interactive QFT

Draw the diagram. Watch the physics follow.

A complete QED course, one real calculation at a time — from Feynman diagrams you draw yourself, through tree level and into renormalization, to numbers an experiment can check.

every lesson rides the same pipeline

  1. Diagrams
  2. Amplitude
  3. Squaring
  4. Spin sum
  5. Phase space
  6. Cross section

QED Part 1 — Foundations

We start where a first QFT course hands you the keys: the fields are already quantized and the Lagrangian is on the table. These lessons turn it into a working rulebook.

  1. 1.1 The QED Lagrangian Where the course begins: the electron field, the photon field, and how demanding local gauge invariance conjures the one interaction term that powers everything that follows. the Dirac & photon fields · local U(1) gauge invariance · the interaction term start →
  2. 1.2 Propagators How disturbances in the fields travel: derive the photon and fermion propagators from the free theory — and meet the gauge-fixing trick that makes the photon’s possible at all. Green functions · gauge fixing & the photon propagator · the fermion propagator start →
  3. 1.3 The Feynman rules & available interactions Assemble the complete rulebook — external legs, propagators, the vertex — then take it to the canvas and discover what QED lets you draw, and what it forbids. the QED vertex · external legs · the feasibility frontier start →

QED Part 2 — Tree level

Every calculation rides the same pipeline; every lesson adds the minimum new physics.

  1. 2.1 e⁺e⁻ → μ⁺μ⁻ at leading order Draw the contributing diagram, watch the amplitude assemble itself, then follow it through squaring, spin sums, and phase space to σ(√s) on a collider plot. Feynman rules · the amplitude · spin sums · phase space → σ(√s) start →
  2. 2.2 Crossing & Mandelstam e⁻μ⁻ → e⁻μ⁻ The same amplitude, read sideways: rotate annihilation into electron–muon scattering, meet s, t, u — and watch q² go spacelike. crossing symmetry · Mandelstam variables · spacelike q² start →
  3. 2.3 Interference e⁺e⁻ → e⁺e⁻ Two diagrams contribute — so amplitudes add before they square. Interference, the relative minus sign from exchanging fermions, and your first two-diagram checkpoint. amplitude superposition · interference · the relative minus sign start →
  4. 2.4 Identical particles e⁻e⁻ → e⁻e⁻ Indistinguishable electrons: the u-channel partner diagram, antisymmetrization, and why the cross section carries a factor of ½. identical fermions · u-channel · symmetry factor ½ start →
  5. 2.5 Propagating fermions & photon polarization e⁻γ → e⁻γ A photon scatters off an electron: the first internal fermion propagator, photon polarization sums, and the Klein–Nishina formula. fermion propagators · polarization sums · Klein–Nishina start →
  6. 2.6 Annihilation — putting it together e⁺e⁻ → γγ Cross Compton into pair annihilation: identical photons in the final state, and a checkpoint that uses everything Part 1 taught. crossing in practice · identical bosons · the spin-statistics sign · a collinear logarithm start →
  7. 2.7 From amplitude to number σ → N Every lesson so far ended at a curve. No experiment measures a curve — it counts events. Flux and why σ is an area, luminosity, and why a detector’s acceptance defines the observable rather than approximating it. Then point it at LEP, and watch tree-level QED miss by a factor of 191.5. the flux factor · luminosity and N = σ∫L dt · acceptance as definition · when a discrepancy is a discovery start →

QED Part 3 — Loops & renormalization

Where the infinities are. Every integral in this part has one or two propagators and an exact closed form — enough to meet a divergence, decide what to do with it, and watch the fine structure constant stop being constant.

