Editing Canonical quantization (section)

===Axioms for quantization===
If {{mvar|Q}} represents the quantization map that acts on functions {{mvar|f}} in classical phase space, then the following properties are usually considered desirable:<ref>{{cite journal | last=Shewell | first=John Robert | title=On the Formation of Quantum-Mechanical Operators | journal=American Journal of Physics | publisher=American Association of Physics Teachers (AAPT) | volume=27 | issue=1 | year=1959 | issn=0002-9505 | doi=10.1119/1.1934740 | pages=16–21| bibcode=1959AmJPh..27...16S }}</ref>
#<math>Q_x \psi = x \psi</math> and <math>Q_p \psi = -i\hbar \partial_x \psi ~~</math>   &nbsp;  (elementary position/momentum operators)
#<math>f \longmapsto Q_f ~~</math> &nbsp;  is a linear map
#<math>[Q_f,Q_g]=i\hbar Q_{\{f,g\}}~~</math> &nbsp;  (Poisson bracket)
#<math>Q_{g \circ f}=g(Q_f)~~</math> &nbsp; (von Neumann rule).

However, not only are these four properties mutually inconsistent, ''any three'' of them are also inconsistent!<ref>{{cite journal | last1=ALI | first1=S. TWAREQUE | last2=Engliš | first2=MIROSLAV | title=Quantization Methods: A Guide for Physicists and Analysts | journal=Reviews in Mathematical Physics | volume=17 | issue=4 | year=2005 | issn=0129-055X | doi=10.1142/s0129055x05002376 | pages=391–490| arxiv=math-ph/0405065 | s2cid=119152724 }}</ref>  As it turns out, the only pairs of these properties that lead to self-consistent, nontrivial solutions are 2 & 3, and possibly 1 & 3 or 1 & 4.  Accepting properties 1 & 2, along with a weaker condition that 3 be true only asymptotically in the limit  {{math|''ħ''→0}} (see [[Moyal bracket]]), leads to [[phase space formulation|deformation quantization]], and some extraneous information must be provided, as in the standard theories utilized in most of physics.  Accepting properties 1 & 2 & 3 but restricting the space of quantizable observables to exclude terms such as the cubic ones in the above example amounts to [[geometric quantization]].