Editing Canonical quantization

{{Short description|Process of converting a classical physical theory into one compatible with quantum mechanics
}}
{{Quantum field theory}}{{Use American English|date=January 2019}}In [[physics]], '''canonical quantization''' is a procedure for [[quantization (physics)|quantizing]] a [[classical theory]], while attempting to preserve the formal structure, such as [[symmetry (physics)|symmetries]], of the classical theory to the greatest extent possible.

Historically, this was not quite [[Werner Heisenberg]]'s route to obtaining [[quantum mechanics]], but [[Paul Dirac]]  introduced it in his 1926 doctoral thesis, the "method of classical analogy" for quantization,<ref>{{Cite journal | last1 = Dirac | first1 = P. A. M. | title = The Fundamental Equations of Quantum Mechanics | doi = 10.1098/rspa.1925.0150 | journal = Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume = 109 | issue = 752 | pages = 642–653 | year = 1925 |bibcode = 1925RSPSA.109..642D | doi-access = free }}</ref>  and detailed it in his classic text ''Principles of Quantum Mechanics''.<ref name="dirac">{{cite book|last=Dirac|first=P. A. M.|author-link=Paul Dirac|title=Principles of Quantum Mechanics|publisher=Oxford University Press|location=USA|isbn=0-19-852011-5|year=1982}}</ref>  The word ''canonical'' arises from the [[Hamiltonian mechanics|Hamiltonian]] approach to classical mechanics, in which a system's dynamics is generated via canonical [[Poisson bracket]]s, a structure which is ''only partially preserved''  in canonical quantization.

This method was further used by Paul Dirac in the context of [[quantum field theory]], in his construction of [[quantum electrodynamics]].  In the field theory context, it is also called the [[second quantization]] of fields, in contrast to the semi-classical [[first quantization]] of single particles.

==History==

When it was first developed, [[quantum physics]] dealt only with the [[quantization (physics)|quantization]] of the [[motion (physics)|motion]] of particles, leaving the [[electromagnetic field]] [[Classical electromagnetism|classical]], hence the name [[quantum mechanics]].<ref name="van der Waerden">{{cite book|last=van der Waerden|first= B.L.|author-link=Bartel Leendert van der Waerden|title=Sources of quantum mechanics|publisher=Dover Publications|location= New York|isbn=0486618811|year=1968}}</ref>

Later the electromagnetic field was also quantized, and even the particles themselves became represented through quantized fields, resulting in the development of [[quantum electrodynamics]] (QED) and [[quantum field theory]] in general.<ref name="Schweber">{{cite book|last=Schweber|first= S.S. |title=QED and the men who made it|url=https://archive.org/details/qedmenwhomadeitd0000schw|url-access=registration|publisher=Princeton University Press|location= Princeton|isbn=0691033277|year=1983}}</ref> Thus, by convention, the original form of particle quantum mechanics is denoted [[first quantization]], while quantum field theory is formulated in the language of [[second quantization]].

==First quantization==
{{Main|First quantization}}

===Single particle systems===

The following exposition is based on [[Paul Dirac|Dirac's]] treatise on quantum mechanics.<ref name="dirac"/>
In the [[classical mechanics]] of a particle, there are dynamic variables which are called coordinates ({{mvar|x}}) and momenta ({{mvar|p}}). These specify the ''state'' of a classical system. The '''canonical structure''' (also known as the [[Symplectic geometry|symplectic]] structure) of [[classical mechanics]] consists of [[Poisson bracket]]s enclosing these variables, such as {{math|1= {''x'', ''p''} = 1}}.  All transformations of variables which preserve  these brackets are allowed as [[canonical transformation]]s in classical mechanics. Motion itself is such a canonical transformation.

By contrast, in [[quantum mechanics]], all significant features of a particle are contained in  a '''state'''  <math>|\psi\rangle</math>, called a [[quantum state]]. Observables are represented by '''operators''' acting on a [[Hilbert space]] of such [[quantum states]].

The eigenvalue of an operator acting on one of its eigenstates represents the value of a measurement on the particle thus represented. For example, the [[energy]] is read off by the [[Hamiltonian (quantum mechanics)|Hamiltonian]] operator <math>\hat{H}</math> acting on  a state <math>|\psi_n\rangle</math>, yielding
<math display="block">\hat{H}|\psi_n\rangle=E_n|\psi_n\rangle,</math>
where {{math|''E<sub>n</sub>''}} is the characteristic energy associated to this <math>|\psi_n\rangle</math> [[eigenstate]].

