Quantum Field Theory II

Gauge theory, symmetry breaking, and the Standard Model assembled

About this course

The second course in quantum field theory: the path integral for fields, generating functionals and the effective action, Ward identities, non-Abelian gauge symmetry and ghosts, the beta function and confinement, the renormalisation group as physics, spontaneous symmetry breaking and the Higgs mechanism, electroweak unification, anomalies and chiral symmetry, and the Standard Model with its precision tests and its gaps.

What you will be able to do

  • Quantise a field theory with the path integral and read the effective action
  • Derive Ward identities and gauge-fix a non-Abelian theory with Faddeev-Popov ghosts
  • Use the beta function and the renormalisation group to explain running and confinement
  • Explain spontaneous symmetry breaking, the Higgs mechanism and electroweak unification
  • Assemble the Standard Model, and state precisely what it does not explain

What this course covers

  • The central question: proton-electron charge equality
  • Anomalies: loops that break symmetries
  • The anomaly ledger and its four exact zeros
  • The pion decay as the one allowed leak
  • Deriving quark charges from the electron alone
  • Pion width vs. color counting: two different instruments
  • Resolution of the opening question
  • Gauge rings and force-field count
  • Three gauge coupling constants
  • Matter field slots and generation structure
  • Resting-field (Higgs) parameters
  • Reducing 54 Yukawa grips to 13 physical parameters
  • Total parameter count: 19 free numbers
  • Z-boson invisible width as a generation counter
  • Fields vs. particles: one number per point
  • The field action: springs and tethers
  • Real-time path integral: arrows of length one
  • Wick rotation: arrows become brightness
  • Typical shapes are jagged; average action grows with lattice points
  • The partition function Z diverges; physical observables are ratios
  • The stationary shape is the crowd's centre, not a typical member
  • Local gauge freedom and the cost of fresh choices
  • The gauge connection on edges
  • Field strength as failure of a closed loop to close
  • Non-commuting turns and the second-order gap
  • Self-interacting gauge field and its vertices
  • Summary: what the hour earned
  • Spontaneous symmetry breaking: the hat and ball
  • Instability of the crown: negative curvature
  • Two directions in the valley: massive and massless modes
  • Goldstone theorem: flatness forced by roundness
  • Massless particle heard on a ring of sites
  • Superfluid helium as a physical realisation
  • Counting flat ways: the Goldstone theorem in full
  • Why a gauge field cannot have a mass by hand
  • Local turns and the gauge field as a link
  • Transferring a ripple from the hat's field to the gauge field
  • The Higgs mechanism: token crossing and the balanced ledger
  • The Higgs particle as the remaining ripple across the valley
  • The real hat: four-component field, four gauge fields, count of 12
  • The muon's magnetic moment G and its excess over two
  • The QED ladder: loop-by-loop contributions to A
  • The hadronic vacuum polarisation rung and the two-route tension
  • The W-boson mass as a second precision test
  • The pion mass puzzle: same quarks, different masses
  • Chiral symmetry and spontaneous breaking by the condensate
  • Massless-quark limit: three massless Goldstone pions
  • Quark mass term as a tilt of the hat
  • Pion mass squared proportional to quark mass (GOR relation)
  • Numerical check with measured inputs
  • Proton mass is insensitive to quark masses; pion mass vanishes without them
  • The free string and its setup
  • Single-poke response and the propagator as a tent
  • Two-point correlation function equals the tent
  • Wick-pairing rule and the generating functional Z(J)
  • Yukawa-like force between two sources
  • Free vs stiffened string: width change and leftover interactions
  • The Standard Model's two kinds of silence
  • The strong CP problem and the color angle
  • The hierarchy problem and gravity's scale
  • Neutrino oscillations and mass splittings
  • The right-handed neutrino row and the extended tally
  • Dark matter has no slot on the grid
  • Summing up: silence versus wrongness
  • The coupling as a number at a scale
  • Quark-pair screening and the matter bar
  • 1/α as a straight line and its slope
  • The gluon bar and asymptotic freedom
  • Quark thresholds and piecewise running
  • Walking the prediction from Z to tau
  • Limits of the approximation and the high-energy regime
  • Classical field energy and source tilting
  • Legendre transform and the effective action
  • One-loop quantum correction to the potential
  • Ultraviolet divergences and renormalisation (counter terms)
  • Coleman–Weinberg potential and radiative symmetry breaking
  • Fermion loops, sign, and limits of one-loop expansion
  • The neutral-field mixing plane
  • Circle of masses and identifying the Z and photon
  • The weak-mixing angle and the W3–Z–photon triangle
  • Particle charge arrows and electric neutrality
  • The Z–W mass triangle: central result
  • Radiative loop corrections and the top-quark mass
  • Gauge invariance on five bars
  • Ward identity: telescoping staircase
  • Leaks in the Ward identity: off-shell ends and wrong charges
  • Photon mass forbidden by a loop with no ends
  • Charge universality from Z1 = Z2
  • The Cabibbo mixing puzzle introduced
  • Why matter cannot be given a mass directly
  • Mass from the resting field: the Yukawa dial
  • Two frames: mass frame vs. W frame
  • The Cabibbo angle and the one-in-twenty shadow
  • Experimental check of the Cabibbo turn
  • Three copies: the CKM matrix, three angles, and one phase
  • The phase, matter–antimatter asymmetry, and open questions
  • Spreading fields, finite separation energy
  • Color flux tube and linear confinement
  • String tension: reading the price per fermi
  • String breaking and quark confinement prediction
  • Rotating string: independent tension measurement
  • The Ising lattice and block-spin coarse-graining
  • Criticality and scale-free structure
  • Measuring couplings after blocking: the coupling plane
  • Renormalization-group flow and the fixed point
  • Renormalizability as infrared survival, not a fundamental law
  • Universal critical exponent and universality

Syllabus

  1. The Path Integral and the Effective Action

    1. L1 · The Path Integral for Fields
    2. L2 · Generating Functionals
    3. L3 · The Effective Action
    4. L4 · Ward Identities

  2. Non-Abelian Gauge Theory

    1. L5 · Non-Abelian Gauge Symmetry
    2. L7 · The Beta Function
    3. L8 · Confinement
    4. L9 · The Renormalisation Group as Physics

  3. Symmetry Breaking and the Electroweak Theory

    1. L10 · Spontaneous Symmetry Breaking
    2. L11 · The Higgs Mechanism
    3. L12 · Electroweak Unification
    4. L13 · Fermion Masses and Mixing

  4. Anomalies and Chiral Symmetry

    1. L14 · Anomalies
    2. L15 · Chiral Symmetry and the Pion

  5. The Standard Model and Its Limits

    1. L16 · The Standard Model, Assembled
    2. L17 · Precision Tests
    3. L18 · What the Standard Model Does Not Explain