View source for Quantization (physics)

{{Short description|Systematic procedure of turning a classical theory into a quantum one}}
{{for|quantization of general classical theories|Canonical quantization}}
{{Other uses|Quantization (disambiguation){{!}}Quantization}}
 
'''Quantization''' (in British English '''quantisation''') is the  systematic transition procedure from a classical understanding of physical phenomena to a newer understanding known as [[quantum mechanics]].  It is a procedure for constructing  [[quantum mechanics]] from [[classical mechanics]].  A generalization involving infinite degrees of freedom  is [[Second quantization|'''field quantization''']], as in the "quantization of the [[electromagnetic field]]", referring to [[photons]] as field "[[Quantum|quanta]]" (for instance as [[light quanta]]). This procedure is basic to theories of [[atomic physics]], chemistry, [[particle physics]], [[nuclear physics]], [[condensed matter physics]], and [[quantum optics]].

== Historical overview == 
In 1901, when [[Max Planck]] was developing the [[distribution function (physics)|distribution function]] of [[statistical mechanics]] to solve the [[ultraviolet catastrophe]] problem, he realized that the properties of [[blackbody radiation]] can be explained by the assumption that the amount of energy must be in countable fundamental units, i.e. amount of energy is not continuous but [[quantum|discrete]]. That is, a minimum unit of energy exists and the following relationship holds  <math>E = h \nu</math>
for the frequency  <math>\nu</math>. Here,  <math>h</math> is called the [[Planck constant]], which represents the amount of the quantum mechanical effect. It means a fundamental change of mathematical model of physical quantities.

In 1905, [[Albert Einstein]] published a paper, "On a heuristic viewpoint concerning the emission and transformation of light", which explained the [[photoelectric effect]] on quantized [[electromagnetic wave|electromagnetic waves]].<ref>{{citation|first=Albrecht|last=Folsing|title=Albert Einstein: A Biography |publisher=trans. Ewald Osers, Viking|year=1997}}</ref> The ''energy quantum'' referred to in this paper was later called "[[photon]]".  In July 1913, [[Niels Bohr]] used quantization to describe the spectrum of a hydrogen atom in his paper "On the constitution of atoms and molecules". 

The preceding theories have been successful, but they are very phenomenological theories.  However, the French mathematician [[Henri Poincaré]] first gave a systematic and rigorous definition of what quantization is in his 1912 paper "Sur la théorie des quanta".<ref>
{{cite journal
 |last=McCormmach |first=Russell
 |title=Henri Poincaré and the Quantum Theory |journal=Isis |volume=58 |issue=1 |pages=37–55 |date=Spring 1967
 |doi=10.1086/350182 |s2cid=120934561
}}</ref><ref>{{cite journal|last=Irons|first=F.E.|title=Poincaré's 1911–12 proof of quantum discontinuity interpreted as applying to atoms|journal=American Journal of Physics|volume=69|issue=8|pages=879–84|date=August 2001|doi=10.1119/1.1356056|bibcode=2001AmJPh..69..879I }}</ref> 

The term "quantum physics" was first used in Johnston's ''Planck's Universe in Light of Modern Physics''.  (1931).

== Canonical quantization ==
{{main|Canonical quantization}}

Canonical quantization  develops [[quantum mechanics]] from [[classical mechanics]].  One introduces a [[commutation relation]]  among [[canonical coordinate]]s. Technically, one converts  coordinates to operators, through combinations of [[creation and annihilation operators]]. The operators act on [[quantum state]]s of the theory. The lowest energy state is called the [[vacuum state]].

== Quantization schemes ==
Even within the setting of canonical quantization, there is difficulty associated to quantizing arbitrary observables on the classical phase space. This is the '''ordering ambiguity''': classically, the position and momentum variables ''x'' and ''p'' commute, but their quantum mechanical operator counterparts do not. Various ''quantization schemes'' have been proposed to resolve this ambiguity,<ref>{{harvnb|Hall|2013}} Chapter 13</ref> of which the most popular is the [[Wigner–Weyl transform|Weyl quantization scheme]]. Nevertheless, the ''Groenewold–van Hove theorem''  dictates that no perfect quantization scheme exists. Specifically, if the quantizations of ''x'' and  ''p''  are taken to be the usual position and momentum operators, then no quantization scheme can perfectly reproduce the Poisson bracket relations among the classical observables.<ref>{{harvnb|Hall|2013}} Theorem 13.13</ref> See [[Canonical quantization#Groenewold.27s theorem|Groenewold's theorem]] for one version of this result.

== Covariant canonical quantization ==

There is a way to perform a canonical quantization without having to resort to the non covariant approach of [[foliation|foliating]] spacetime and choosing a [[Hamiltonian (quantum mechanics)|Hamiltonian]]. This method is based upon a classical action, but is different from the functional integral approach.

