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Quantum Electrodynamics

The Story Behind the Math

Richard Feynman called it "the jewel of physics — our proudest possession." Quantum electrodynamics (QED) is the quantum theory of light and matter: electrons, positrons, and photons, and every way they can interact. It is the most precisely tested theory humanity has ever produced.

But it was nearly stillborn. When physicists in the 1930s tried to calculate anything beyond the crudest approximation, the answers came out infinite. The self-energy of the electron, the corrections to its charge — all infinite. For more than a decade the theory was a beautiful idea that produced nonsense numbers.

Two experiments in 1947 forced the issue. Willis Lamb measured a tiny shift between two hydrogen energy levels that the Dirac equation said should be identical — the "Lamb shift." And Polykarp Kusch measured the electron's magnetic moment and found it was not exactly the predicted value, but larger by about 0.1%. These were small effects, but they were real, and they demanded a theory that could calculate them.

The answer arrived almost simultaneously from three directions. In Japan, Sin-Itiro Tomonaga had quietly developed the method during the war. In the US, Julian Schwinger built an austere, powerful formalism and computed the magnetic moment correction by hand. And Richard Feynman invented something completely different: little pictures — now called Feynman diagrams — that turned impossible integrals into a bookkeeping game of lines and vertices. The three approaches looked nothing alike, until Freeman Dyson proved they were mathematically identical. Tomonaga, Schwinger, and Feynman shared the 1965 Nobel Prize.

The trick that defeated the infinities is called renormalization: the "bare" mass and charge in the equations are themselves infinite and unobservable; only the combination that includes the infinite corrections — the measured mass and charge — is finite. Feynman was uneasy about it to the end ("a dippy process"), but it works to a degree that is almost unreasonable.

How well? The theory's prediction for the electron's magnetic moment and the experimental measurement agree to about twelve decimal places. Feynman's analogy: it is like measuring the distance from New York to Los Angeles and getting it right to the thickness of a human hair.

Why It Matters

  • It is the template for the entire Standard Model. The "gauge principle" that builds QED (below) is reused to build the weak and strong forces.
  • It explains light and matter — every chemical bond, every photon emitted or absorbed, the behavior of all electronics, at the most fundamental level.
  • Renormalization was born here — the conceptual tool that makes all of quantum field theory predictive.
  • It is the precision benchmark of physics, agreeing with experiment better than any other theory.

Prerequisites

The Formula

The entire theory is contained in one Lagrangian density:

\[ \mathcal{L}_{\text{QED}} = \bar\psi\left(i\gamma^\mu D_\mu - m\right)\psi - \tfrac{1}{4}F_{\mu\nu}F^{\mu\nu} \]

with the covariant derivative and field-strength tensor

\[ D_\mu = \partial_\mu + ieA_\mu, \qquad F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \]

Here \(\psi\) is the electron field, \(A_\mu\) is the photon (electromagnetic) field, \(e\) is the electron's charge, and \(\gamma^\mu\) are the Dirac matrices. The remarkable claim of this page is that you do not have to postulate the photon or its coupling — you can derive their necessity from a single demand of symmetry.

Derivation: The Gauge Principle

We will start with a free electron and show that insisting on a local symmetry forces the photon into existence, fully determining how it couples. This is one of the most beautiful arguments in physics, due originally to Hermann Weyl.

Step 1: The free electron

A free relativistic electron is described by the Dirac Lagrangian:

\[ \mathcal{L}_0 = \bar\psi\left(i\gamma^\mu\partial_\mu - m\right)\psi \]

No photons, no forces — just an electron propagating through spacetime.

Step 2: A symmetry we happen to notice (global \(U(1)\))

The quantum phase of a wavefunction is unobservable, so \(\mathcal{L}_0\) is unchanged if we rotate the field's phase by a constant angle \(\theta\):

\[ \psi \to e^{i\theta}\psi, \qquad \bar\psi \to e^{-i\theta}\bar\psi \]

The two phases cancel in \(\bar\psi\psi\) and in \(\bar\psi\gamma^\mu\partial_\mu\psi\) (since \(\theta\) is constant, \(\partial_\mu\) passes through). By Noether's theorem this symmetry corresponds to conservation of electric charge. So far, nothing is forced.

