Editing Feynman diagram (section)

== Canonical quantization formulation ==
The [[probability amplitude]] for a transition of a quantum system (between asymptotically free states) from the initial state {{math|{{ket|i}}}} to the final state {{math|{{ket| f }}}} is given by the matrix element

:<math>S_{\rm fi}=\langle \mathrm{f}|S|\mathrm{i}\rangle\;,</math>

where {{mvar|S}} is the [[S-matrix|{{mvar|S}}-matrix]]. In terms of the [[time-evolution operator]] {{mvar|U}}, it is simply

:<math>S=\lim _{t_{2}\rightarrow +\infty }\lim _{t_{1}\rightarrow -\infty }U(t_2, t_1)\;.</math>

In the [[interaction picture]], this expands to

:<math>S = \mathcal{T}e^{-i\int _{-\infty}^{+\infty}d\tau H_V(\tau )}.</math>

where {{mvar|H<sub>V</sub>}} is the interaction Hamiltonian and {{mvar|T}} signifies the [[time-ordered product]] of operators. [[Dyson series|Dyson's formula]] expands the time-ordered [[matrix exponential]] into a perturbation series in the powers of the interaction Hamiltonian density,

:<math>S=\sum_{n=0}^{\infty}\frac{(-i)^n}{n!} \left(\prod_{j=1}^n \int d^4 x_j\right) \mathcal{T}\left\{\prod_{j=1}^n \mathcal{H}_V\left(x_j\right)\right\} \equiv\sum_{n=0}^{\infty}S^{(n)}\;.</math>

Equivalently, with the interaction Lagrangian {{mvar|L<sub>V</sub>}}, it is

:<math>S=\sum_{n=0}^{\infty}\frac{i^n}{n!} \left(\prod_{j=1}^n \int d^4 x_j\right) \mathcal{T}\left\{\prod_{j=1}^n \mathcal{L}_V\left(x_j\right)\right\} \equiv\sum_{n=0}^{\infty}S^{(n)}\;.</math>

A Feynman diagram is a graphical representation of a single summand in the [[Wick's theorem|Wick's expansion]] of the time-ordered product in the {{mvar|n}}th-order term {{math|''S''<sup>(''n'')</sup>}} of the [[Dyson series]] of the {{mvar|S}}-matrix,

:<math>\mathcal{T}\prod_{j=1}^n\mathcal{L}_V\left(x_j\right)=\sum_{\text{A}}(\pm)\mathcal{N}\prod_{j=1}^n\mathcal{L}_V\left(x_j\right)\;,</math>

where {{mvar|''N''}} signifies the [[Normal order|normal-ordered product]] of the operators and (±) takes care of the possible sign change when commuting the fermionic operators to bring them together for a contraction (a [[propagator]]) and {{mvar|''A''}} represents all possible contractions.

=== Feynman rules ===
The diagrams are drawn according to the Feynman rules, which depend upon the interaction Lagrangian. For the [[Quantum electrodynamics|QED]] interaction Lagrangian
:<math>L_v=-g\bar\psi\gamma^\mu\psi A_\mu</math>
describing the interaction of a fermionic field {{mvar|ψ}} with a bosonic gauge field {{mvar|A<sub>μ</sub>}}, the Feynman rules can be formulated in coordinate space as follows:

* Each integration coordinate {{mvar|x<sub>j</sub>}} is represented by a point (sometimes called a vertex);
* A bosonic [[propagator]] is represented by a wiggly line connecting two points;
* A fermionic propagator is represented by a solid line connecting two points;
* A bosonic field <math>A_\mu(x_i)</math> is represented by a wiggly line attached to the point {{mvar|x<sub>i</sub>}};
* A fermionic field {{math|''ψ''(''x<sub>i</sub>'')}} is represented by a solid line attached to the point {{mvar|x<sub>i</sub>}} with an arrow toward the point;
* An anti-fermionic field {{math|{{overline|''ψ''}}(''x<sub>i</sub>'')}} is represented by a solid line attached to the point {{mvar|x<sub>i</sub>}} with an arrow away from the point;

=== Example: second order processes in QED ===
The second order perturbation term in the {{mvar|S}}-matrix is

:<math>S^{(2)}=\frac{(ie)^2}{2!}\int d^4x\, d^4x'\, T\bar\psi(x)\,\gamma^\mu\,\psi(x)\,A_\mu(x)\,\bar\psi(x')\,\gamma^\nu\,\psi(x')\,A_\nu(x').\;</math>

==== Scattering of fermions ====
    {|align="right"
    |[[File:Feynman-diagram-ee-scattering.png|class=skin-invert-image|right|thumb|360px|The Feynman diagram of the term <math>N\bar\psi(x)ie\gamma^\mu\psi(x)\bar\psi(x')ie\gamma^\nu\psi(x')A_\mu(x)A_\nu(x')</math>]]
    |}
The [[Wick's theorem|Wick's expansion]] of the integrand gives (among others) the following term

:<math>N\bar\psi(x)\gamma^\mu\psi(x)\bar\psi(x')\gamma^\nu\psi(x')\underline{A_\mu(x)A_\nu(x')}\;,</math>

where

:<math>\underline{A_\mu(x)A_\nu(x')}=\int\frac{d^4k}{(2\pi)^4}\frac{-ig_{\mu\nu}}{k^2+i0}e^{-ik(x-x')}</math>

is the electromagnetic contraction (propagator) in the Feynman gauge. This term is represented by the Feynman diagram at the right. This diagram gives contributions to the following processes:
# e<sup>−</sup> e<sup>−</sup> scattering (initial state at the right, final state at the left of the diagram);
# e<sup>+</sup> e<sup>+</sup> scattering (initial state at the left, final state at the right of the diagram);
# e<sup>−</sup> e<sup>+</sup> scattering (initial state at the bottom/top, final state at the top/bottom of the diagram).

==== Compton scattering and annihilation/generation of e<sup>−</sup> e<sup>+</sup> pairs ====
Another interesting term in the expansion is

:<math>N\bar\psi(x)\,\gamma^\mu\,\underline{\psi(x)\,\bar\psi(x')}\,\gamma^\nu\,\psi(x')\,A_\mu(x)\,A_\nu(x')\;,</math>

where

:<math>\underline{\psi(x)\bar\psi(x')}=\int\frac{d^4p}{(2\pi)^4}\frac{i}{\gamma p-m+i0}e^{-ip(x-x')}</math>

is the fermionic contraction (propagator).