# Quantum electrodynamics

The quantum electrodynamics ( QED ) is in the context of quantum physics , the quantum field theoretical description of electromagnetism .

## General

The QED gives a description of all phenomena that are caused by charged point particles , such as electrons or positrons , and by photons . It contains classical electrodynamics as a borderline case of strong fields or high energies, in which the possible measured values can be viewed as continuous . Of deeper interest, however, is its application to microscopic objects, where it explains quantum phenomena, such as the structure of atoms and molecules . It also includes processes of high energy physics , such as the generation of particles by an electromagnetic field . One of her best results is the calculation of the anomalous magnetic moment of the electron, which corresponds to 11  decimal places with the experimentally determined value ( Landé factor ). This makes QED one of the most precisely experimentally verified theories today.

The QED describes the interaction of a spinor field with charge  -e , which describes the electron, with a calibration field, which describes the photon. It receives its motion equations of electrodynamics by quantization of Maxwell's equations . Quantum electrodynamics explains with high accuracy the electromagnetic interaction between charged particles (e.g. electrons, muons , quarks ) by means of the exchange of virtual photons and the properties of electromagnetic radiation .

QED was the first quantum field theory in which the difficulties of a consistent quantum theoretical description of fields and the creation and extinction of particles were satisfactorily solved. The creators of this theory, developed in the 1940s, were honored with the award of the Nobel Prize in Physics to Richard P. Feynman , Julian Schwinger and Shin'ichirō Tomonaga in 1965.

## Lagrangian density

The fundamental function of quantum field theory is the Lagrangian : ${\ displaystyle {\ mathcal {L}}}$

${\ displaystyle {\ mathcal {L}} _ {\ text {QED}} = \ sum _ {n} {\ bar {\ psi}} _ {n} (i \ gamma ^ {\ mu} \ partial _ { \ mu} -m_ {n}) \ psi _ {n} - {\ frac {1} {4}} F _ {\ mu \ nu} F ^ {\ mu \ nu} - \ sum _ {n} q_ { n} {\ bar {\ psi}} _ {n} \ gamma ^ {\ mu} A _ {\ mu} \ psi _ {n}.}$

In the formula:

• The free spinor field obeys the Dirac equation and describes fermions like electrons or quarks.${\ displaystyle \ psi}$
• The photon field obeys Maxwell's equations .${\ displaystyle A ^ {\ mu}}$
• The field strength tensor is an abbreviation for .${\ displaystyle F _ {\ mu \ nu}}$${\ displaystyle \ partial _ {\ mu} A _ {\ nu} - \ partial _ {\ nu} A _ {\ mu}}$

The physical free parameters of quantum electrodynamics are

• the (bare) masses of the individual objects${\ displaystyle m_ {n}}$
• their (bare) coupling constants , which in the case of quantum electrodynamics correspond to the classical electrical charge .${\ displaystyle q_ {n}}$

The Lagrangian of quantum electrodynamics is designed in such a way that it arises from the Lagrangian of the free spinor field and the free photon field, if the local gauge invariance is additionally required, which manifests itself in a coupling term (see Dirac equation ).

In particular, the Lagrangian of quantum electrodynamics is the maximum expression that all u. G. Criteria met, d. H. no term can be added that does not violate the conditions.

Quantum electrodynamics is a relativistic gauge theory based on the unitary group ( circle group ), so that the following conditions must be met: ${\ displaystyle U (1)}$

• Invariance among transformations of the Poincaré group , which includes the Lorentz transformations ,
• Invariance under a local gauge transformation and the field operators and${\ displaystyle \ psi \ to \ psi '= e ^ {\ mathrm {i} q \ alpha (x)} \ psi}$${\ displaystyle A _ {\ mu} \ to A _ {\ mu} '= A _ {\ mu} + \ partial _ {\ mu} \ alpha (x)}$ ${\ displaystyle \ psi}$${\ displaystyle A}$
• Renormalizability in the context of a perturbative calculation

