Physics notes

Quantum field theory (QFT) is the framework that QED and QCD are built in. It combines quantum mechanics with special relativity. The basic objects are fields, one for each kind of particle, filling all of space. A particle is a quantum of excitation of its field, in the same way a photon is a quantum of the electromagnetic field.

Why fields. Relativity allows particles to be created and destroyed, and single-particle quantum mechanics cannot describe that. In QFT the field operators create and annihilate particles, so states with different particle numbers live in the same space. Pair production, decays and scattering are all described by the same equations.

Symmetry fixes the forces. Requiring the theory to be unchanged by a local phase rotation of the electron field (a gauge symmetry) forces a photon field to exist and fixes how it couples. The same principle with the symmetry group SU(3) gives QCD and its eight gluons; with SU(2)×U(1) it gives the weak and electromagnetic forces together. The Standard Model is the QFT built from these symmetries plus the Higgs field.

Calculating. Predictions are made with perturbation theory, organised by Feynman diagrams. Loop diagrams give divergent integrals; renormalisation absorbs the divergences into the measured mass and charge, and the finite remainder is the prediction. A consequence is that couplings run with energy scale, which is why α grows slowly and α_s shrinks.

In the chamber. Every event on the home page is a QFT process: a photon converting to a pair (QED), a beam proton breaking a nucleus into pions (QCD), a pion or muon decaying (weak interaction). The bubbles themselves come from ionisation, the lowest-order QED interaction between the passing charge and the atomic electrons.

Further reading: Lancaster and Blundell, Quantum Field Theory for the Gifted Amateur; Zee, Quantum Field Theory in a Nutshell; Peskin and Schroeder, An Introduction to Quantum Field Theory.

QED is the quantum field theory of the electron field and the photon field. Every track on the home page is an electromagnetic process: a charged particle moving through liquid hydrogen transfers energy to the electrons of the atoms it passes, and photons carry that interaction.

Fields. Electrons and photons are excitations of two quantum fields. Electrons interact by emitting and absorbing photons; a photon exchanged between two electrons transfers momentum, which is the electromagnetic force.

Feynman diagrams. Every way a process can happen gets a diagram and a complex number called an amplitude. The amplitudes for all diagrams are summed and the result squared to get a probability. Simple diagrams dominate because each photon vertex contributes a factor of the fine-structure constant, roughly 1/137, so a diagram with two extra vertices is suppressed by about 137².

Precision. QED predicts the electron's anomalous magnetic moment to about twelve significant figures, in agreement with experiment. It is the most precisely tested theory in physics.

In the chamber. A gamma ray is invisible until it converts into an electron and a positron near a nucleus, the V-shaped pair in the field guide. The electron then radiates photons as it curls (bremsstrahlung), which is why electron spirals tighten quickly.

Further reading: Feynman, QED: The Strange Theory of Light and Matter.

Protons, neutrons, pions and the sprays of tracks from a beam collision are made of quarks, held together by the strong force. QCD is the theory of that force. Its structure is similar to QED, but its behaviour is very different.

Colour charge. Quarks carry one of three colour charges (labelled red, green and blue), and the force carriers are eight gluons. Unlike photons, gluons carry colour charge themselves, so they interact with one another.

Asymptotic freedom and confinement. Because gluons interact with gluons, the strong coupling weakens at short distances and high energies, so quarks inside a proton behave almost as free particles. At large distances the field between two quarks forms a flux tube whose energy grows with separation, until it is energetically cheaper to create a new quark–antiquark pair from the vacuum than to stretch further. This is confinement: isolated quarks are never observed, only hadrons.

Mass. The Higgs field gives the up and down quarks only a few MeV each, yet a proton has a mass of 938 MeV. Almost all of it is the energy of the gluon field and the quarks' motion, confined inside the proton.

In the chamber. When a beam proton hits a hydrogen nucleus, quarks scatter and hadronise into a spray of pions and protons: the star of tracks in the simulation. The pions then decay through the weak force into the muons and electrons that kink off their ends.

Further reading: Frank Wilczek, The Lightness of Being.

Write the wavefunction in the Schrödinger equation as a magnitude times a phase, ψ = √ρ · e^{iS/ħ}, and separate the real and imaginary parts. The result, found by Erwin Madelung in 1927, is two equations: a continuity equation for a probability fluid of density ρ, and an Euler equation for its velocity v = ∇S/m, identical to classical fluid dynamics except for one extra term.

The quantum potential. It depends on the shape of the density rather than its magnitude, so a dilute fluid feels it as strongly as a dense one. In this formulation it accounts for the stability of atoms, interference fringes and tunnelling.

Superfluids. Superfluid helium below 2.17 K and Bose–Einstein condensates are fluids described by a single macroscopic wavefunction, so these are their equations of motion. Because the velocity is the gradient of a phase, the fluid cannot rotate smoothly; it rotates only through quantised vortices, each carrying one quantum of circulation, h/m. A rotating superfluid forms a lattice of identical vortices.

Connection to QCD. The quark–gluon plasma produced in heavy-ion collisions at RHIC and the LHC behaves as a nearly perfect fluid, with a viscosity-to-entropy ratio close to the bound ħ/4πk_B. Relativistic hydrodynamics is one of the main tools used to compare QCD with those experiments.

Further reading: Madelung (1927), Quantentheorie in hydrodynamischer Form; Donnelly, Quantized Vortices in Helium II.