Step 3: Demand the symmetry locally

Here is the deep question. Why should the phase convention be the same at every point in spacetime? A choice of phase here and a choice of phase on the other side of the galaxy ought to be independent. So let us demand invariance under a phase that varies from point to point:

\[ \psi \to e^{i\theta(x)}\psi \]

This is a local (gauge) symmetry. It is a much stronger requirement.

Step 4: The derivative breaks it

Apply the derivative to the transformed field:

\[ \partial_\mu\!\left(e^{i\theta(x)}\psi\right) = e^{i\theta(x)}\left(\partial_\mu\psi + i\,\partial_\mu\theta(x)\,\psi\right) \]

That extra term \(i\,(\partial_\mu\theta)\psi\) has nowhere to cancel — it ruins the invariance. The plain derivative compares the field at neighboring points, but those points now have different phase conventions, so the comparison is illegitimate.

Step 5: Repair it by introducing a new field

To compare phases at different points we need a "connection" — a field \(A_\mu\) that tells us how the convention changes from point to point. Demand that it transforms as:

\[ A_\mu \to A_\mu - \tfrac{1}{e}\,\partial_\mu\theta(x) \]

and replace the ordinary derivative with the covariant derivative:

\[ D_\mu = \partial_\mu + ieA_\mu \]

Now check: under the local transformation,

\[ D_\mu\psi \to \left(\partial_\mu + ieA_\mu - i\,\partial_\mu\theta\right)e^{i\theta}\psi = e^{i\theta}\left(\partial_\mu\psi + i\,\partial_\mu\theta\,\psi + ieA_\mu\psi - i\,\partial_\mu\theta\,\psi\right) = e^{i\theta}\,D_\mu\psi \]

The unwanted \(\partial_\mu\theta\) terms cancel exactly. The combination \(\bar\psi\,i\gamma^\mu D_\mu\,\psi\) is now locally invariant.

Step 6: The new field is the photon

Expanding the covariant derivative reveals what we have created:

\[ \bar\psi\,i\gamma^\mu D_\mu\,\psi = \underbrace{\bar\psi\,i\gamma^\mu\partial_\mu\,\psi}_{\text{free electron}} \;-\; \underbrace{e\,\bar\psi\,\gamma^\mu\psi\,A_\mu}_{\text{interaction}} \]

A new term has appeared: the electron field couples to \(A_\mu\) with strength \(e\). To make \(A_\mu\) a real, propagating field — a dynamical particle — we give it the only gauge-invariant kinetic term possible, \(-\tfrac14 F_{\mu\nu}F^{\mu\nu}\). That field \(A_\mu\) is precisely the electromagnetic potential, and its quantum is the photon.

Final Result

\[ \boxed{\;\mathcal{L}_{\text{QED}} = \bar\psi\left(i\gamma^\mu D_\mu - m\right)\psi - \tfrac{1}{4}F_{\mu\nu}F^{\mu\nu}\;} \]

We did not assume electromagnetism — we derived it. The mere insistence that the electron's phase can be chosen independently at each point in spacetime forces the existence of the photon and fixes exactly how it interacts with charged matter. Electromagnetism is the price (and the gift) of a local symmetry.

Key Properties

  • Gauge invariance. The redundancy \(\psi \to e^{i\theta(x)}\psi\), \(A_\mu \to A_\mu - \tfrac1e\partial_\mu\theta\) is built into the theory. It is the abelian group \(U(1)\) — its transformations all commute.
  • The coupling is weak. Interactions are governed by the fine-structure constant
\[ \alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c} \approx \frac{1}{137} \]

Because \(\alpha \ll 1\), each extra photon in a process is suppressed by another factor of \(\alpha\), so perturbation theory (Feynman diagrams) converges beautifully. - Renormalizable. The infinities can be absorbed into the measured mass and charge, leaving finite, testable predictions. - Feynman diagrams. Every process is a sum of diagrams; each vertex (electron emits/absorbs a photon) contributes a factor \(\sim\sqrt{\alpha}\).

