Introduction to QED

Quantum electrodynamics is the most precisely tested theory in all of science, and yet its entire content can be built from one instruction: wherever an ordinary derivative $\partial_\mu$ appears in the Dirac equation, replace it with $\partial_\mu + iqA_\mu$ . That single substitution — minimal coupling — turns a free electron into one that feels, emits, and absorbs light. Everything else follows. The replacement fixes the shape of the electron–photon vertex, the propagator carries the influence of the photon from one charged particle to another, and the two together generate every electromagnetic amplitude, from an electron scattering off a muon to the anomalous wobble of the electron's own magnetism. This article assembles QED from parts the reader already owns: the Dirac equation's spinors $u(p)$ and gamma matrices, the Feynman-diagram picture of exchanged virtual quanta, the Yukawa propagator $1/(q^2-m^2)$ , and the adjoint spinor $\bar\psi=\psi^\dagger\gamma^0$ that keeps the amplitude Lorentz invariant. The plan is to take the scalar particle-exchange amplitude $\mathcal M = \langle\psi_c|V|\psi_a\rangle\,\tfrac{1}{q^2-m_X^2}\,\langle\psi_d|V|\psi_b\rangle$ and upgrade its placeholder scalar vertex into the genuine electromagnetic one for a spin-½ fermion, then account for the photon being a massless spin-1 particle by summing over its polarizations. What emerges is the invariant current–current amplitude $\mathcal M = -Q_1Q_2 e^2\,(j_1\cdot j_2)/q^2$ and, behind it, QED as a gauge theory whose one free parameter is $\alpha\approx 1/137$ .

Key takeaways

QED is the quantum field theory of charged spin-½ fermions interacting by exchanging photons; the photon is the massless, spin-1 mediator.
The interaction is generated by the minimal substitution $\partial_\mu \to \partial_\mu + iqA_\mu$ applied to the Dirac equation — no new force law is added by hand.
That substitution fixes the electron–photon vertex factor as $iQ_f e\,\gamma^\mu$ (electron $Q_f=-1$ , so $-ie\gamma^\mu$ ), sandwiched between an adjoint spinor and a spinor.
The photon carries the propagator $-ig_{\mu\nu}/q^2$ ; the polarization sum $\sum_\lambda \varepsilon_\mu^{(\lambda)}\varepsilon_\nu^{(\lambda)*}=-g_{\mu\nu}$ is what collapses it into this form.
The amplitude becomes the Lorentz-invariant current–current product $\mathcal M = -\dfrac{Q_1Q_2 e^2\,(j_1\cdot j_2)}{q^2}$ , with $j^\mu=\bar u(p')\gamma^\mu u(p)$ .
Because the photon is massless the Yukawa potential loses its exponential and becomes the infinite-range Coulomb $1/r$ ; the coupling strength is $\alpha=e^2/4\pi\varepsilon_0\hbar c = 1/137.036$ .
Key idea: electromagnetism is inserted into Dirac theory by the single replacement $\partial_\mu \to \partial_\mu + iqA_\mu$ ; this minimal coupling fixes the photon–fermion vertex as $iQ_f e\gamma^\mu$ and, combined with the photon propagator $-ig_{\mu\nu}/q^2$ , generates every QED amplitude.

The core insight

Electromagnetism is introduced into the Dirac theory by the single replacement $\partial_\mu \to \partial_\mu + iqA_\mu$ ; this minimal coupling fixes the photon–fermion vertex as $iQ_f e\,\gamma^\mu$ and, combined with the photon propagator $-ig_{\mu\nu}/q^2$ , generates every QED amplitude — with one measured coupling, $\alpha\approx 1/137$ .

Where this lives today

QED is the standard against which precision physics measures itself. The flagship test is the electron's anomalous magnetic moment: the 2023 Harvard experiment (Fan, Myers, Sukra, Gabrielse) trapped a single electron in a cryogenic Penning trap and measured $a_e = 1.159\,652\,180\,59(13)\times10^{-3}$ , a precision of 0.13 parts per billion. The QED prediction — a five-loop calculation of thousands of Feynman diagrams built from the very vertex derived above — agrees to that precision, making $a_e$ the most stringently confirmed prediction in physics and a route to the world's best independent value of $\alpha$ . The muon's anomaly $a_\mu$ , measured at Fermilab's Muon $g-2$ experiment, is enhanced by $(m_\mu/m_e)^2\approx4\times10^4$ in its sensitivity to heavy virtual particles, and its comparison with theory (now hinging on lattice-QCD evaluations of the hadronic contribution) is one of the sharpest live probes for physics beyond the Standard Model. The same coupling $\alpha$ is seen to run: at the $Z$ -boson mass its effective value climbs from $1/137.036$ to about $1/127.95$ , a change measured at LEP and a direct sighting of vacuum polarization — the photon propagator dressed by virtual electron–positron pairs. And at the practical end, QED underwrites all of atomic and molecular structure and the Lamb shift that anchors the optical clocks now reaching $10^{-18}$ fractional accuracy.