Any state could be represented as a [[linear combination]] of eigenstates of energy; for example,
<math display="block">|\psi\rangle=\sum_{n=0}^{\infty} a_n |\psi_n\rangle ,</math>where {{math|''a<sub>n</sub>''}} are constant coefficients.

As in classical mechanics, all dynamical operators can be represented by functions of the position and momentum ones, <math>\hat{X}</math> and <math>\hat{P}</math>, respectively. The connection between this representation and the more usual [[wavefunction]] representation is given by the eigenstate of the position operator <math>\hat{X}</math> representing a particle at position <math>x</math>, which is  denoted by an element <math>|x\rangle</math> in the Hilbert space, and which satisfies <math>\hat{X}|x\rangle = x|x\rangle</math>. Then, <math>\psi(x)= \langle x|\psi\rangle</math>.

Likewise, the eigenstates <math>|p\rangle</math> of  the momentum operator <math>\hat{P}</math> specify the [[Position and momentum space|momentum representation]]: <math>\psi(p)= \langle p|\psi\rangle</math>.

The central relation between these operators is a quantum  analog  of the above [[Poisson bracket]] of classical mechanics, the  '''[[canonical commutation relation]]''',
<math display="block">[\hat{X},\hat{P}] = \hat{X}\hat{P}-\hat{P}\hat{X} = i\hbar.</math>

This relation encodes (and formally leads to) the [[uncertainty principle]], in the form  {{math|Δ''x'' Δ''p'' ≥ ''ħ''/2}}.  This algebraic structure may be thus considered as the quantum analog of the ''canonical structure'' of classical mechanics.

===Many-particle systems===

When turning to N-particle systems, i.e., systems containing N [[identical particles]] (particles characterized by the same [[quantum numbers]] such as [[mass]], [[Electric charge|charge]] and [[Spin (physics)|spin]]), it is necessary to extend the single-particle state function <math>\psi(\mathbf{r})</math> to the N-particle state function <math>\psi(\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_N)</math>. A fundamental difference between classical and quantum mechanics concerns the concept of [[Identical particles|indistinguishability]] of identical particles. Only two species of particles are thus possible in quantum physics, the so-called [[bosons]] and [[fermions]] which obey the following rules for each kind of particle:

* for bosons: <math display="block">\psi(\mathbf{r}_1,\dots,\mathbf{r}_j,\dots,\mathbf{r}_k,\dots,\mathbf{r}_N)=+\psi(\mathbf{r}_1,\dots,\mathbf{r}_k,\dots,\mathbf{r}_j,\dots,\mathbf{r}_N),</math>
* for fermions: <math display="block">\psi(\mathbf{r}_1,\dots,\mathbf{r}_j,\dots,\mathbf{r}_k,\dots,\mathbf{r}_N)=-\psi(\mathbf{r}_1,\dots,\mathbf{r}_k,\dots,\mathbf{r}_j,\dots,\mathbf{r}_N),</math>

where we have interchanged two coordinates <math>(\mathbf{r}_j, \mathbf{r}_k)</math> of the state function. The usual wave function is obtained using the [[Slater determinant]] and the [[identical particles]] theory. Using this basis, it is possible to solve various many-particle problems.

==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
|indent =::
|equation =   <math>\{A,B\} \longmapsto \tfrac{1}{i \hbar} [\hat{A},\hat{B}] ~.</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}

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]].

==Second quantization: field theory==
{{Main|Second quantization}}
[[Quantum mechanics]] was successful at describing non-relativistic systems with fixed numbers of particles, but a new framework was needed to describe systems in which particles can be created or destroyed, for example, the electromagnetic field, considered as a collection of photons. It was eventually realized that [[special relativity]] was inconsistent with single-particle quantum mechanics, so that all particles are now described relativistically by [[quantum field]]s.

When the canonical quantization procedure is applied to a field, such as the electromagnetic field, the classical [[field (physics)|field]] variables become ''[[quantum operator]]s''.  Thus, the normal modes comprising the amplitude of the field are simple oscillators, each of which is  [[quantum oscillator|quantized]] in standard first quantization, above, without ambiguity. The resulting quanta are identified with individual particles or excitations. For example, the quanta of the electromagnetic field are identified with photons. Unlike first quantization, conventional second quantization is completely unambiguous, in effect a [[functor]], since the constituent set of its oscillators are quantized unambiguously.

