}_{2}\right)+\left(\hat{u}_{1}\left\{\hat{u}_{2}, \hat{v}_{1}\right\}+\left\{\hat{u}_{1}, \hat{v}_{1}\right\} \hat{u}_{2}\right) \hat{v}_{2}\end{aligned}\tag{3.5}$$Thus:$$\left\{\hat{u}_{1}, \hat{v}_{1}\right\}\left(\hat{v}_{2} \hat{u}_{2}-\hat{u}_{2} \hat{v}_{2}\right)=\left(\hat{v}_{1} \hat{u}_{1}-\hat{u}_{1} \hat{v}_{1}\right)\left\{\hat{u}_{2}, \hat{v}_{2}\right\}\tag{3.6}$$Since this has to hold for all possible choices of $ \hat{u}_{1}, \hat{u}_{2}, \hat{v}_{1}, \hat{v}_{2} $, we require all expressions $ \hat{u}, \hat{v} $ in the position and momentum variables to obey:$$\hat{v} \hat{u}-\hat{u} \hat{v}=k\{\hat{u}, \hat{v}\}\tag{3.7}$$Here, $ k $ must be independent of $ \hat{u} $ and $ \hat{v} $ and must be commuting with everything. But what value does $ k $ take?
Can it be considered as origin of the canonical quantization procedure?
