Editing Quantum field theory (section)

===Canonical quantization===
{{Main|Canonical quantization}}

The quantization procedure for the above classical field to a quantum operator field is analogous to the promotion of a classical harmonic oscillator to a [[quantum harmonic oscillator]].

The displacement of a classical harmonic oscillator is described by
:<math>x(t) = \frac{1}{\sqrt{2\omega}} a e^{-i\omega t} + \frac{1}{\sqrt{2\omega}} a^* e^{i\omega t},</math>
where {{math|''a''}} is a complex number (normalized by convention), and {{math|''ω''}} is the oscillator's frequency. Note that {{math|''x''}} is the displacement of a particle in simple harmonic motion from the equilibrium position,  not to be confused with the spatial label {{math|'''x'''}} of a quantum field.

For a quantum harmonic oscillator, {{math|''x''(''t'')}} is promoted to a [[linear operator]] <math>\hat x(t)</math>:
:<math>\hat x(t) = \frac{1}{\sqrt{2\omega}} \hat a e^{-i\omega t} + \frac{1}{\sqrt{2\omega}} \hat a^\dagger e^{i\omega t}.</math>
Complex numbers {{math|''a''}} and {{math|''a''<sup>*</sup>}} are replaced by the [[annihilation operator]] <math>\hat a</math> and the [[creation operator]] <math>\hat a^\dagger</math>, respectively, where {{math|†}} denotes [[Hermitian conjugation]]. The [[commutation relation]] between the two is
:<math>\left[\hat a, \hat a^\dagger\right] = 1.</math>
The [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the simple harmonic oscillator can be written as
:<math>\hat H = \hbar\omega \hat{a}^\dagger \hat{a} +\frac{1}{2}\hbar\omega.</math>
The [[vacuum state]] <math>|0\rang</math>, which is the lowest energy state, is defined by
:<math>\hat a|0\rang = 0</math>
and has energy <math>\frac12\hbar\omega.</math>
One can easily check that <math>[\hat H, \hat{a}^\dagger]=\hbar\omega\hat{a}^\dagger,</math> which implies that <math>\hat{a}^\dagger</math> increases the energy of the simple harmonic oscillator by <math>\hbar\omega</math>. For example, the state <math>\hat{a}^\dagger|0\rang</math> is an eigenstate of energy <math>3\hbar\omega/2</math>.
Any energy eigenstate state of a single harmonic oscillator can be obtained from <math>|0\rang</math> by successively applying the creation operator <math>\hat a^\dagger</math>:{{r|peskin|page1=20}} and any state of the system can be expressed as a linear combination of the states
:<math>|n\rang \propto \left(\hat a^\dagger\right)^n|0\rang.</math>

A similar procedure can be applied to the real scalar field {{math|''ϕ''}}, by promoting it to a quantum field operator <math>\hat\phi</math>, while the annihilation operator <math>\hat a_{\mathbf{p}}</math>, the creation operator <math>\hat a_{\mathbf{p}}^\dagger</math> and the angular frequency <math>\omega_\mathbf {p}</math>are now for a particular {{math|'''p'''}}:
:<math>\hat \phi(\mathbf{x}, t) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_{\mathbf{p}}}}\left(\hat a_{\mathbf{p}} e^{-i\omega_{\mathbf{p}}t + i\mathbf{p}\cdot\mathbf{x}} + \hat a_{\mathbf{p}}^\dagger e^{i\omega_{\mathbf{p}}t - i\mathbf{p}\cdot\mathbf{x}}\right).</math>
Their commutation relations are:{{r|peskin|page1=21}}
:<math>\left[\hat a_{\mathbf p}, \hat a_{\mathbf q}^\dagger\right] = (2\pi)^3\delta(\mathbf{p} - \mathbf{q}),\quad \left[\hat a_{\mathbf p}, \hat a_{\mathbf q}\right] = \left[\hat a_{\mathbf p}^\dagger, \hat a_{\mathbf q}^\dagger\right] = 0,</math>
where {{math|''δ''}} is the [[Dirac delta function]]. The vacuum state <math>|0\rang</math> is defined by
:<math>\hat a_{\mathbf p}|0\rang = 0,\quad \text{for all }\mathbf p.</math>
Any quantum state of the field can be obtained from <math>|0\rang</math> by successively applying creation operators <math>\hat a_{\mathbf{p}}^\dagger</math> (or by a linear combination of such states), e.g. {{r|peskin|page1=22}}
:<math>\left(\hat a_{\mathbf{p}_3}^\dagger\right)^3 \hat a_{\mathbf{p}_2}^\dagger \left(\hat a_{\mathbf{p}_1}^\dagger\right)^2 |0\rang.</math>

While the state space of a single quantum harmonic oscillator contains all the discrete energy states of one oscillating particle, the state space of a quantum field contains the discrete energy levels of an arbitrary number of particles. The latter space is known as a [[Fock space]], which can account for the fact that particle numbers are not fixed in relativistic quantum systems.<ref>{{cite journal |last1=Fock |first1=V. |author-link=Vladimir Fock |date=1932-03-10 |title=Konfigurationsraum und zweite Quantelung |journal=Zeitschrift für Physik |volume=75 |issue=9–10 |pages=622–647 |doi=10.1007/BF01344458 |language=de |bibcode=1932ZPhy...75..622F |s2cid=186238995 }}</ref> The process of quantizing an arbitrary number of particles instead of a single particle is often also called [[second quantization]].{{r|peskin|page1=19}}

The foregoing procedure is a direct application of non-relativistic quantum mechanics and can be used to quantize (complex) scalar fields, [[Dirac field]]s,{{r|peskin|page1=52}} [[vector field]]s (''e.g.'' the electromagnetic field), and even [[string theory|strings]].<ref>{{cite book |last1=Becker |first1=Katrin |last2=Becker |first2=Melanie|author-link2=Melanie Becker |last3=Schwarz |first3=John H. |date=2007 |title=String Theory and M-Theory |url=https://archive.org/details/stringtheorymthe00beck_649 |url-access=limited |publisher=Cambridge University Press |page=[https://archive.org/details/stringtheorymthe00beck_649/page/n53 36] |isbn=978-0-521-86069-7 |author-link3=John Henry Schwarz }}</ref> However, creation and annihilation operators are only well defined in the simplest theories that contain no interactions (so-called free theory). In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations of motion into a sum of normal modes. To perform calculations on any realistic interacting theory, [[perturbation theory (quantum mechanics)|perturbation theory]] would be necessary.

The Lagrangian of any quantum field in nature would contain interaction terms in addition to the free theory terms. For example, a [[quartic interaction]] term could be introduced to the Lagrangian of the real scalar field:{{r|peskin|page1=77}}
:<math>\mathcal{L} = \frac 12 (\partial_\mu\phi)\left(\partial^\mu\phi\right) - \frac 12 m^2\phi^2 - \frac{\lambda}{4!}\phi^4,</math>
where {{math|''μ''}} is a spacetime index, <math>\partial_0 = \partial/\partial t,\ \partial_1 = \partial/\partial x^1</math>, etc. The summation over the index {{math|''μ''}} has been omitted following the [[Einstein notation]]. If the parameter {{math|''λ''}} is sufficiently small, then the interacting theory described by the above Lagrangian can be considered as a small perturbation from the free theory.