The physical picture

QED has one elementary event: a charged fermion kinks its worldline and emits or absorbs a photon. That three-line junction — two fermion legs, one photon leg — is the vertex, and every diagram is built by wiring copies of it together with propagators. The simplest process needing two of them is one charged particle scattering off another, $e^-\mu^- \to e^-\mu^-$ : the electron emits a virtual photon at one vertex, the muon absorbs it at another, and the propagator carries the momentum transfer $q=p_1-p_3$ between them. The first figure is that diagram, each piece labelled by the algebraic factor it contributes.

Figure

The t-channel photon exchange e⁻μ⁻ → e⁻μ⁻: each vertex contributes iQ_f e γ^μ, the virtual photon carries the propagator −i g_μν /q², and stitching them together gives the invariant amplitude ℳ = −Q_e Q_μ e² (j_e·j_μ)/q² with currents j^μ = ū(p′)γ^μ u(p)

The coupling at each vertex is weak — the quantitative fact that makes QED work as a perturbation theory. Every vertex brings one factor of the electric charge $e$ , and in natural units $e=\sqrt{4\pi\alpha}\approx 0.303$ , so a two-vertex tree amplitude scales as $e^2\propto\alpha\approx 1/137$ and each extra photon loop costs another $\alpha$ . The second quantitative fact is that because the photon is massless, the propagator's $1/(q^2-m^2)$ has no mass gap: Fourier-transformed to position space it gives an infinite-range $1/r$ potential, which is why electromagnetism reaches across a room while the weak force does not. The second figure shows that contrast — Coulomb from massless exchange against the cut-off Yukawa potential of a massive mediator.

Figure

Massless exchange gives the infinite-range Coulomb potential V(r) = −Q₁Q₂ α ℏc /r (blue), with α = 1/137.036 and αℏc = 1.440 eV·nm — evaluated at the Bohr radius this is 27.2 eV. A massive mediator of mass m gives the Yukawa form −α e^(−r/R)/r cut off at range R = ℏ/mc (amber); a W boson m = 80.4 GeV gives R = 0.0025 fm

The core equations

The whole theory fits on one line. Modern QED is defined by a Lagrangian density with three pieces: a free Dirac fermion, a free electromagnetic field, and the coupling between them buried inside a single covariant derivative. Nothing else is put in by hand — the interaction is forced by demanding invariance under local phase rotations of the electron field.

\mathcal L_{\mathrm{QED}} = \bar\psi\,(i\gamma^\mu D_\mu - m)\,\psi - \tfrac14 F_{\mu\nu}F^{\mu\nu},\qquad D_\mu = \partial_\mu + iqA_\mu \qquad (1)

where: $\psi$ — the Dirac spinor field, mass $m$ , charge $q$ ; $\bar\psi=\psi^\dagger\gamma^0$ — its adjoint; $A_\mu$ — the photon (electromagnetic four-potential); $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ — the field-strength tensor; $D_\mu$ — the gauge-covariant derivative. Expanding $D_\mu$ splits $\mathcal L$ into a free part plus the interaction $-q\,\bar\psi\gamma^\mu\psi\,A_\mu$ — one photon coupling to the fermion current, the seed of every vertex. Undoing the compact notation of (1) shows where the coupling comes from: start from the free Dirac equation and make the minimal substitution $\partial_\mu\to D_\mu$ ; the extra term is the interaction.

(i\gamma^\mu\partial_\mu - m)\psi = q\,\gamma^\mu A_\mu\,\psi \quad\Longrightarrow\quad \hat V_D = q\,\gamma^0\gamma^\mu A_\mu \qquad (2)

where: the left equation is the Dirac equation for a fermion of charge $q$ in an external field $A_\mu$ ; multiplying by $\gamma^0$ and reading off the term added to the free Hamiltonian gives the potential operator $\hat V_D$ . Its time-like piece ( $\mu=0$ , $A_0=\phi$ ) reduces to $q\phi$ — the electrostatic potential energy, the check that the substitution really is electromagnetism. Sandwiching $\hat V_D$ between the initial and final fermion states, and stripping off the external-field factor, leaves the vertex factor — the Feynman rule for the electron–photon junction.

