Quantum Field Theory I

From free fields to Feynman diagrams, loops and renormalisation

About this course

Canonical quantisation of scalar, Dirac and electromagnetic fields, the S-matrix and Feynman rules, tree-level QED processes, and one-loop renormalisation through to the running coupling and the anomalous magnetic moment.

What you will be able to do

  • Quantise free scalar, Dirac and electromagnetic fields and derive their propagators
  • Derive Feynman rules from Wick's theorem and the S-matrix
  • Calculate tree-level cross sections for Compton scattering and electron-positron annihilation
  • Apply crossing, unitarity and the optical theorem to scattering amplitudes
  • Regularise and renormalise one-loop divergences and compute the running coupling

What this course covers

  • Five borrowed premises and the hour's question
  • Relativistic energy-momentum relation and its two roots
  • Energy cost of confinement and the reduced Compton wavelength
  • Pair-production threshold from halving the box
  • Particle number is not conserved; charge is
  • The quantum field as the forced object
  • Causality preserved by cancellation, not prohibition
  • Course outlook: the field's law via stationary action
  • The Interaction Picture
  • The Cut Is a Choice: Particles Depend on the Split
  • Time-Ordering and the Exponential Approximation
  • Linear Coupling: One Switching Priced
  • Three Assumptions Behind 'Free at Both Ends'
  • Ordering Costs More Than a Phase for Non-Linear Couplings
  • The S-Matrix and What It Actually Says
  • The divergent loop sum introduced
  • Volume of shells in four-dimensional momentum space
  • Pricing one term: the cancellation of sixteen against sixteen
  • The superficial degree of divergence
  • Nested-diagram failure of the naive count
  • Exact cancellation and the list of two divergent shapes
  • The bracket that decides renormalisability
  • Why two Feynman diagrams are required
  • Gauge test: two diagrams form one object
  • Compton wavelength shift (kinematics)
  • Classical radius as a length from charge and mass
  • Klein-Machina differential cross-section
  • Forward-direction identity at all energies
  • Total cross-section, Thomson limit, and numerical result
  • Feynman diagram enumeration for electron-positron annihilation
  • Conservation laws and rest-mass non-conservation
  • Angular distribution of the squared amplitude
  • Annihilation cross-section and its energy dependence
  • The R-ratio: a dimensionless count of quark charges
  • Quark color: three copies per charge from the R-ratio staircases
  • Residual excess and what the calculation leaves open
  • Regularization: the cost of stopping a divergent sum
  • Scheme-dependence: the gap between two stopping methods
  • Separation of the answer into two brackets
  • Scheme-independence of the energy-dependent bracket
  • Symmetry breaking by a sharp cutoff: the rename test
  • The unabsorbed bracket and the next problem
  • Normal ordering and the contraction of two fields
  • The contraction as a propagator with time-ordering signature
  • Wick's theorem: pairing expansion for many fields
  • Counting complete pairings: the double-factorial ladder
  • Feynman diagrams as classes of pairings
  • The full theorem away from the vacuum: incomplete pairings
  • Disconnected diagrams and vacuum cancellation
  • The Action as a Single Number for a Field History
  • Stationary Action and Which History the Field Takes
  • Deriving the Euler–Lagrange Field Equation
  • Relativistic Dispersion Relation as a Derived Result
  • Noether's Theorem: Symmetry Yields a Conserved Current and Charge
  • Symmetry Breaking, Current Ambiguity, and the Classical Charge's Limits
  • Counting terms: splitting four legs into pairs
  • Three invariants on one kinematic plane
  • Reality and complexity of the amplitude
  • Unitarity bound and the circle in the complex plane
  • The unitarity ceiling on cross-section
  • Optical theorem: forward amplitude encodes total cross-section
  • Interconnection of channels via the total width
  • The unmeasured split: written-down value vs. instrument reading
  • Written-down value's dependence on stopping point
  • Spending one measurement: stopping point leaves all predictions
  • Three theories, one curve: parallel shift under stopping-point change
