Born-Jordan Pseudodifferential Operators and the Dirac Correspondence: Beyond the Groenewold-van Hove Theorem

Abstract

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.