\text{vertex:}\quad i\,Q_f\,e\,\gamma^\mu \qquad (3)

where: $Q_f$ — the fermion's charge in units of the positron charge $e>0$ (electron $Q_f=-1$ , so the vertex is $-ie\gamma^\mu$ ; up quark $Q_f=+\tfrac23$ ); $\gamma^\mu$ — a Dirac gamma matrix carrying the spinor indices. Its single Lorentz index $\mu$ is the leftover that must contract with the photon. The two vertices are joined by the photon propagator. For the massless spin-1 photon in Feynman gauge it is diagonal in the Lorentz indices, its numerator being exactly the polarization sum.

\frac{-i\,g_{\mu\nu}}{q^2+i\epsilon},\qquad \sum_{\lambda}\varepsilon_\mu^{(\lambda)}\varepsilon_\nu^{(\lambda)*} = -g_{\mu\nu} \qquad (4)

where: $q$ — the four-momentum carried by the virtual photon; $g_{\mu\nu}$ — the Minkowski metric; $\varepsilon^{(\lambda)}$ — the photon polarization vectors summed over states $\lambda$ ; $+i\epsilon$ — the Feynman causal prescription (droppable at tree level). The absence of a mass term in the denominator ( $m_\gamma=0$ ) is what makes the force long-ranged. Assembling vertex–propagator–vertex for two-body scattering gives the central result: a Lorentz-invariant product of two currents divided by the momentum transfer squared.

\mathcal M = \Big[\,\bar u(p_3)\,iQ_1 e\gamma^\mu\, u(p_1)\Big]\frac{-i g_{\mu\nu}}{q^2}\Big[\,\bar u(p_4)\,iQ_2 e\gamma^\nu\, u(p_2)\Big] = -\frac{Q_1 Q_2\, e^2}{q^2}\,(j_1\cdot j_2) \qquad (5)

where: $j_1^\mu=\bar u(p_3)\gamma^\mu u(p_1)$ and $j_2^\mu=\bar u(p_4)\gamma^\mu u(p_2)$ — the transition currents of the two fermions; $q=p_1-p_3$ . Each current transforms as a contravariant four-vector, so their contraction $j_1\cdot j_2$ is a Lorentz scalar and the amplitude looks the same in every frame — the QED analogue of the scalar-exchange matrix element, with the placeholder vertices replaced by genuine electromagnetic ones. Finally, the strength of the whole edifice is set by one dimensionless number. The two factors of $e$ in every tree amplitude combine into the fine-structure constant.

\alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c} = \frac{1}{137.035\,999} \approx 7.297\times10^{-3},\qquad e=\sqrt{4\pi\alpha}\approx 0.303\ \text{(natural units)} \qquad (6)

where: $\alpha$ — the fine-structure constant, the QED expansion parameter; its smallness is why perturbation theory converges and each extra loop costs about $\alpha/\pi\approx 0.0023$ . Everything measurable in QED is a power series in this one number.

Derivation AMinimal coupling fixes the electron–photon vertex

The goal is to show that the entire electron–photon interaction — the vertex factor $iQ_f e\gamma^\mu$ at the heart of every QED diagram — is not an independent postulate but the forced consequence of one replacement in the free Dirac equation. We start from the free equation, perform the minimal substitution $\partial_\mu\to\partial_\mu+iqA_\mu$ , isolate the interaction term, confirm it reproduces ordinary electrostatics in its time-like component, and read off the factor left between the spinors once the external field is stripped.

Step 1. The free Dirac equation is $(i\gamma^\mu\partial_\mu-m)\psi=0$ . Apply the minimal substitution by promoting the derivative to the covariant derivative $D_\mu=\partial_\mu+iqA_\mu$ , giving $i\gamma^\mu(\partial_\mu+iqA_\mu)\psi-m\psi=0$ , which rearranges to $(i\gamma^\mu\partial_\mu-m)\psi = q\gamma^\mu A_\mu\psi$ . Why: this is the same rule that turns $\mathbf p\to\mathbf p-q\mathbf A$ in non-relativistic mechanics — the minimal way to couple a charge to a field while keeping the equation first-order and covariant.

