  Quantization procedures play an essential role in microlocal analysis,
time-frequency analysis and, of course, in quantum mechanics. Roughly speaking
the basic idea, due to Dirac, is to associate to any symbol, or observable,
$a(x,\xi)$ an operator $\mathrm{Op}(a)$, according to some axioms dictated by
physical considerations. This led to the introduction of a variety of
quantizations. They all agree when the symbol $a(x,\xi)=f(x)$ depends only on
$x$ or $a(x,\xi)=g(\xi)$ depends only on $\xi$: \[
\mathrm{Op}(f\otimes1)u=fu,\quad\mathrm{Op}(1\otimes g)u=\mathcal{F}%
^{-1}(g\mathcal{F}u) \] where $\mathcal{F}$ stands for the Fourier transform.
Now, Dirac aimed at finding a quantization satisfying, in addition, the key
correspondence \[
\lbrack\mathrm{Op}(a),\operatorname*{Op}(b)]=i\mathrm{Op}(\{a,b\}) \] where
$[\,,\,]$ stands for the commutator and $\{\,,\}$ for the Poisson brackets,
which would represent a tight link between classical and quantum mechanics.
Unfortunately, the famous Groenewold-van Hove theorem states that such a
quantization does not exist, and indeed most quantization rules satisfy this
property only approximately.
  Now, in this note we show that the above commutator rule in fact holds for
the Born-Jordan quantization, at least for symbols of the type $f(x)+g(\xi)$.
Moreover we will prove that, remarkably, this property completely characterizes
this quantization rule, making it the quantization which best fits the Dirac
dream.
