Editing Canonical quantization (section)

===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.