Step 2. Cast the result in Hamiltonian form by multiplying through by $\gamma^0$ . The free part becomes $i\partial_t\psi=(\boldsymbol\alpha\cdot\hat{\mathbf p}+\beta m)\psi$ with $\boldsymbol\alpha=\gamma^0\boldsymbol\gamma$ , $\beta=\gamma^0$ , and the new term supplies an added potential $\hat V_D=q\gamma^0\gamma^\mu A_\mu$ . Why: writing $i\partial_t\psi=(H_0+\hat V_D)\psi$ exposes $\hat V_D$ as the perturbing interaction scattering theory feeds into the amplitude.

Step 3. Verify the identification. Take the time-like piece $\mu=0$ : since $(\gamma^0)^2=\mathbb 1$ and $A_0=\phi$ , the term is $q\gamma^0\gamma^0 A_0=q\phi$ , exactly the electrostatic potential energy of a charge $q$ in a potential $\phi$ . Why: this is the sanity check that the abstract substitution really encodes electromagnetism — its slow-particle, static limit must reproduce the Coulomb energy every first course derives, and it does.

Step 4. Compute a vertex. The first-order amplitude for a fermion to go from spinor state $u(p_1)$ to $u(p_3)$ is $\langle\psi_3|\hat V_D|\psi_1\rangle \propto u^\dagger(p_3)\,q\gamma^0\gamma^\mu\,u(p_1)\,A_\mu$ . Package $u^\dagger\gamma^0=\bar u$ to get $q\,\bar u(p_3)\gamma^\mu u(p_1)\,A_\mu$ ; writing $q=Q_f e$ and stripping the external field $A_\mu$ leaves the vertex factor $iQ_f e\gamma^\mu$ (the $i$ being the standard Feynman-rule convention). Why: the adjoint spinor $\bar u=u^\dagger\gamma^0$ is precisely what makes $\bar u\gamma^\mu u$ transform as a four-vector, so the leftover object is a genuine current and the vertex carries exactly one Lorentz index to hand to the photon.

Go deeper · Derivation B · 1 minThe photon propagator and the current–current amplitude

Derivation BThe photon propagator and the current–current amplitude

The aim is to assemble the lowest-order amplitude for two charged fermions scattering by photon exchange, and to show why the photon's spin-1 nature collapses into the simple numerator $-g_{\mu\nu}$ . We attach a vertex to each fermion line, connect them with the exchanged photon, sum over its polarization states, and read off the invariant amplitude as a Lorentz scalar built from two four-vector currents.

Step 1. Write down the two vertices and the two currents. At the electron line the factor is $\bar u(p_3)\,iQ_1 e\gamma^\mu\,u(p_1)\equiv iQ_1 e\,j_1^\mu$ ; at the muon line, $\bar u(p_4)\,iQ_2 e\gamma^\nu\,u(p_2)\equiv iQ_2 e\,j_2^\nu$ . The exchanged photon must connect index $\mu$ to index $\nu$ . Why: each vertex, by the previous derivation, contributes one free Lorentz index; the photon is the only object that can soak up both, which is exactly the role of its propagator.

Step 2. Build the propagator from the polarization sum. A real photon has two transverse polarizations, but a virtual (off-shell) photon needs all four to be exchanged; summing $\sum_\lambda\varepsilon_\mu^{(\lambda)}\varepsilon_\nu^{(\lambda)*}=-g_{\mu\nu}$ over these states and dividing by $q^2$ gives the propagator $-g_{\mu\nu}/q^2$ (Feynman gauge). Why: we never measure the intermediate photon, so its polarization is summed, not fixed — and the completeness relation turns the sum into the clean metric numerator.

Step 3. Contract everything. Multiplying vertex × propagator × vertex gives $\mathcal M = (iQ_1 e\,j_1^\mu)\,\dfrac{-ig_{\mu\nu}}{q^2}\,(iQ_2 e\,j_2^\nu)$ . The three factors of $i$ and the metric combine to $\mathcal M = -\dfrac{Q_1Q_2 e^2}{q^2}\,g_{\mu\nu}j_1^\mu j_2^\nu = -\dfrac{Q_1Q_2 e^2}{q^2}\,(j_1\cdot j_2)$ . Why: the metric $g_{\mu\nu}$ lowers an index and performs the four-vector dot product, packaging the answer as a single scalar — the amplitude a physicist can square without worrying about frame.