  • The same move applied to the charge
  • Counting purchased measurements: renormalizability
  • Written-down values are scheme-dependent, not merely unknown
  • What this hour settles and what it hands on
  • Field as a collection of independent modes
  • Quantizing one harmonic oscillator: zero-point energy
  • Particles as ladder rungs; indistinguishability from the ladder
  • Vacuum energy divergence
  • Boundary dependence makes vacuum energy observable
  • Casimir effect — finite attractive pressure between plates
  • What the Casimir measurement does and does not prove
  • Dimensional analysis of the coupling constant
  • Vertex factor and the half-cancellation
  • Rate as a slope over a continuum of end states
  • Identical-particle factor of one-half
  • Phase-space measure after conservation conditions
  • Four-factor decay rate and dimensional ledger
  • Three readings of the decay rate and limits of first-order perturbation theory
  • The running coupling: charge depends on distance
  • Vacuum polarization as the screening mechanism
  • The beta function: rate of running per decade
  • Particle thresholds and slope steps
  • Reading the particle count from the curve
  • The quark contribution: calculated vs. measured blocks
  • Recovery of the collider R-ratio and the Landau pole
  • Causality as a commutator condition
  • Normalization of the equal-time commutator
  • Commutator as a difference of two amplitudes
  • Lorentz boost reversing spacelike separations
  • Commutator behaviour inside vs outside the light cone
  • The Feynman propagator: time-ordered amplitude and pole prescription
  • Propagator outside the light cone and limits of the result
  • Five given foundations of the calculation
  • The anomalous magnetic moment and its measurement
  • The one-loop vertex correction: the central derivation
  • Mass independence of the one-loop result
  • Higher-order corrections and the perturbative ladder
  • What limits the comparison: the charge measured elsewhere
  • The even perturbative ladder and its spacing
  • Borrowed results and the spin-half setup
  • Energy of the spin-half field and the reversed-order problem
  • The anti-commutator repair and its cost: negative-length states
  • The causality test: bracket of fields outside the light cone
  • Odd vs. even amplitude combinations and bracket selection
  • The spin-statistics connection: the two-by-two table
  • Exclusion principle as a consequence of the anti-commutator
  • The electromagnetic operator's universal singularity
  • Redundancy versus symmetry
  • Gauge fixing and residual freedom
  • Polarisation count on the light cone
  • Cavity counting as independent verification
  • Gauge-fixed propagator and the gauge parameter
  • Measurement over algebra as an epistemic principle
  • Five foundational loans from prior hours
  • Experimental measurement of the Lamb shift
  • Degeneracy of hydrogen 2S and 2P½ levels in the Dirac formula
  • Vacuum jitter shifts only the level that reaches the nucleus
  • The basic unit of the Lamb shift calculation
  • Logarithmic structure of the jitter sum and the objection answered
  • Vacuum charge screening lowers the shifted level
  • Anomalous magnetic moment contributions to both levels
  • The full one-loop Lamb shift ledger and its result
  • What remains in the half-percent gap
  • Cross-check via the ground-state Lamb shift ratio
  • Why a radio receiver, not a spectrograph

Syllabus

  1. Free Fields

    1. L1 · Why Fields Are Forced
    2. L2 · The Classical Field and Its Action
    3. L3 · Quantising the Free Scalar
    4. L4 · Causality and the Propagator
    5. L5 · The Dirac Field, Quantised
    6. L6 · The Electromagnetic Field, Quantised

  2. Interactions and Feynman Rules

    1. L7 · Interactions and the S-Matrix
    2. L8 · Wick's Theorem and the Feynman Rules
    3. L9 · The First Diagram, End to End

  3. Tree-Level Processes

    1. L10 · Compton Scattering
    2. L11 · Electron-Positron Annihilation
    3. L12 · Crossing, Unitarity, the Optical Theorem

  4. Loops and Renormalisation

    1. L13 · Loops, and Where the Infinity Comes From
    2. L14 · Regularisation
    3. L15 · Renormalisation, and What “Measured” Means
    4. L16 · The Running Coupling
    5. L17 · The Anomalous Magnetic Moment
    6. L18 · The Lamb Shift, and What the Vacuum Does