Editing Quantum electrodynamics (section)

==Mathematical formulation==
=== QED action ===
Mathematically, QED is an [[abelian group|abelian]] [[gauge theory]] with the symmetry group [[U(1)]], defined on [[Minkowski space]] (flat spacetime). The [[gauge field]], which mediates the interaction between the charged [[Spin (physics)|spin-1/2]] [[field (physics)|field]]s, is the [[electromagnetic field]].
The QED [[Lagrangian (field theory)|Lagrangian]] for a spin-1/2 field interacting with the electromagnetic field in natural units gives rise to the action<ref name=Peskin>{{cite book | last1 =Peskin | first1 =Michael | last2 =Schroeder | first2 =Daniel | title =An introduction to quantum field theory | publisher =Westview Press | edition =Reprint | date =1995 | isbn =978-0201503975 | url-access =registration | url =https://archive.org/details/introductiontoqu0000pesk }}</ref>{{rp|78}}
{{Equation box 1
|title='''QED Action'''
|indent=:
|equation = <math>S_{\text{QED}} = \int d^4x\,\left[-\frac{1}{4}F^{\mu\nu}F_{\mu\nu} + \bar\psi\,(i\gamma^\mu D_\mu - m)\,\psi\right]</math>
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|border colour =#50C878
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}}

where
*<math> \gamma^\mu </math> are [[Dirac matrices]].
*<math>\psi</math> a [[bispinor]] [[field (physics)|field]] of [[spin-1/2]] particles (e.g. [[electron]]–[[positron]] field).
*<math>\bar\psi\equiv\psi^\dagger\gamma^0</math>, called "psi-bar", is sometimes referred to as the [[Dirac adjoint]].
*<math>D_\mu \equiv \partial_\mu+ieA_\mu+ieB_\mu </math> is the [[gauge covariant derivative]].
**''e'' is the [[Fine-structure constant|coupling constant]], equal to the [[electric charge]] of the bispinor field.
**<math>A_\mu</math> is the [[Lorentz covariance|covariant]] [[four-potential]] of the electromagnetic field generated by the electron itself. It is also known as a gauge field or a <math>\text{U}(1)</math> connection.
**<math>B_\mu</math> is the external field imposed by external source.
*''m'' is the mass of the electron or positron.
*<math>F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu </math> is the [[electromagnetic field tensor]]. This is also known as the curvature of the gauge field.

Expanding the covariant derivative reveals a second useful form of the Lagrangian (external field <math>B_\mu</math> set to zero for simplicity)
:<math>\mathcal{L} = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \bar\psi(i\gamma^\mu \partial_\mu - m)\psi - ej^\mu A_\mu</math>
where <math>j^\mu</math> is the conserved <math>\text{U}(1)</math> current arising from Noether's theorem. It is written
:<math>j^\mu = \bar\psi\gamma^\mu\psi.</math>

=== Equations of motion ===
Expanding the covariant derivative in the Lagrangian gives

:<math>\mathcal{L} = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i \bar\psi \gamma^\mu \partial_\mu \psi - e\bar{\psi}\gamma^\mu A_\mu \psi -m \bar{\psi} \psi </math>
:<math> = - \frac{1}{4}F_{\mu\nu}F^{\mu\nu} + i \bar\psi \gamma^\mu \partial_\mu \psi -m \bar{\psi} \psi - ej^\mu A_\mu .</math>
For simplicity, <math>B_\mu</math> has been set to zero, with no loss of generality. Alternatively, we can absorb <math>B_\mu</math> into a new gauge field <math>A'_\mu = A_\mu + B_\mu</math> and relabel the new field as <math>A_\mu.</math>

From this Lagrangian, the equations of motion for the <math>\psi</math> and <math>A_\mu</math> fields can be obtained.

