Editing Canonical quantization (section)

==Issues and limitations==
===Classical and quantum brackets===
Dirac's book<ref name="dirac"/> details his popular rule  of  supplanting  [[Poisson bracket]]s   by  [[commutator]]s:
{{Equation box 1
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|equation =   <math>\{A,B\} \longmapsto \tfrac{1}{i \hbar} [\hat{A},\hat{B}] ~.</math>
|cellpadding= 6
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|border colour = #0073CF
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One might interpret this proposal as saying that we should seek a "quantization map" <math>Q</math> mapping a function <math>f</math> on the classical phase space to an operator <math>Q_f</math> on the quantum Hilbert space such that
<math display="block">Q_{\{f,g\}} = \frac{1}{i\hbar}[Q_f,Q_g]</math>
It is now known that there is no reasonable such quantization map satisfying the above identity exactly for all functions <math>f</math> and {{nowrap|<math>g</math>.}}{{Citation needed|date=April 2023}}

===Groenewold's theorem===
One concrete version of the above impossibility claim is Groenewold's theorem (after Dutch theoretical physicist [[Hilbrand J. Groenewold]]), which we describe for a system with one degree of freedom for simplicity. Let us accept the following "ground rules" for the map <math>Q</math>. First, <math>Q</math> should send the constant function 1 to the identity operator. Second, <math>Q</math> should take <math>x</math> and <math>p</math> to the usual position and momentum operators <math>X</math> and <math>P</math>. Third, <math>Q</math> should take a polynomial in <math>x</math> and <math>p</math> to a "polynomial" in <math>X</math> and <math>P</math>, that is, a finite linear combinations of products of <math>X</math> and <math>P</math>, which may be taken in any desired order. In its simplest form, Groenewold's theorem says that there is no map satisfying the above ground rules and also the bracket condition
<math display="block">Q_{\{f,g\}} = \frac{1}{i\hbar} [Q_f,Q_g]</math>
for all polynomials <math>f</math> and <math>g</math>.

Actually, the nonexistence of such a map occurs already by the time we reach polynomials of degree four. Note that the Poisson bracket of two polynomials of degree four has degree six, so it does not exactly make sense to require a map on polynomials of degree four to respect the bracket condition. We ''can'', however, require that the bracket condition holds when <math>f</math> and <math>g</math> have degree three. Groenewold's theorem<ref>{{harvnb|Hall|2013}} Theorem 13.13</ref> can be stated as follows:
{{math theorem
| math_statement = There is no quantization map <math>Q</math> (following the above ground rules) on polynomials of degree less than or equal to four that satisfies
<math display="block"> Q_{ \{f, g\} } = \frac{1}{i\hbar}[Q_f,Q_g]</math>
whenever <math>f</math> and <math>g</math> have degree less than or equal to three. (Note that in this case, <math>\{f,g\}</math> has degree less than or equal to four.)
}}

The proof can be outlined as follows.<ref>{{cite journal | last=Groenewold | first=H.J. | title=On the principles of elementary quantum mechanics | journal=Physica | publisher=Elsevier BV | volume=12 | issue=7 | year=1946 | issn=0031-8914 | doi=10.1016/s0031-8914(46)80059-4 | pages=405–460| bibcode=1946Phy....12..405G }}</ref><ref>{{harvnb|Hall|2013}} Section 13.4</ref> Suppose we first try to find a quantization map on polynomials of degree less than or equal to three satisfying the bracket condition whenever <math>f</math> has degree less than or equal to two and <math>g</math> has degree less than or equal to two. Then there is precisely one such map, and it is the [[Weyl quantization]]. The impossibility result now is obtained by writing the same polynomial of degree four as a Poisson bracket of polynomials of degree three ''in two different ways''. Specifically, we have
<math display="block">x^2 p^2 = \frac{1}{9} \{x^3,p^3\} = \frac{1}{3} \{x^2p,xp^2\}</math>
On the other hand, we have already seen that if there is going to be a quantization map on polynomials of degree three, it must be the Weyl quantization; that is, we have already determined the only possible quantization of all the cubic polynomials above.

The argument is finished by computing by brute force that
<math display="block">\frac{1}{9}[Q(x^3),Q(p^3)]</math>
does not coincide with
<math display="block">\frac{1}{3}[Q(x^2p),Q(xp^2)].</math>
Thus, we have two incompatible requirements for the value of <math>Q(x^2p^2)</math>.

===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]].