### Significance of the calibration transformations

The transformation is the classic local calibration transformation of the electromagnetic potentials and , which does not change the value of the electric field or the magnetic flux density . ${\ displaystyle A _ {\ mu} \ to A _ {\ mu} '= A _ {\ mu} + \ partial _ {\ mu} \ alpha (x)}$ ${\ displaystyle \ Phi}$${\ displaystyle {\ vec {A}}}$ ${\ displaystyle {\ vec {E}} = - {\ vec {\ nabla}} \ Phi - \ partial _ {t} {\ vec {A}}}$ ${\ displaystyle {\ vec {B}} = {\ vec {\ nabla}} \ times {\ vec {A}}}$

The corresponding transformation, on the other hand, describes a local change in phase without a direct analogue in classical physics. The invariance of the Lagrangian under this phase change leads, according to Noether's theorem , to the conservation quantity of the Dirac current with the continuity equation . ${\ displaystyle \ psi \ to \ psi '= e ^ {\ mathrm {i} q \ alpha (x)} \ psi}$${\ displaystyle j _ {\ mu} = {\ bar {\ psi}} \ gamma _ {\ mu} \ psi}$ ${\ displaystyle \ partial ^ {\ mu} j _ {\ mu} = 0}$

The requirements for gauge invariance, Lorentz invariance and renormalizability of the Lagrangian also lead to the statement that the photon is massless , since a renormalizable scalar mass term for the photon is not gauge invariant . ${\ displaystyle A _ {\ mu} m _ {\ gamma} ^ {2} A ^ {\ mu}}$

### Equations of motion

The Lagrange density leads via the Lagrange equation to the equations of motion for the field operators:

${\ displaystyle (\ mathrm {i} \ gamma ^ {\ mu} \ partial _ {\ mu} -m) \ psi = q \ gamma ^ {\ mu} A _ {\ mu} \ psi}$
${\ displaystyle \ partial _ {\ mu} F ^ {\ mu \ nu} = j ^ {\ nu}}$

The second system of equations precisely represents the Maxwell equations in potential form, with the classic electromagnetic four-fold current density being replaced by the Dirac current.

## Classification of quantum electrodynamics

 Fundamental interactions and their descriptions Strong interaction Electromagnetic interaction Weak interaction Gravity classic Electrostatics & magnetostatics , electrodynamics Newton's law of gravitation , general relativity quantum theory Quantum ( standard model ) Quantum electrodynamics Fermi theory Quantum gravity  ? Electroweak Interaction ( Standard Model ) Big Unified Theory  ? World formula ("theory of everything")? Theories at an early stage of development are grayed out.

## literature

• Richard P. Feynman : QED. The strange theory of light and matter. Piper-Verlag, Munich et al. 1988, ISBN 3-492-03103-X (popular science textbook).
• Franz Mandl, Graham Shaw: Quantum Field Theory. Aula-Verlag, Wiesbaden 1993, ISBN 3-89104-532-8 (introductory textbook).
• Silvan S. Schweber : QED and the men who made it. Dyson, Feynman, Schwinger, and Tomonaga. Princeton University Press, Princeton NJ 1994, ISBN 0-691-03685-3 .
• G. Scharf: Finite Quantum Electrodynamics. The causal approach. 2nd Edition. Jumper. Berlin et al. 1995, ISBN 3-540-60142-2
• Peter W. Milonni: The quantum vacuum. An introduction to quantum electrodynamics. Academic Press, Boston et al. 1994, ISBN 0-12-498080-5 .
• Walter Dittrich, Holger Gies: Probing the Quantum Vacuum. Perturbative Effective Action Approach in Quantum Electrodynamics and its Application (= Springer Tracts in modern Physics 166). Springer, Berlin et al. 2000, ISBN 3-540-67428-4 .
• Giovanni Cantatore: Quantum electrodynamics and physics of the vacuum (= AIP Conference Proceedings 564). American Institute of Physics, Melville NY 2001, ISBN 0-7354-0000-8 .