The method does not apply to all possible actions (for instance, actions with a noncausal structure or actions with [[analysis of flows|gauge "flows"]]). It starts with the classical algebra of all (smooth) functionals over the configuration space. This algebra is quotiented over by the ideal generated by the [[Euler–Lagrange equation]]s. Then, this quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action, called the [[Peierls bracket]]. This Poisson algebra is then ℏ -deformed in the same way as in canonical quantization.

In [[quantum field theory]], there is also a way to quantize actions with [[analysis of flows|gauge "flows"]]. It involves the [[Batalin–Vilkovisky formalism]], an extension of the [[BRST formalism]].

== 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.

== Geometric quantization ==
{{main|Geometric quantization}}

In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory and the quantum theory remain manifest. For example, the similarity between the Heisenberg equation in the Heisenberg picture of quantum mechanics and the Hamilton equation in classical physics should be built in.

A more geometric approach to quantization, in which the classical phase space can be a general symplectic manifold, was developed in the 1970s by [[Bertram Kostant]] and [[Jean-Marie Souriau]]. The method proceeds in two stages.<ref>{{harvnb|Hall|2013}} Chapters 22 and 23</ref> First, once constructs a "prequantum Hilbert space" consisting of square-integrable functions (or, more properly, sections of a line bundle) over the phase space. Here one can construct operators satisfying commutation relations corresponding exactly to the classical Poisson-bracket relations. On the other hand, this prequantum Hilbert space is too big to be physically meaningful. One then restricts to functions (or sections) depending on half the variables on the phase space, yielding the quantum Hilbert space.

== Path integral quantization ==
{{further|Path integral formulation}}
{{see also|Lattice gauge theory}}
A classical mechanical theory is given by an [[action (physics)|action]] with the permissible configurations being the ones which are extremal with respect to functional [[Calculus of variations|variations]] of the action. A quantum-mechanical description of the classical system can also be constructed from the action of the system by means of the [[path integral formulation]].

== Other types ==
* [[Loop quantum gravity]] (loop quantization)
* [[Uncertainty principle]] (quantum statistical mechanics approach)
* [[Schwinger's quantum action principle]]

== See also ==
{{cols}}
* [[First quantization]]
* [[Feynman path integral]]
* [[Light front quantization]]
* [[Photon polarization]]
* [[Quantum decoherence]]
* [[Quantum Hall effect]]
* [[Quantum number]]
* [[Stochastic quantization]]
{{colend}}

== References ==

* Ali, S. T., & Engliš, M. (2005). "Quantization methods: a guide for physicists and analysts". ''Reviews in Mathematical Physics''  '''17''' (04), 391-490. {{arXiv|math-ph/0405065}}{{doi|10.1142/S0129055X05002376}}
* Abraham, R. & Marsden (1985): ''Foundations of Mechanics'', ed. Addison–Wesley, {{ISBN|0-8053-0102-X}}
* {{citation|first=Brian C.|last=Hall|title=Quantum Theory for Mathematicians|series=Graduate Texts in Mathematics|volume=267 |publisher=Springer|year=2013|bibcode=2013qtm..book.....H }}
*{{Cite book |last=Woodhouse |first=Nicholas M. J. |title=Geometric quantization |date=2007 |publisher=Clarendon Press |isbn=978-0-19-850270-8 |edition=2. ed., repr |series=Oxford mathematical monographs |location=Oxford}}
*{{cite arXiv |last=Landsman |first=N. P. |title=Between classical and quantum |date=2005-07-25  |eprint=quant-ph/0506082}}
* M. Peskin, D. Schroeder, ''An Introduction to Quantum Field Theory'' (Westview Press, 1995) {{ISBN|0-201-50397-2}}
* Weinberg, Steven, ''The Quantum Theory of Fields'' (3 volumes)
* {{Cite journal | last1 = Curtright | first1 = T. L. | last2 = Zachos | first2 = C. K. | doi = 10.1142/S2251158X12000069 | title = Quantum Mechanics in Phase Space | journal = Asia Pacific Physics Newsletter | volume = 01 | pages = 37–46 | year = 2012 | arxiv = 1104.5269 | s2cid = 119230734 }} 
* G. Giachetta, L. Mangiarotti, [[Gennadi Sardanashvily|G. Sardanashvily]], ''Geometric and Algebraic Topological Methods in Quantum Mechanics'' (World Scientific, 2005) {{ISBN|981-256-129-3}}
* {{cite arXiv |eprint=1206.3116 |last1=Todorov |first1=Ivan |title="Quantization is a mystery" |date=2012 |class=math-ph }}

== Notes ==
{{reflist}}

[[Category:Physical phenomena]]
[[Category:Quantum field theory]]
[[Category:Mathematical quantization]]
[[Category:Mathematical physics]]