Step 4. Confirm the invariance and the coupling. Each $j^\mu=\bar u\gamma^\mu u$ is a contravariant four-vector (the adjoint spinor guarantees it), so $j_1\cdot j_2$ is a Lorentz scalar and $\mathcal M$ is frame-independent; the prefactor $Q_1Q_2 e^2$ is $4\pi\alpha\,Q_1Q_2$ , exhibiting the fine-structure constant. Why: manifest Lorentz invariance is the payoff of the whole construction — it guarantees the same cross-section in the lab and centre-of-mass frames, and it isolates $\alpha$ as the one number that sets the scale.

Go deeper · Derivation C · 1 minFrom massless exchange to the Coulomb potential

Derivation CFrom massless exchange to the Coulomb potential

The purpose is to connect the abstract propagator to something tangible: to show that exchanging a massless photon between two static charges reproduces the ordinary Coulomb $1/r$ potential, while a massive mediator yields the short-range Yukawa potential. We take the non-relativistic static limit of the amplitude, recognise the propagator as the Fourier transform of the potential, and invert it — the same Born-approximation logic that connects any exchange amplitude to a force law.

Step 1. In the static non-relativistic limit only the time-like ( $\mu=\nu=0$ ) part of the current survives, with $\bar u\gamma^0 u\to 2m$ (the density normalisation), and the momentum transfer becomes purely spatial, $q^2\to-|\mathbf q|^2$ . The amplitude reduces to $\mathcal M \propto \dfrac{Q_1Q_2 e^2}{|\mathbf q|^2}$ . Why: a slow, heavy charge is essentially a static source, so its current is just its charge density; the spatial momentum transfer is what a potential scattering sees.

Step 2. In the Born approximation the scattering amplitude off a potential $V(\mathbf r)$ is proportional to the Fourier transform $\tilde V(\mathbf q)=\int V(\mathbf r)\,e^{-i\mathbf q\cdot\mathbf r}\,d^3r$ . Matching to Step 1 identifies $\tilde V(\mathbf q)\propto Q_1Q_2 e^2/|\mathbf q|^2$ for the massless case, and $\propto Q_1Q_2 g^2/(|\mathbf q|^2+m_X^2)$ if the mediator has mass $m_X$ . Why: the propagator and the potential are Fourier partners, so reading off the momentum-space amplitude is reading off the potential — no new dynamics, just a transform.

Step 3. Invert the transforms, using the standard results $\displaystyle\int \frac{e^{i\mathbf q\cdot\mathbf r}}{|\mathbf q|^2}\frac{d^3q}{(2\pi)^3}=\frac{1}{4\pi r}$ and $\displaystyle\int \frac{e^{i\mathbf q\cdot\mathbf r}}{|\mathbf q|^2+m_X^2}\frac{d^3q}{(2\pi)^3}=\frac{e^{-m_X r}}{4\pi r}$ . The massive case gives the Yukawa potential $V(r)=-\dfrac{g^2}{4\pi}\dfrac{e^{-m_X r}}{r}$ ; setting $m_X\to0$ removes the exponential, leaving $V(r)=\dfrac{Q_1Q_2 e^2}{4\pi\varepsilon_0 r}=\dfrac{Q_1Q_2\,\alpha\,\hbar c}{r}$ . Why: the exponential $e^{-m_X r}$ is the mediator's mass expressing itself as a finite range $R=\hbar/m_X c$ ; a massless photon has infinite range, which is the deep reason electromagnetism is a long-range force.

Step 4. Read off the numbers. With $\alpha\hbar c = 1.440\ \text{eV}\,\cdot\,\text{nm}$ , two unit charges at the Bohr radius $a_0=0.0529\ \text{nm}$ sit at $V=27.2\ \text{eV}$ (one Hartree). A mediator as heavy as the $W$ boson ( $m_W=80.4\ \text{GeV}$ ) has range $R=\hbar/m_W c = 0.0025\ \text{fm}$ — subnuclear, which is why the weak force appears point-like. Why: the same propagator explains both the everyday reach of electricity and the extreme brevity of the weak interaction.