Historically, quantizing the classical theory of a single  particle gave rise to a wavefunction. The classical equations of motion of a field are typically identical in form to the (quantum) equations for the wave-function of ''one of its quanta''. For example, the [[Klein–Gordon equation]] is the classical equation of motion for a free scalar field, but also the quantum equation for a scalar particle wave-function. This meant that quantizing a field ''appeared'' to be similar to quantizing a theory that was already quantized, leading to the fanciful term '''second quantization''' in the early literature, which is still used to describe field quantization, even though the modern interpretation detailed is different.

One drawback to canonical quantization for a relativistic field is that by relying on the Hamiltonian to determine time dependence, [[relativistic invariance]] is no longer manifest. Thus it is necessary to check that [[relativistic invariance]] is not lost. Alternatively, the [[Path integral formulation|Feynman integral approach]]  is available for quantizing relativistic fields, and is manifestly invariant.  For non-relativistic field theories, such as those used in [[condensed matter physics]], Lorentz invariance is not an issue.

===Field operators===

Quantum mechanically, the variables of a field (such as the field's amplitude at a given point) are represented by operators on a [[Hilbert space]]. In general, all observables are constructed as operators on the Hilbert space, and the time-evolution of the operators is governed by the [[Hamiltonian (quantum mechanics)|Hamiltonian]], which must be a positive operator. A state <math>|0\rangle</math> annihilated by the Hamiltonian must be identified as the [[vacuum state]], which is the basis for building all other states. In a non-interacting (free) field theory, the vacuum is normally identified as a state containing zero particles. In a theory with interacting particles, identifying the vacuum is more subtle, due to [[vacuum polarization]], which implies that the physical vacuum in quantum field theory is never really empty. For further elaboration, see the articles on [[vacuum#The quantum-mechanical vacuum|the quantum mechanical vacuum]] and [[QCD vacuum|the vacuum of quantum chromodynamics]].  The details of the canonical quantization depend on the field being quantized, and whether it is free or interacting.

====Real scalar field====
A [[scalar field theory]] provides a good example of the canonical quantization procedure.<ref>This treatment is based primarily on Ch. 1 in {{cite book|last1=Connes|first1=Alain|author1-link=Alain Connes| last2=Marcolli|first2=Matilde| author2-link=Matilde Marcolli| title=Noncommutative Geometry, Quantum Fields, and Motives|publisher=American Mathematical Society | year=2008|isbn=978-0-8218-4210-2|url=http://www.alainconnes.org/docs/bookwebfinal.pdf|access-date=2010-05-16| archive-date=2009-12-29|archive-url=https://web.archive.org/web/20091229123920/http://www.alainconnes.org/docs/bookwebfinal.pdf | url-status=dead}}</ref>  Classically, a scalar field is a collection of an infinity of [[simple harmonic oscillator|oscillator]] [[normal mode]]s. It suffices to consider a 1+1-dimensional space-time <math>\mathbb{R} \times S_1,</math> in which the spatial direction is [[one-point compactification|compactified]] to a circle of circumference 2{{mvar|π}}, rendering the momenta discrete.

The classical [[Lagrangian (field theory)|Lagrangian]] density describes an [[Quantum harmonic oscillator#Harmonic oscillators lattice: phonons|infinity of coupled harmonic oscillators]], labelled by {{mvar|x}} which is now a ''label'' (and not the  displacement dynamical variable to be quantized), denoted by the classical field {{mvar|φ}},
<math display="block">\mathcal{L}(\phi) = \tfrac{1}{2}(\partial_t \phi)^2 - \tfrac{1}{2}(\partial_x \phi)^2  - \tfrac{1}{2} m^2\phi^2 - V(\phi),</math>
where {{math|''V''(''φ'')}} is a potential term, often taken to be a polynomial or monomial of degree 3 or higher. The action functional is
<math display="block">S(\phi) = \int \mathcal{L}(\phi) dx dt = \int L(\phi, \partial_t\phi) dt \, .</math>The canonical momentum obtained via the [[Legendre transformation]] using the action {{mvar|L}} is <math>\pi = \partial_t\phi</math>, and the classical [[Hamiltonian mechanics#Mathematical formalism|Hamiltonian]] is found to be
<math display="block">H(\phi,\pi) = \int dx \left[\tfrac{1}{2} \pi^2 + \tfrac{1}{2} (\partial_x \phi)^2 + \tfrac{1}{2} m^2 \phi^2 + V(\phi)\right].</math>

