Quantum Electrodynamics

Quantum electrodynamics (QED) is the quantum field theory of electromagnetic interactions. It combines quantum mechanics with special relativity to describe how particles carrying electric charge interact with photons, the quanta of the electromagnetic field. Its simplest formulation contains an electron field and a photon field, encompassing electrons, their antiparticles, and electromagnetic radiation. QED provides a framework for calculating scattering, radiation, and corrections to atomic properties, and is distinguished by the precision with which many predictions can be tested experimentally. (damtp.cam.ac.uk)

Historical development

QED emerged from efforts to give electromagnetic radiation a quantum description. In 1927, Paul Dirac formulated a theory of radiation interacting with matter in which the electromagnetic field was quantized. Early calculations successfully described processes such as photon scattering, but higher-order corrections frequently produced infinite results, preventing straightforward comparison with measurements. (nobelprize.org)

Experiments in 1947 exposed small departures from predictions based on the relativistic Dirac equation. Measurements of the electron’s magnetic moment and the Lamb shift in hydrogen demonstrated the importance of electromagnetic corrections beyond that equation’s simplest applications. Hans Bethe’s calculation of the Lamb shift helped establish a route toward finite predictions. (nobelprize.org)

During the 1940s, Sin-Itiro Tomonaga, Julian Schwinger, and Richard Feynman developed formulations that made these corrections calculable. Freeman Dyson helped connect their methods and establish systematic diagrammatic calculations. Tomonaga, Schwinger, and Feynman jointly received the 1965 Nobel Prize in Physics for their fundamental work in QED. (nobelprize.org)

Fields, symmetry, and mathematical structure

In QED, particles are excitations of fields rather than permanently identifiable objects following definite trajectories. The electron field also describes the positron, which has the electron’s mass but opposite electric charge. Quantizing both matter and radiation allows the theory to describe processes in which particle numbers change, including electron–positron annihilation into photons. (damtp.cam.ac.uk)

QED is an Abelian gauge theory with local $U(1)$ symmetry. “Abelian” means that the group’s transformations commute. Gauge freedom is a redundancy in the mathematical description: different potentials and corresponding matter-field phases can represent the same physical situation. Observable predictions must not depend on this choice. (damtp.cam.ac.uk)

For one charged Dirac field, the Lagrangian density can be written, in units where $\hbar=c=1$ , as

\mathcal L=-\frac14F_{\mu\nu}F^{\mu\nu} +\bar\psi(i\gamma^\mu D_\mu-m)\psi, \qquad D_\mu=\partial_\mu+iqA_\mu .

Here $\psi$ is the matter field, $m$ its mass, $q$ its signed charge, and $A_\mu$ the electromagnetic potential. The field strength is $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ . The covariant derivative introduces the interaction while preserving local gauge invariance. Additional charged fields can be included with their own masses and charges. (damtp.cam.ac.uk)

Perturbation theory and diagrams

Most practical QED calculations use perturbation theory, expanding predictions in powers of the electromagnetic coupling. Its strength is expressed by the dimensionless fine-structure constant,

\alpha=\frac{e^2}{4\pi\varepsilon_0\hbar c},

which is approximately $1/137$ at very low momentum transfer. Its small value makes successive corrections useful in many ordinary applications. (pml.nist.gov)

Feynman diagrams organize terms in this expansion. Lines represent propagators, and vertices represent interactions. A diagram specifies a mathematical contribution to an amplitude; it is not a literal picture of a particle’s path. Internal photon lines are often described as virtual photons, which are not independently detected radiation. Contributions must be combined as amplitudes before extracting a probability. (damtp.cam.ac.uk)

Representative processes include electron–electron scattering, Compton scattering of photons by electrons, and electron–positron annihilation. At leading order, photon exchange between charged particles reproduces the familiar electromagnetic interaction, including the Coulomb potential in the appropriate nonrelativistic limit. Higher-order diagrams supply radiative corrections. (damtp.cam.ac.uk)

Renormalization and vacuum polarization

Loop calculations involve integrations over internal momenta and can produce ultraviolet divergences. Renormalization relates the parameters appearing in intermediate calculations to physically specified masses and charges. A regulator makes divergent expressions manageable, while counterterms absorb their divergent parts. QED is perturbatively renormalizable: a finite set of parameter and field redefinitions suffices to handle ultraviolet divergences order by order. (damtp.cam.ac.uk)

Vacuum polarization modifies photon propagation through virtual charged-particle contributions. It produces charge screening and makes the effective electromagnetic coupling depend on the momentum scale being probed. Consequently, the familiar low-energy value of $\alpha$ is not the interaction strength at every scale; the effective coupling increases at higher momentum transfers. (pml.nist.gov)

Experimental tests and scope

An important test is the electron’s anomalous magnetic moment, conventionally written $a_e=(g-2)/2$ . The simplest Dirac prediction gives $g=2$ , while QED adds radiative corrections. Its leading contribution is $a_e=\alpha/(2\pi)$ . Comparing precise magnetic-moment measurements with calculations and independently determined values of $\alpha$ tests the theory; conversely, assuming QED permits a determination of $\alpha$ from the measured moment. (damtp.cam.ac.uk)

The Lamb shift provides a complementary test through spectroscopy of the hydrogen atom. It includes a splitting between the $2S_{1/2}$ and $2P_{1/2}$ levels that are degenerate in the simplest Dirac treatment. Electron self-energy and vacuum-polarization corrections contribute to the observed difference. (damtp.cam.ac.uk)

Within the Standard Model, electromagnetic interactions belong to the broader electroweak theory, which also incorporates the weak interaction. QED remains the appropriate electromagnetic framework when other interactions can be neglected or treated separately. It does not by itself describe the strong interaction or gravitation. (cds.cern.ch)