Limiting cases

Case	Result	Meaning
Massless mediator ( $m_\gamma=0$ )	$V(r)\propto 1/r$ , infinite range	photon exchange gives the Coulomb law; electromagnetism reaches to infinity
Massive mediator ( $m_X\neq0$ )	$V(r)\propto e^{-m_X r}/r$	Yukawa potential of range $R=\hbar/m_X c$ ; the weak force ( $m_W,m_Z$ ) is short-ranged
Static, non-relativistic	$\bar u\gamma^0 u\to 2m$ , only $j^0$ survives	the current becomes a charge density; QED reduces to electrostatics and the Coulomb potential
Weak coupling ( $\alpha\ll1$ )	amplitude is a power series in $\alpha\approx 1/137$	tree level dominates; each extra loop costs $\sim\alpha/\pi\approx 0.0023$ , so QED is calculable to many digits
Neutral particle ( $Q_f=0$ )	vertex factor vanishes	neutrinos and photons do not couple at the QED vertex; there is no direct photon–photon vertex in QED

Common trapQED "explains" the Coulomb force by literal photon bullets flying between charges

A common cartoon says two electrons repel because they toss real photons back and forth like ice-skaters throwing a ball. It is misleading. The exchanged photon in $e^-e^-$ scattering is virtual — off the mass shell ( $q^2\neq0$ ), never observed, merely a term in a perturbation series. There are no little light bullets; the "force" is the low-energy, static limit of a scattering amplitude, and the same math gives an attractive Coulomb potential for opposite charges, which no ball-throwing analogy can reproduce. Two further traps ride along: the negative-energy Dirac solutions are not discarded but reinterpreted as positive-energy antiparticles; and the final-state factor is the adjoint $\bar u=u^\dagger\gamma^0$ , never a naive complex conjugate $u^*(p_3)$ , because these are four-component spinors, not scalars.

Worked examples

1 · The size of the QED coupling (fully worked)

From the measured fine-structure constant $\alpha^{-1}=137.035\,999$ , find the electron charge in natural (Heaviside–Lorentz) units, the strength of a single vertex, the scale of a tree amplitude, and the leading loop correction to the electron magnetism.

Step 1. The coupling. $\alpha = 1/137.035\,999 = 7.2974\times10^{-3}$ . In natural units the electromagnetic coupling that appears at a vertex is $e=\sqrt{4\pi\alpha}=\sqrt{4\pi\times7.2974\times10^{-3}}=\sqrt{0.09170}=\mathbf{0.3028}$ . The often-quoted " $e$ " of QED Feynman rules is this pure number, not $1.6\times10^{-19}\ \mathrm{C}$ — the Coulomb value is $e$ in SI units, tied to $\alpha$ by equation (6).

Step 2. One vertex. Each vertex contributes one factor of $e$ , so the amplitude strength per vertex is $\sqrt{\alpha}=\sqrt{7.2974\times10^{-3}}=\mathbf{0.0854}$ . This is why QED is perturbative: a vertex is an 8.5% effect, not an order-unity one.

Step 3. A tree amplitude. The simplest scattering ( $e^-\mu^-\to e^-\mu^-$ ) has two vertices, so $\mathcal M\propto e^2 = 4\pi\alpha$ , and the cross-section carries $|\mathcal M|^2\propto\alpha^2 = 5.3\times10^{-5}$ — the reason low-energy electromagnetic cross-sections are so small.

Step 4. The leading loop. Adding one internal photon multiplies the amplitude by $\sim\alpha/2\pi$ . For the electron's magnetic moment this is exactly Schwinger's 1948 result: the anomaly $a_e=(g-2)/2 = \alpha/2\pi = \mathbf{1.1614\times10^{-3}}$ . Measured value: $a_e=1.159\,652\times10^{-3}$ — agreement to three digits from one loop alone, and to twelve digits once higher orders are included. The chain $\alpha=1/137 \to e=0.303 \to$ vertex $=0.085 \to$ loop $=\alpha/2\pi=1.16\times10^{-3}$ is the numerical spine of QED.

2 · The Coulomb energy from photon exchange (fully worked)

Two electrons are held at a separation equal to the Bohr radius $a_0=0.0529\ \mathrm{nm}$ . Using the potential extracted from massless photon exchange, find their electrostatic potential energy and check it against atomic physics.

Step 1. The potential from the propagator (Derivation C) is $V(r)=Q_1Q_2\,\alpha\hbar c/r$ . Both are electrons, $Q_1=Q_2=-1$ , so $Q_1Q_2=+1$ and the potential is positive (repulsive), as it must be for like charges.

Step 2. Insert $\alpha\hbar c$ . With $\hbar c = 197.327\ \mathrm{eV\,nm}$ and $\alpha=1/137.036$ , $\alpha\hbar c = 197.327/137.036 = \mathbf{1.440\ eV\,nm}$ — the familiar $e^2/4\pi\varepsilon_0$ in convenient units.