==== Equation of motion for ψ ====
These arise most straightforwardly by considering the Euler-Lagrange equation for <math>\bar\psi</math>. Since the Lagrangian contains no <math>\partial_\mu\bar\psi</math> terms, we immediately get
:<math>\frac{\partial \mathcal{L}}{\partial(\partial_\mu \bar\psi)} = 0</math>
so the equation of motion can be written
<math>(i\gamma^\mu\partial_\mu-m)\psi = e\gamma^\mu A_\mu\psi.</math>

==== Equation of motion for A<sub>μ</sub> ====
* Using the Euler–Lagrange equation for the <math>A_\mu</math> field,
{{NumBlk2||<math display="block"> \partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) - \frac{\partial \mathcal{L}}{\partial A_\mu} = 0,</math>|3}}

the derivatives this time are
<math display="block">\partial_\nu \left( \frac{\partial \mathcal{L}}{\partial ( \partial_\nu A_\mu )} \right) = \partial_\nu \left( \partial^\mu A^\nu - \partial^\nu A^\mu \right),</math>
<math display="block">\frac{\partial \mathcal{L}}{\partial A_\mu} = -e\bar{\psi} \gamma^\mu \psi.</math>

Substituting back into ({{EquationNote|3}}) leads to

:<math>\partial_\mu F^{\mu\nu} = e\bar\psi \gamma^\nu \psi</math>
which can be written in terms of the <math>\text{U}(1)</math> current <math>j^\mu</math> as
{{Equation box 1
|indent =:
|equation = <math>\partial_\mu F^{\mu \nu} = e j^\nu.</math>
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Now, if we impose the [[Lorenz gauge condition]]
<math display="block">\partial_\mu A^\mu = 0,</math>
the equations reduce to
<math display="block">\Box A^\mu = ej^\mu,</math>
which is a [[wave equation]] for the four-potential, the QED version of the classical [[Maxwell equations]] in the [[Lorenz gauge]]. (The square represents the [[wave operator]], <math>\Box = \partial_\mu \partial^\mu</math>.)

===Interaction picture===
This theory can be straightforwardly quantized by treating bosonic and fermionic sectors{{clarify|reason=Definition needed.|date=April 2015}} as free. This permits us to build a set of asymptotic states that can be used to start computation of the probability amplitudes for different processes. In order to do so, we have to compute an [[Hamiltonian (quantum mechanics)|evolution operator]], which for a given initial state <math>|i\rangle</math> will give a final state <math>\langle f|</math> in such a way to have<ref name=Peskin/>{{rp|5}}

<math display="block">M_{fi} = \langle f|U|i\rangle.</math>

This technique is also known as the [[S-matrix]]. The evolution operator is obtained in the [[interaction picture]], where time evolution is given by the interaction Hamiltonian, which is the integral over space of the second term in the Lagrangian density given above:<ref name=Peskin/>{{rp|123}}

<math display="block">V = e \int d^3 x\, \bar\psi \gamma^\mu \psi A_\mu,</math>

Which can also be written in terms of an integral over the interaction Hamiltonian density <math>\mathcal{H}_I = e \overline \psi \gamma^\mu \psi A_\mu</math>. Thus, one has<ref name=Peskin/>{{rp|86}}

<math display="block">U = T \exp\left[-\frac{i}{\hbar} \int_{t_0}^t dt'\, V(t')\right],</math>

where ''T'' is the [[Path-ordering|time-ordering]] operator. This evolution operator only has meaning as a series, and what we get here is a [[Perturbation theory (quantum mechanics)|perturbation series]] with the [[fine-structure constant]] as the development parameter. This series expansion of the probability amplitude <math>M_{fi}</math> is called the [[Dyson series]], and is given by:

<math display="block">
M_{fi} = \langle f | U |i\rangle 
 =\left\langle f\left|\sum _{n=0}^{\infty }{\frac {(-i)^{n}}{n!}}\int d^4x_{1} \cdots \int d^4x_{n} T \bigg\{ \mathcal{H}(x_{1})\cdots \mathcal {H}(x_{n}) \bigg \} \right|i\right\rangle
</math>