Variables Explained

Symbol Name Description
\(\psi\) Electron field Dirac spinor field whose quanta are electrons/positrons
\(A_\mu\) Photon field The electromagnetic four-potential; its quantum is the photon
\(D_\mu\) Covariant derivative \(\partial_\mu + ieA_\mu\); makes the Lagrangian locally invariant
\(F_{\mu\nu}\) Field strength \(\partial_\mu A_\nu - \partial_\nu A_\mu\); encodes \(\mathbf E\) and \(\mathbf B\)
\(\gamma^\mu\) Dirac matrices \(4\times4\) matrices encoding spin and relativity
\(e\) Coupling The electron's electric charge
\(\alpha\) Fine-structure constant \(\approx 1/137\); strength of the electromagnetic interaction

Worked Examples

Example 1: Why "\(1/137\)" controls everything

The probability amplitude for an electron to emit or absorb a photon is proportional to \(e\), so each vertex in a Feynman diagram carries \(\sqrt{\alpha}\). A process probability squares the amplitude, so the simplest scattering goes like \(\alpha\), the next correction like \(\alpha^2\), and so on:

\[ \alpha \approx 0.0073, \quad \alpha^2 \approx 5.3\times10^{-5}, \quad \alpha^3 \approx 3.9\times10^{-7} \]

Each extra loop is ~137 times smaller. That tiny ratio is why QED is so accurately calculable — the series converges fast.

Example 2: The electron's magnetic moment (\(g-2\))

The Dirac equation predicts the electron's gyromagnetic ratio is exactly \(g = 2\). QED says the electron can emit and reabsorb a virtual photon, nudging this value. Schwinger's famous 1948 one-loop calculation gave the first correction:

\[ \frac{g-2}{2} = \frac{\alpha}{2\pi} \approx \frac{0.0073}{6.283} \approx 0.00116 \]

engraved on Schwinger's tombstone. Higher-order terms refine it, and theory now matches experiment to about twelve digits — the most precise agreement in science.

Example 3: Coulomb's law as photon exchange

In QED, the static electric force between two charges is the result of them exchanging virtual photons. Summing the simplest one-photon-exchange diagram in the low-energy limit reproduces exactly the \(1/r^2\) force of Coulomb's law. The classical inverse-square law is the shadow of a single photon being passed back and forth.

Common Mistakes

  • Thinking the gauge field is optional. It is not bolted on — local symmetry forces it. Remove \(A_\mu\) and you cannot have a locally phase-invariant theory of charged matter.
  • Confusing gauge symmetry with a physical symmetry. Gauge invariance is a redundancy in our description (many \(A_\mu\) describe the same physics), not a symmetry that relates distinct states.
  • Forgetting that virtual particles are not "real." The photons exchanged in a force are internal lines in a diagram — calculational tools, not detectable particles obeying \(E^2=p^2+m^2\).
  • Expecting perturbation theory to always work. It works in QED because \(\alpha\approx1/137\) is small. In QCD the coupling is large at low energy and this entire approach breaks down.

History

  • 1927–28 — Dirac quantizes the EM field and writes the relativistic electron equation
  • 1930s — Calculations beyond leading order give infinities; the theory stalls
  • 1947 — The Lamb shift and the anomalous magnetic moment are measured
  • 1948 — Schwinger computes \(g-2 = \alpha/\pi\); Feynman introduces diagrams; Tomonaga's wartime work surfaces
  • 1949 — Dyson proves the three formulations are equivalent and lays out renormalization
  • 1965 — Tomonaga, Schwinger, and Feynman share the Nobel Prize
  • today — Theory and experiment for \(g-2\) agree to ~12 significant figures

References

  • Feynman, R. P. (1985). QED: The Strange Theory of Light and Matter. Princeton University Press.
  • Griffiths, D. J. (2008). Introduction to Elementary Particles (2nd ed.). Wiley-VCH.
  • Peskin, M. E., & Schroeder, D. V. (1995). An Introduction to Quantum Field Theory. Addison-Wesley.
  • Schwinger, J. (1948). On Quantum-Electrodynamics and the Magnetic Moment of the Electron. Physical Review, 73(4), 416.