Canonical quantization treats the variables  {{mvar|φ}} and  {{mvar|π}} as operators with [[canonical commutation relations]] at time  {{mvar|t}}= 0,  given by
<math display="block">[\phi(x),\phi(y)] = 0, \ \ [\pi(x), \pi(y)] = 0, \ \ [\phi(x),\pi(y)] = i\hbar \delta(x-y).</math>
Operators constructed from  {{mvar|φ}} and  {{mvar|π}} can then formally be defined at other times via the time-evolution generated by the Hamiltonian,
<math display="block"> \mathcal{O}(t) = e^{itH} \mathcal{O} e^{-itH}.</math>

However, since {{mvar|φ}} and {{mvar|π}} no longer commute, this expression is ambiguous at the quantum level.  The problem is to construct a representation of the relevant operators <math>\mathcal{O}</math> on a [[Hilbert space]] <math>\mathcal{H}</math>  and to construct a positive operator {{mvar|H}} as a [[quantum operator]] on this Hilbert space in such a way that it gives this evolution for the operators <math>\mathcal{O}</math> as given by the preceding equation, and to show that <math>\mathcal{H}</math> contains a vacuum state <math>|0\rangle</math> on which {{mvar|H}} has zero eigenvalue.  In practice, this construction is a difficult problem for interacting field theories, and has been solved completely only in a few simple cases via the methods of [[constructive quantum field theory]]. Many of these issues can be sidestepped using the Feynman integral as described for a particular {{math|''V''(''φ'')}} in the article on [[scalar field theory]].

In the case of a free field, with  {{math|1=''V''(''φ'') = 0}}, the quantization procedure is relatively straightforward.  It is convenient to [[Fourier transform]] the fields, so that
<math display="block">  \phi_k = \int \phi(x) e^{-ikx} dx, \ \ \pi_k = \int \pi(x) e^{-ikx} dx. </math>
The reality of the fields implies that
<math display="block">\phi_{-k} = \phi_k^\dagger,  ~~~ \pi_{-k} = \pi_k^\dagger .</math>The classical Hamiltonian may be expanded in Fourier modes as
<math display="block"> H=\frac{1}{2}\sum_{k=-\infty}^{\infty}\left[\pi_k \pi_k^\dagger + \omega_k^2\phi_k\phi_k^\dagger\right],</math>
where <math>\omega_k = \sqrt{k^2+m^2}</math>.

This Hamiltonian is thus recognizable as an infinite sum of classical [[normal mode]] oscillator excitations {{math|''φ<sub>k</sub>''}}, each one of which is quantized  in the [[quantum harmonic oscillator|standard]] manner, so the free quantum Hamiltonian looks identical. It is the {{math|''φ<sub>k</sub>''}}s that have become operators obeying the standard commutation relations, {{math|1=[''φ<sub>k</sub>'', ''π<sub>k</sub>''<sup>†</sup>] = [''φ<sub>k</sub>''<sup>†</sup>, ''π<sub>k</sub>''] = ''iħ''}}, with all others vanishing. The collective Hilbert space of all these oscillators is thus constructed using creation and annihilation operators constructed from these modes,
<math display="block"> a_k = \frac{1}{\sqrt{2\hbar\omega_k}}\left(\omega_k\phi_k + i\pi_k\right), \ \ a_k^\dagger = \frac{1}{\sqrt{2\hbar\omega_k}}\left(\omega_k\phi_k^\dagger - i\pi_k^\dagger\right), </math>
for which {{math|1=[''a<sub>k</sub>'', ''a<sub>k</sub>''<sup>†</sup>] = 1}} for all {{mvar|k}}, with all other commutators vanishing.