===Feynman diagrams===
Despite the conceptual clarity of the Feynman approach to QED, almost no early textbooks follow him in their presentation. When performing calculations, it is much easier to work with the [[Fourier transform]]s of the [[propagator]]s. Experimental tests of quantum electrodynamics are typically scattering experiments. In scattering theory, particles' [[Momentum|momenta]] rather than their positions are considered, and it is convenient to think of particles as being created or annihilated when they interact. Feynman diagrams then ''look'' the same, but the lines have different interpretations. The electron line represents an electron with a given energy and momentum, with a similar interpretation of the photon line. A vertex diagram represents the annihilation of one electron and the creation of another together with the absorption or creation of a photon, each having specified energies and momenta.

Using [[Wick's theorem]] on the terms of the Dyson series, all the terms of the [[S-matrix]] for quantum electrodynamics can be computed through the technique of [[Feynman diagrams]]. In this case, rules for drawing are the following<ref name=Peskin/>{{rp|801–802}}

[[Image:qed rules.jpg|488px|center]]

[[Image:qed2e.jpg|488px|center]]

To these rules we must add a further one for closed loops that implies an integration on momenta <math display="inline">\int d^4p/(2\pi)^4</math>, since these internal ("virtual") particles are not constrained to any specific energy–momentum, even that usually required by special relativity (see [[Propagator#Propagators in Feynman diagrams|Propagator]] for details). The signature of the metric <math>\eta_{\mu \nu }</math> is <math>{\rm diag}(+---)</math>.

From them, computations of [[probability amplitude]]s are straightforwardly given. An example is [[Compton scattering]], with an [[electron]] and a [[photon]] undergoing [[elastic scattering]]. Feynman diagrams are in this case<ref name=Peskin/>{{rp|158–159}}

[[Image:compton qed.jpg|300px|center]]

and so we are able to get the corresponding amplitude at the first order of a [[Perturbation theory (quantum mechanics)|perturbation series]] for the [[S-matrix]]:

<math display="block">M_{fi} = (ie)^2 \overline{u}(\vec{p}', s')\epsilon\!\!\!/\,'(\vec{k}',\lambda')^* \frac{p\!\!\!/ + k\!\!\!/ + m_e}
{(p + k)^2 - m^2_e} \epsilon\!\!\!/(\vec{k}, \lambda) u(\vec{p}, s) + (ie)^2\overline{u}(\vec{p}', s')\epsilon\!\!\!/(\vec{k},\lambda) \frac{p\!\!\!/ - k\!\!\!/' + m_e}{(p - k')^2 - m^2_e} \epsilon\!\!\!/\,'(\vec{k}', \lambda')^* u(\vec{p}, s),</math>

from which we can compute the [[Cross section (physics)|cross section]] for this scattering.

===Nonperturbative phenomena===
The predictive success of quantum electrodynamics largely rests on the use of perturbation theory, expressed in Feynman diagrams. However, quantum electrodynamics also leads to predictions beyond perturbation theory. In the presence of very strong electric fields, it predicts that electrons and positrons will be spontaneously produced, so causing the decay of the field. This process, called the [[Schwinger effect]],<ref name="Schwinger">{{cite journal | last=Schwinger | first=Julian | title=On Gauge Invariance and Vacuum Polarization | journal=Physical Review | publisher=American Physical Society (APS) | volume=82 | issue=5 | date=1951-06-01 | issn=0031-899X | doi=10.1103/physrev.82.664 | pages=664–679| bibcode=1951PhRv...82..664S }}</ref> cannot be understood in terms of any finite number of Feynman diagrams and hence is described as [[Non-perturbative|nonperturbative]]. Mathematically, it can be derived by a semiclassical approximation to the [[Path integral formulation|path integral]] of quantum electrodynamics.