Step 3. Evaluate at $r=a_0$ : $V = 1.440\ \mathrm{eV\,nm}\,/\,0.0529\ \mathrm{nm} = \mathbf{27.2\ eV}$ . This is exactly one Hartree ( $=2$ Rydberg $=27.211\ \mathrm{eV}$ ), the natural energy scale of atomic binding. The 27.2 eV that sets the whole scale of chemistry is the low-energy shadow of the QED propagator $-g_{\mu\nu}/q^2$ — the same vertex-and-propagator machinery that computes relativistic scattering, taken to its static limit, hands back the Coulomb energy a first-year student already knows.

Physics of the portrait

Quantum electrodynamics has no single author, but it has a single origin: Paul Dirac. He wrote the field-quantization paper that named the subject a year before he wrote the equation that supplies its electrons — both load-bearing in every line above.

Origins · 1 minPaul Dirac and how this was discovered

Paul Dirac

1902–1984 · the birth of quantum electrodynamics

In 1927, at twenty-five, Paul Dirac published "The Quantum Theory of the Emission and Absorption of Radiation," the paper that quantized the electromagnetic field into photons and founded what he called quantum electrodynamics — the first quantum field theory. A year later came the equation that bears his name: seeking a relativistic wave equation linear in time (to keep a positive probability density, unlike Klein–Gordon), Dirac factorized $E^2=p^2+m^2$ and found the price was anticommuting matrices — the gamma matrices — acting on a four-component spinor. The equation predicted the electron's spin and its $g=2$ magnetism for free, plus negative-energy solutions that Dirac reinterpreted in 1931 as antiparticles, predicting the positron Anderson found in 1932. Everything this article uses — the spinors $u(p)$ , the adjoint $\bar\psi$ , the minimal coupling, the vertex $\gamma^\mu$ — is Dirac's inheritance. His theory was left with catastrophic infinities in its loop corrections; it took Tomonaga, Schwinger, Feynman, and Dyson two decades to tame them by renormalization, turning the framework into the most accurately verified theory ever written — though Dirac himself, ever the aesthete, remained uneasy about renormalization to the end.

History

1927

— Dirac quantizes the electromagnetic field and coins "quantum electrodynamics"

1928

— Dirac's relativistic wave equation for the electron predicts spin and g = 2

1932

— Anderson discovers the positron, confirming Dirac's antiparticle prediction

1947

— Lamb and Retherford measure the Lamb shift; Kusch and Foley find g slightly above 2

1948

— Schwinger computes a = α/2π; Feynman, Tomonaga, and Dyson complete renormalized QED

1965

— Tomonaga, Schwinger, and Feynman share the Nobel Prize for quantum electrodynamics

Going further

Texts. Thomson, Modern Particle Physics, §5.3 (Introduction to QED), eqs. 5.10–5.18 — the vertex-and-propagator construction followed here · Griffiths, Introduction to Elementary Particles, 2nd ed., Ch. 7 §7.1–7.2 (the Dirac equation and its solutions) and Ch. 7 §7.5 (Feynman rules) · Peskin & Schroeder, An Introduction to Quantum Field Theory, §4.1–4.8 (the QED Lagrangian, Feynman rules, and first amplitudes) · Halzen & Martin, Quarks and Leptons, Ch. 6 · Schwartz, Quantum Field Theory and the Standard Model, Ch. 9 (QED as a gauge theory).

Provenance. The four-component spinors $u(p)$ , the gamma matrices, and the factorization of $E^2=p^2+m^2$ are recalled by name from aq-pp-dirac-equation; the adjoint $\bar\psi=\psi^\dagger\gamma^0$ and the bilinear currents from aq-pp-dirac-adjoint-bilinear-covariants; the virtual-particle exchange picture and the scalar propagator $1/(q^2-m_X^2)$ from aq-pp-feynman-diagrams-virtual; and the massive-mediator potential from aq-pp-yukawa-potential. This article does not re-derive those; it upgrades the scalar vertex to the electromagnetic one. A verifier should confirm: (1) the minimal substitution yields $\hat V_D=q\gamma^0\gamma^\mu A_\mu$ with time-like piece $q\phi$ ; (2) the polarization sum $\sum_\lambda\varepsilon\varepsilon^*=-g_{\mu\nu}$ collapses the propagator to $-g_{\mu\nu}/q^2$ ; (3) $\alpha=e^2/4\pi\varepsilon_0\hbar c=1/137.036$ and $e=\sqrt{4\pi\alpha}=0.3028$ ; (4) $\alpha\hbar c=1.440\ \mathrm{eV\,nm}$ gives $27.2\ \mathrm{eV}$ at the Bohr radius; (5) the Schwinger anomaly $\alpha/2\pi=1.16\times10^{-3}$ .