The vacuum <math>|0\rangle</math> is taken to be annihilated by all of the {{math|''a<sub>k</sub>''}}, and <math>\mathcal{H}</math> is the Hilbert space constructed by applying any combination of the infinite collection of creation operators {{math|''a<sub>k</sub>''}}<sup>†</sup>  to <math>|0\rangle</math>.  This Hilbert space is called [[Fock space]].  For each  {{mvar|k}}, this construction is identical to a [[quantum harmonic oscillator]]. The quantum field is an infinite array of quantum oscillators. The quantum Hamiltonian then amounts to
<math display="block"> H = \sum_{k=-\infty}^{\infty} \hbar\omega_k a_k^\dagger a_k = \sum_{k=-\infty}^{\infty} \hbar\omega_k N_k ,</math>where {{math|''N<sub>k</sub>''}} may be interpreted as the ''[[number operator]]'' giving the [[number of particles]] in a state with momentum {{mvar|k}}.

This Hamiltonian differs from the previous expression by the subtraction of the zero-point energy {{math| ''ħω<sub>k</sub>''/2}} of each harmonic oscillator. This satisfies the condition that {{mvar|H}} must annihilate the vacuum, without affecting the time-evolution of operators via the above exponentiation operation.  This subtraction of the zero-point energy may be considered to be a resolution of the quantum operator ordering ambiguity, since it is equivalent to requiring that ''all creation operators appear to the left of annihilation operators'' in the expansion of the Hamiltonian. This procedure is known as [[Wick ordering]] or '''normal ordering'''.

====Other fields====

All other fields can be quantized by a generalization of this procedure. Vector or tensor fields simply have more components, and independent creation and destruction operators must be introduced for each independent component. If a field has any [[internal symmetry]], then creation and destruction operators must be introduced for each component of the field related to this symmetry as well. If there is a [[gauge symmetry]], then the number of independent components of the field must be carefully analyzed to avoid over-counting equivalent configurations, and [[gauge-fixing]] may be applied if needed.

It turns out that commutation relations are useful only for quantizing ''bosons'', for which the occupancy number of any state is unlimited. To quantize ''fermions'', which satisfy the [[Pauli exclusion principle]], anti-commutators are needed.  These are defined by {{math|{''A'', ''B''} {{=}} ''AB'' + ''BA''}}.

When quantizing fermions, the fields are expanded in creation and annihilation operators, {{math|''θ<sub>k</sub>''<sup>†</sup>}}, {{math|''θ<sub>k</sub>''}}, which satisfy
<math display="block">\{\theta_k,\theta_l^\dagger\} = \delta_{kl}, \ \ \{\theta_k, \theta_l\} = 0, \ \ \{\theta_k^\dagger, \theta_l^\dagger\} = 0. </math>

The states are constructed on a vacuum <math>|0\rangle</math> annihilated by the {{math|''θ<sub>k</sub>''}}, and the [[Fock space]] is built by applying all products of creation operators {{math|''θ<sub>k</sub>''<sup>†</sup>}} to {{ket|0}}.  Pauli's exclusion principle is satisfied, because <math>(\theta_k^\dagger)^2|0\rangle = 0</math>, by virtue of the anti-commutation relations.

===Condensates===

The construction of the scalar field states above assumed that the potential was minimized at {{mvar|φ}} = 0, so that the vacuum minimizing the Hamiltonian satisfies {{math|1=⟨''φ''⟩ = 0}}, indicating that the [[vacuum expectation value]] (VEV) of the field is zero. In cases involving [[spontaneous symmetry breaking]], it is possible to have a non-zero VEV, because the potential is minimized for a value  {{mvar|φ}} = {{mvar|v}} .  This occurs for example, if {{math|''V''(''φ'') {{=}} ''gφ''<sup>4</sup> − 2''m''<sup>2</sup>''φ''<sup>2</sup>}} with {{math|''g'' > 0}} and {{math|''m''<sup>2</sup> > 0}}, for which the minimum energy is found at {{math|''v'' {{=}} ±''m''/{{radic|''g''}}}}. The value of {{mvar|v}} in one of these vacua may be considered as ''condensate'' of the field {{mvar|φ}}. Canonical quantization then can be carried out for the ''shifted field'' {{math| ''φ''(''x'',''t'') − ''v''}}, and particle states with respect to the shifted vacuum are defined by quantizing the shifted field.  This construction is utilized in the [[Higgs mechanism]] in the [[standard model]] of [[particle physics]].