  1. 3.1 Why loops diverge A closed cycle leaves one momentum undetermined, and you integrate over it. Count the powers of that momentum and you know in advance which diagrams diverge — a list that turns out to be short, and to contain every loop the canvas will let you draw. loop momenta · superficial degree of divergence · the divergent list · choosing a regulator start →
  2. 3.2 Poles, contours, and continuation Two promises come due at once. The $+i\varepsilon$ you were told to carry silently turns out to be what licenses swinging a contour, and the analytic continuation dimensional regularization runs on is a theorem nobody has shown you yet. Residues, arcs that die, and a function that outlives the integral defining it. residues and contours · Jordan's lemma · the iε prescription · analytic continuation start →
  3. 3.3 Evaluating a loop integral The recipe, once, in the abstract: combine the denominators with Feynman parameters, shift the loop momentum, rotate the contour into Euclidean space, and do the angular integral in d dimensions. What comes back is a Γ function with a pole sitting exactly at d = 4. Feynman parameters · Wick rotation · the master formula · expanding in ε start →
  4. 3.4 Vacuum polarization The first complete calculation. A photon spends part of its life as an electron–positron pair; the Ward identity fixes the shape of the answer before you compute it, and above 4m² the result goes complex — because up there the pair can be real. the photon self-energy · transversality from the Ward identity · the branch cut at 4m² · the optical theorem start →
  5. 3.5 The one-loop masters You have just computed a bubble at equal masses. Here is the same object with any masses, in every kinematic region, beside the tadpole that accompanies it — the two integrals every one-loop calculation in this part is assembled from. the tadpole A₀ · the bubble B₀ · thresholds & branch cuts · a basis of masters start →
  6. 3.6 Renormalization What to do with a 1/ε. Counterterms reorganize the Lagrangian rather than patch its output, Z factors connect the parameters you wrote down to the ones you can measure, and a scale μ appears that no observable is allowed to remember. counterterms · Z₃ and the photon field · on-shell vs MS-bar · the scale μ start →
  7. 3.7 Mass renormalization The electron dresses itself in its own field. The self-energy moves the pole of the propagator, so the mass in the Lagrangian is not the mass anybody measures — and there is more than one honest answer to which one deserves the name. the self-energy −iΣ(p) · Z₂ and field renormalization · pole mass vs running mass evaluates locked
  8. 3.8 The running coupling Sum the polarization bubbles and the coupling starts to move: vacuum screening makes the charge you measure depend on how closely you look. The prediction is checkable against real experiments — and where it falls short is where the rest of the Standard Model is hiding. resummation · screening · α(q²) · the Landau pole locked
  9. 3.9 The renormalization group Turn “no observable may remember μ” into a differential equation and it starts predicting. The β function runs any coupling, not just this one — so run all three of the Standard Model’s, and watch them very nearly meet at an energy nobody will ever build a machine to reach. the RG equation · the β function · running any coupling · the near-miss at 10¹⁶ GeV locked

QED Part 4 — Precision QED

Where the integrals stop being solvable one at a time. Triangles and boxes reduce onto a basis of masters — and out of the vertex fall the electron’s anomalous magnetic moment and the cancellation that makes a cross section finite.

  1. 4.1 The vertex & tensor reduction Three propagators, and a numerator carrying loop momenta upstairs. Passarino–Veltman reduction decomposes it onto scalar integrals, and what survives is two form factors — everything a photon is able to learn about an electron’s structure. tensor reduction · the triangle C₀ · form factors F₁, F₂ · Ward identity Z₁ = Z₂ evaluates locked
  2. 4.2 g − 2 F₂(0) = α/2π. One line of algebra at the end of a long calculation predicts that the electron is not quite as magnetic as Dirac said — and the measurement agrees to more decimal places than almost anything else in physics. the anomalous magnetic moment · Schwinger’s α/2π · prediction vs measurement locked
  3. 4.3 The box Two photons exchanged at once: four propagators around a loop, and the last master integral. Reducing it closes the one-loop toolkit — past four points, nothing new is ever needed. the box D₀ · two-photon exchange · reduction to lower points locked
  4. 4.4 Soft photons & the infrared A photon of arbitrarily low energy costs arbitrarily little to emit, and the integral says so. Where the ultraviolet divergence came from short distances, this one comes from long ones — the same regulator, doing the opposite job. the eikonal approximation · the soft factor · UV ε vs IR ε · collinear divergences locked
  5. 4.5 Real emission & the cancellation e⁺e⁻ → μ⁺μ⁻(γ) Ask for two muons and the answer is infinite. Ask for two muons plus any photon too soft to see, and it is a number. The virtual and real divergences cancel against each other — and what makes them cancel is the detector’s own resolution. Bloch–Nordsieck · virtual + real cancellation · resolution as definition · finite σ_NLO locked
  6. 4.6 One-loop curiosities — Furry’s theorem & light-by-light Two amplitudes that power counting says should diverge, and do not. Charge conjugation kills every odd-photon loop before you compute it, and gauge invariance rescues light-by-light — symmetry doing the work of an integral. Furry’s theorem · light-by-light · symmetry as computation evaluates locked