Quantum Electrodynamics (QED) is the quantum field theory that describes how light and matter interact. It underlies electricity, magnetism, and light, and it is among the most precisely tested theories in science: its prediction for the electron's magnetic moment matches experiment to about one part in a trillion.
The Central Idea: Fields and Their Quanta
In QED, the electromagnetic field is no longer a classical smooth field. It is a quantum field: an operator-valued function of spacetime that can be in superpositions of different configurations. Excitations of this field are photons. Similarly, the electron field's excitations are electrons (and their antimatter partners, positrons).
Every interaction in QED comes down to a single primitive vertex: a charged particle such as an electron emits or absorbs a photon. All of electromagnetism, from charge attraction and repulsion to the photoelectric effect and Compton scattering, emerges from combinations of this one act.
Feynman Diagrams
Richard Feynman invented a pictorial language for systematically calculating interaction probabilities. Each diagram represents a term in a perturbation series expansion:
- Straight lines with arrows represent electrons (arrow forward = electron, arrow backward = positron)
- Wavy lines represent photons
- Each vertex where these lines meet contributes a factor proportional to the electron charge
e
The simplest process, two electrons repelling each other, is drawn as two electron lines exchanging a single photon. This photon is virtual: it exists only as an intermediate state and is not directly observable. Virtual particles are "off-shell", meaning they do not satisfy the ordinary energy-momentum relation E^2 = (pc)^2 + (mc^2)^2. The time-energy uncertainty principle is the usual heuristic for why this is allowed.
Higher-order corrections add more vertices and loops. One one-loop correction to electron scattering has the exchanged photon briefly split into a virtual electron-positron pair. Each additional loop multiplies the contribution by roughly a factor of the fine structure constant:
alpha = e^2 / (4*pi*eps_0*hbar*c) ≈ 1/137.036
Since alpha is small, the first few orders of corrections shrink rapidly and give extremely accurate predictions. (The full series is asymptotic rather than convergent: Dyson showed in 1952 that it must eventually diverge.)
The Fine Structure Constant
alpha is the most important dimensionless number in QED. It sets the strength of the electromagnetic coupling: the probability amplitude for an electron to emit a photon is proportional to sqrt(alpha).
Current physics has no explanation for why alpha ≈ 1/137. Feynman called it "one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man."
The Path Integral Formulation
Feynman's path integral approach provides a deep intuition for quantum mechanics. Rather than a particle traveling a single definite trajectory, every possible path through spacetime contributes to the final probability amplitude. Each path is weighted by:
exp(i*S/hbar)
where S is the classical action (integral of the Lagrangian) along that path. Paths near the classical trajectory interfere constructively because neighboring paths have nearly the same phase. Paths far from the classical trajectory interfere destructively and cancel. As hbar approaches zero, only the classical path survives, which is how quantum mechanics recovers classical mechanics in the limit.
In QED, the Lagrangian that enters this path integral is:
L = psi-bar*(i*gamma^mu*D_mu - m)*psi - (1/4)*F_mu_nu*F^mu_nu
The first term describes the electron and its coupling to the electromagnetic field, which enters through the covariant derivative D_mu. The second term is the kinetic energy of the electromagnetic field itself, expressed through the field strength tensor F.
Renormalization: Taming the Infinities
QED's perturbation series produces apparent infinities. Loop diagrams involve integrals over all possible momenta of the virtual particles, and these integrals diverge at high momenta. For years this seemed to make the theory meaningless.
The resolution is renormalization. What we call the electron's mass and charge are not the bare parameters in the Lagrangian; they include all quantum corrections. The infinities can be absorbed into the definition of these physical parameters, and the theory then makes finite predictions for measurable quantities.
This procedure, developed by Julian Schwinger, Sin-Itiro Tomonaga, and Richard Feynman (with Freeman Dyson showing that their approaches are equivalent), reflects the fact that only measured values are meaningful, and that the effective charge and mass depend on the energy scale of the measurement.
The Anomalous Magnetic Moment
The most famous test of QED is the anomalous magnetic moment of the electron, denoted g - 2. The Dirac equation predicts g = 2. QED predicts corrections from virtual photon loops:
Theory: g/2 = 1.001 159 652 181 78 (± 0.00000000000077)
Experiment: g/2 = 1.001 159 652 180 59 (± 0.00000000000013)
The two values agree to 12 significant figures, within their combined uncertainty. The theory value depends on a measured value of alpha, so the comparison also tests which alpha measurement is consistent.
Glossary
Quantum field theory — A framework in physics that describes matter and forces as ripples in underlying fields that permeate all of spacetime, in place of particles moving through space. QED is the quantum field theory of electromagnetism.
Field — A physical quantity that has a value at every point in space and time. Temperature is a familiar example: every location in a room has a temperature, and it can change from moment to moment. In modern physics, fields are considered the fundamental entities, and particles are concentrated disturbances, or excitations, within them.
Electromagnetic field — The field responsible for electricity, magnetism, and light. In QED, it is a quantum object whose smallest possible disturbance is a photon.
Photon — The particle of light. In QED, photons are the carriers of the electromagnetic force: when two electrons repel each other, they do so by exchanging photons.
Electron — A fundamental negatively charged particle found in all atoms. In QED, it is described as an excitation of the electron field rather than a tiny classical ball.
Positron — The antimatter counterpart of the electron: same mass, opposite charge. When an electron and positron meet, they annihilate and convert entirely into photons.
Perturbation series — A method of building up an answer step by step, starting with the simplest possible interaction and adding successively smaller corrections. In QED, each additional correction is proportional to the fine structure constant, which is small enough that the first few corrections give very accurate results.
Lagrangian — A mathematical function that encodes all the physics of a system in a single compact expression: what particles exist, what fields they interact with, and how they move. Think of it as the rulebook of the theory. In QED, the Lagrangian contains two terms: one describing how electrons behave and couple to the electromagnetic field, and one describing the field itself. Everything the theory predicts, including Feynman diagrams and renormalization, is derived from it.