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# Quantum electrodynamics

Quantum electrodynamics (QED) is a quantum field theory of electromagnetism. QED describes all phenomena involving electrically charged particles interacting by means of the electromagnetic force and has been called "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic moment of the muon, and the Lamb shift of the energy levels of hydrogen.

Mathematically, QED has the structure of an Abelian gauge theory with a U(1) gauge group. The gauge field which mediates the interaction between the charged spin-1/2 fields is the electromagnetic field.

Physically, this translates to the picture of charged particles (and their antiparticles) interacting with each other by the exchange of photons. The magnitude of these interactions can be can be computed using perturbation theory; these rather complex formulas have a remarkable pictorial representation as Feynman diagrams. QED was historically the theory to which Feynman diagrams were first applied.

QED was the first quantum field theory in which the difficulties of building a consistent, fully quantum description of fields and creation and annihilation of quantum particles were satisfactorily resolved. Sin-Itiro Tomonaga, Julian Schwinger and Richard Feynman received the 1965 Nobel Prize in Physics for its development, their contributions involving a covariant and gauge invariant prescription for the calculation of observable quantities. Feynman's mathematical technique, based on his diagrams, initially seemed very different from the field-theoretic, operator-based approach of Schwinger and Tomonaga, but was later shown to be equivalent. The renormalization procedure for making sense of some of the infinite predictions of quantum field theory also found its first successful implementation in quantum electrodynamics.

The QED Lagrangian for the interaction of electrons and positrons through photons is

$\mathcal{L}=\bar\psi(i\gamma_\mu D^\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ .

$\ \psi$ and its Dirac adjoint $\bar\psi$ are the fields representing electrically charged particles, specifically electron and positron fields represented as Dirac spinors.

$D_\mu = \partial_\mu+ieA_\mu \,\!$ is the gauge covariant derivative, with $\ e$ the coupling strength (equal to the elementary charge), $\ A_\mu$ the covariant vector potential of the electromagnetic field and $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \,\!$ the electromagnetic field tensor. Also, γμ are Dirac matrices .

The part of the Lagrangian containing the electromagnetic field tensor describes the free evolution of the electromagnetic field, whereas the Dirac-like equation with the gauge covariant derivative describes the free evolution of the electron and positron fields as well as their interaction with the electromagnetic field.

## References

• R. P. Feynman, QED: The strange theory of light and matter [ISBN 0691024170]
• Claude Cohen-Tannoudji, Jacques Dupont-Roc, Gilbert Grynberg, Photons and Atoms : Introduction to Quantum Electrodynamics (John Wiley & Sons, 1997) [ISBN 0471184330]
• J. M. Jauch, F. Rohrlich, The Theory of Photons and Electrons (Springer-Verlag, 1980)
• R. P. Feynman, Quantum Electrodynamics (Perseus Publishing, 1998) [ISBN 0201360756]

Last updated: 10-19-2005 21:21:44
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