Prerequisites. The Dirac Equation · Feynman Diagrams and Virtual Particles · The Yukawa Potential · Adjoint Spinors and Bilinear Covariants. Leads to. Feynman Rules for QED · e⁺e⁻ → μ⁺μ⁻ · The Photon Propagator · Spin Averaging and Trace Techniques.

Ask the tutor: Ask me to run the Feynman rules through a real process — e.g. "walk me through $e^+e^-\to\mu^+\mu^-$ from the vertex to the cross-section" — and we will assemble, contract, and square the amplitude step by step.

Is QED really derived from just one substitution, or is that a simplification?

It is genuinely that economical, though the deep reason is gauge symmetry. The minimal substitution $\partial_\mu\to\partial_\mu+iqA_\mu$ is what you are forced to do if you demand that the Dirac Lagrangian be invariant under a local phase rotation $\psi\to e^{-iq\theta(x)}\psi$ : the ordinary derivative spoils the invariance, and the only cure is to introduce $A_\mu$ transforming as $A_\mu\to A_\mu+\partial_\mu\theta$ and bundle it into the covariant derivative. So the one substitution is the visible face of local $U(1)$ gauge invariance — the principle that dictates the vertex, forbids a photon mass, and guarantees charge conservation.

Why does the photon carry four polarizations in the propagator but only two in reality?

A real, on-shell photon is transverse: it has exactly two physical polarization states, because gauge invariance and masslessness remove the timelike and longitudinal modes. But the photon exchanged inside a diagram is virtual — off the mass shell, $q^2\neq0$ — and to represent an arbitrary momentum transfer the propagator must formally include all four components, producing the clean numerator $-g_{\mu\nu}$ in Feynman gauge. The unphysical pieces never appear in a measurable answer: they cancel by the Ward identity, a consequence of gauge invariance and current conservation $q_\mu j^\mu=0$ .

What is the difference between the vertex factor and a coupling constant?

The vertex factor $iQ_f e\gamma^\mu$ is the complete Feynman rule for the junction: it carries the coupling strength (the charge $Q_f e$ ), the spinor structure ( $\gamma^\mu$ ), and the Lorentz index that hands off to the photon. The coupling constant proper is just the strength, best expressed dimensionlessly as $\alpha=e^2/4\pi\approx1/137$ — the number that tells you how strong the interaction is and how fast the series converges. The vertex factor is the full object you insert into an amplitude: number, matrix, and index together.

Why is the amplitude written with adjoint spinors ū instead of u†?

Because only $\bar u=u^\dagger\gamma^0$ produces objects with definite Lorentz transformation properties. The naive bilinear $u^\dagger\Gamma u$ is not a tensor under boosts, because the Dirac spinor representation is not unitary; inserting the $\gamma^0$ fixes this, so that $\bar u u$ is a scalar and $\bar u\gamma^\mu u$ is a four-vector. In the QED current $j^\mu=\bar u(p')\gamma^\mu u(p)$ that four-vector property is exactly what lets $j_1\cdot j_2$ be a Lorentz scalar and the whole amplitude be frame-independent. Writing $u^\dagger$ instead of $\bar u$ is a genuine error, not a notational preference.

If α "runs," which value do I use — 1/137 or 1/128?

Use the value at the energy scale of your process. The number $1/137.036$ is the coupling measured at essentially zero momentum transfer — the Thomson limit, appropriate for atomic physics and low-energy scattering. As you probe shorter distances (higher $q^2$ ), virtual electron–positron pairs screen the bare charge less effectively, so the effective $\alpha$ grows; at the $Z$ mass it reaches about $1/127.95$ . For a hydrogen atom, $1/137$ is correct. The running itself is a triumph of the theory — vacuum polarization, the photon propagator dressed by loops, seen directly in experiment.

Does QED describe antimatter, and where do the positrons enter?

Yes, completely and automatically. The same Dirac field $\psi$ that annihilates electrons creates positrons, and the negative-energy solutions Dirac worried about become positive-energy antiparticle states. At the level of Feynman rules, a positron is simply an electron line running backwards in time through the same vertex $iQ_f e\gamma^\mu$ ; no new rule is needed. This is why $e^+e^-\to\mu^+\mu^-$ — the archetypal QED process — uses exactly the vertex and propagator built here, with antiparticles represented by $\bar v$ spinors.