==Mathematical quantization==
===Deformation quantization===
The classical theory is described using a [[spacelike]]  [[foliation]] of [[spacetime]] with the state at each slice being described by an element of a [[symplectic manifold]] with the time evolution given by the [[symplectomorphism]] generated by a [[Hamiltonian mechanics|Hamiltonian]] function over the symplectic manifold. The ''quantum algebra'' of "operators" is an {{mvar|ħ}}-[[deformation quantization|deformation of the algebra of smooth functions]] over the symplectic space such that the '''leading term''' in the Taylor expansion over {{mvar|ħ}} of the [[commutator]] {{math| [''A'', ''B'']}}  expressed in the [[phase space formulation]] is {{math|''iħ''{''A'', ''B''} }}.  (Here, the curly braces denote the [[Poisson bracket]]. The subleading terms are all encoded in the [[Moyal bracket]], the suitable quantum deformation of the Poisson bracket.) In general, for the quantities (observables) involved,
and providing the arguments of such brackets,  ''ħ''-deformations are highly nonunique—quantization is an "art", and is specified by the physical context.
(Two ''different'' quantum systems may represent two different, inequivalent, deformations of the same [[classical limit]], {{math| ''ħ'' → 0}}.)

Now, one looks for [[unitary representation]]s of this quantum algebra. With respect to such a unitary representation, a symplectomorphism in the classical theory would now deform to a (metaplectic) [[unitary transformation]]. In particular, the time evolution symplectomorphism generated by the classical Hamiltonian deforms to a unitary transformation generated by the corresponding quantum Hamiltonian.

A further generalization is to consider a [[Poisson manifold]] instead of a symplectic space for the classical theory and perform an ''ħ''-deformation of the corresponding [[Poisson algebra]] or even [[Poisson supermanifold]]s.

===Geometric quantization===
{{main|Geometric quantization}}
In contrast to the theory of deformation quantization described above, geometric quantization seeks to construct an actual Hilbert space and operators on it. Starting with a symplectic manifold <math>M</math>, one first constructs a prequantum Hilbert space consisting of the space of square-integrable sections of an appropriate line bundle over <math>M</math>. On this space, one can map ''all'' classical observables to operators on the prequantum Hilbert space, with the commutator corresponding exactly to the Poisson bracket. The prequantum Hilbert space, however, is clearly too big to describe the quantization of <math>M</math>.

One then proceeds by choosing a polarization, that is (roughly), a choice of <math>n</math> variables on the <math>2n</math>-dimensional phase space. The ''quantum'' Hilbert space is then the space of sections that depend only on the <math>n</math> chosen variables, in the sense that they are covariantly constant in the other <math>n</math> directions. If the chosen variables are real, we get something like the traditional Schrödinger Hilbert space. If the chosen variables are complex, we get something like the [[Segal–Bargmann space]].

==See also==
*[[Correspondence principle]]
*[[Creation and annihilation operators]]
*[[Dirac bracket]]
*[[Moyal bracket]]
*[[Phase space formulation]] (of quantum mechanics)
*[[Geometric quantization]]

==References==
<references/>

===Historical References===

*[[Silvan S. Schweber]]: ''QED and the men who made it'', Princeton Univ. Press, 1994, {{ISBN|0-691-03327-7}}

===General Technical References===

*Alexander Altland, Ben Simons: ''Condensed matter field theory'', Cambridge Univ. Press, 2009, {{ISBN|978-0-521-84508-3}}
*James D. Bjorken, Sidney D. Drell: ''Relativistic quantum mechanics'', New York, McGraw-Hill, 1964
* {{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 |isbn=978-1461471158}}.
*''An introduction to quantum field theory'', by M.E. Peskin and H.D. Schroeder, {{ISBN|0-201-50397-2}}
*Franz Schwabl: ''Advanced Quantum Mechanics'', Berlin and elsewhere, Springer, 2009 {{ISBN|978-3-540-85061-8}}

==External links==
*[http://www.quantumfieldtheory.info Pedagogic Aides to Quantum Field Theory]  Click on the links for Chaps. 1 and 2 at this site to find an extensive, simplified introduction to second quantization. See Sect. 1.5.2 in Chap. 1. See Sect. 2.7 and the chapter summary in Chap. 2.

{{DEFAULTSORT:Canonical Quantization}}
[[Category:Quantum field theory]]
[[Category:Mathematical quantization]]