Editing Quantization (physics) (section)

== Deformation quantization ==
{{main| Phase space formulation}}
{{see also|Weyl quantization|Moyal product|Wigner quasi-probability distribution}}

One of the earliest attempts at a natural quantization was Weyl quantization, proposed by Hermann Weyl in 1927.<ref>
{{cite journal
 |last=Weyl |first=H.
 |year=1927
 |title=Quantenmechanik und Gruppentheorie
 |journal=[[Zeitschrift für Physik]]
 |volume=46 |issue= 1–2|pages=1–46
 |bibcode=1927ZPhy...46....1W
 |doi=10.1007/BF02055756
 |s2cid=121036548
}}</ref>   Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. In 1946, H. J. Groenewold<ref name="Groenewold1946">{{cite journal|last1=Groenewold|first1=H.J.|title=On the principles of elementary quantum mechanics|journal=Physica|volume=12|issue=7|year=1946|pages=405–460|issn=0031-8914|doi=10.1016/S0031-8914(46)80059-4|bibcode=1946Phy....12..405G}}</ref> considered the product of a pair of such observables and asked what the corresponding function would be on the classical phase space. This led him to discover the phase-space star-product of a pair of functions.
More generally, this technique leads to deformation quantization, where the ★-product is taken to be a deformation of the algebra of functions on a symplectic manifold or Poisson manifold. However, as a natural quantization scheme (a [[functor]]), Weyl's map is not satisfactory. 

For example, the Weyl map of the classical angular-momentum-squared is not just the quantum angular momentum squared operator, but it further contains a constant term {{sfrac|3ħ<sup>2</sup>|2}}. (This extra term offset  is pedagogically significant, since it accounts for the nonvanishing angular momentum of the ground-state Bohr orbit in the hydrogen atom, even though the standard QM ground state of the atom has vanishing {{mvar|l}}.)<ref name="DahlSchleich2002">{{cite journal|last1=Dahl|first1=Jens Peder|last2=Schleich|first2=Wolfgang P.|title=Concepts of radial and angular kinetic energies|journal=Physical Review A|volume=65|issue=2|year=2002|page=022109|issn=1050-2947|doi=10.1103/PhysRevA.65.022109|arxiv=quant-ph/0110134|bibcode=2002PhRvA..65b2109D|s2cid=39409789}}</ref>

As a mere ''representation change'', however, Weyl's map is useful and important, as it underlies the alternate ''equivalent'' [[phase space formulation]] of conventional quantum mechanics.