Lectures of Sidney Coleman on Quantum Field Theory

The following sections are included:

• Dedication
• Contents
• Foreword
• Preface
• Frequently cited references
• Index of useful formulae
• A note on the problems

The following sections are included:

• Adding special relativity to quantum mechanics
  • Introductory remarks
  • Theory of a single free, spinless particle of mass μ
  • Determination of the position operator X
• The simplest, many-particle theory
  • First steps in describing a many-particle state
  • Occupation number representation
  • Operator formalism and the harmonic oscillator
  • The operator formalism applied to Fock space
• Constructing a scalar quantum field
  • Ensuring relativistic causality
  • Conditions to be satisfied by a scalar quantum field
  • The explicit form of the scalar quantum field
  • Turning the argument around: the free scalar field as the fundamental object
  • A hint of things to come
• Problems 1
• Solutions 1
• Tho method of the missing box
  • Classical particle mechanics
  • Quantum particle mechanics
  • Classical field theory
  • Quantum field theory
  • Normal ordering
• Symmetries and conservation laws I. Spacetime symmetries
  • Symmetries and conservation laws in classical particle mechanics
  • Extension to quantum particle mechanics
  • Extension to field theory
  • Conserved currents are not uniquely defined
  • Calculation of currents from spacetime translations
  • Lorentz transformations, angular momentum and something else
• Problems 2
• Solutions 2
• Symmetries and conservation laws II. Internal symmetries
  • Continuous symmetries
  • Lorentz transformation properties of the charges
  • Discrete symmetries
• Introduction to perturbation theory and scattering
  • The Schrödinger and Heisenberg pictures
  • The interaction picture
  • Dyson’s formula
  • Scattering and the S-matrix
• Problem 3
• Solution 3
• Perturbation theory I. Wick diagrams
  • Three model field theories
  • Wick’s theorem
  • Dyson’s formula expressed in Wick diagrams
  • Connected and disconnected Wick diagrams
  • The exact solution of Model I
• Problem 4
• Solution 4
• Perturbation theory II. Divergences and counterterms
  • The need for a counterterm in Model 2
  • Evaluating the S matrix in Model 2
  • Computing the Model 2 ground state energy
  • The ground state wave function in Model 2
  • An infrared divergence
• Problem 5
• Solution 5
• Mass renormalization and Feynman diagrams
  • Mass renormalization in Model 3
  • Feynman rules in Model 3
  • Feynman diagrams in Model 3 to order g²
  • $\mathcal{O}$(g²) nucleon-nucleon scattering in Model 3
• Scattering I. Mandelstam variables, CPT and phase space
  • Nucleon-antinucleon scattering
  • Nucleon-meson scattering and meson pair creation
  • Crossing symmetry and CPT invariance
  • Phase space and the S matrix
• Scattering II. Applications
  • Decay processes
  • Differential cross-section for a two-particle initial state
  • The density of final states for two particles
  • The Optical Theorem
  • The density of final states for three particles
  • A question and a preview
• Problems 6
• Solutions 6
• Green’s functions and Heisenberg fields
  • The graphical definition of ${\tilde{G}}^{(n)}(k_i)$
  • The generating functional Z[ρ] for G⁽ⁿ⁾(x_i)
  • Scattering without an adiabatic function
  • Green’s functions in the Heisenberg picture
  • Constructing in and out states
• Problems 7
• Solutions 7
• The LSZ formalism
  • Two-particle states
  • The proof of the LSZ formula
  • Model 3 revisited
  • Guessing the Feynman rules for a derivative interaction
• Problem 8
• Solution 8
• Renormalization I. Determination of counterterms
  • The perturbative determination of A
  • The Källén-Lehmann spectral representation
  • The renormalized meson propagator ${\tilde{D}}^{\prime }$
  • The meson self-energy to $\mathcal{O}$(g²)
  • A table of integrals for one loop
• Renormalization II. Generalization and extension
  • The meson self-energy to $\mathcal{O}$(g²), completed
  • Feynman parametrization for multiloop graphs
  • Coupling constant renormalization
  • Are all quantum field theories renormalizable?
• Problems 9
• Solutions 9
• Unstable particles
  • Calculating the propagator ${\tilde{D}}^{\prime }$ for μ > 2m
  • The Breit-Wigner formula
  • A first look at the exponential decay law
  • Obtaining the decay law by stationary phase approximation
• Representations of the Lorentz Group
  • Defining the problem: Lorentz transformations in general
  • Irreducible representations of the rotation group
  • Irreducible representations of the Lorentz group
  • Properties of the SO(3) representations D^(s)
  • Properties of the SO(3,1) representations D^{(s+, s−)}
• Problems 10
• Solutions 10
• The Dirac Equation I. Constructing a Lagrangian
  • Building vectors out of spinors
  • A Lagrangian for Weyl spinors
  • The Weyl equation
  • The Dirac equation
• The Dirac Equation II. Solutions
  • The Dirac basis
  • Plane wave solutions
  • Pauli’s theorem
  • The γ matrices
  • Bilinear spinor products
  • Orthogonality and completeness
• Problems 11
• Solutions 11
• The Dirac Equation III. Quantization and Feynman Rules
  • Canonical quantization of the Dirac field
  • Wick’s theorem for Fermi fields
  • Calculating the Dirac propagator
  • An example: Nucleon-meson scattering
  • The Feynman rules for theories involving fermions
  • Summing and averaging over spin states
• Problems 12
• Solutions 12
• CPT and Fermi fields
  • Parity and Fermi fields
  • The Majorana representation
  • Charge conjugation and Fermi fields
  • PT invariance and Fermi fields
  • The CPT theorem and Fermi fields
• Renormalization of spin-½ theories
  • Lessons from Model
  • The renormalized Dirac propagator ${\tilde{S}}^{\prime }$
  • The spectral representation of ${\tilde{S}}^{\prime }$
  • The nucleon self-energy ${\tilde{\Sigma }}^{\prime }$
  • The renormalized coupling constant
• Problems 13
• Solutions 13
• Isospin
  • Field theoretic constraints on coupling constants
  • The nucleon and pion as isospin multiplets
  • Experimental consequences of isospin conservation
  • Hypercharge and G-parity
• Coping with infinities: regularization and renormalization
  • Regularization
  • The BPHZ algorithm
  • Applying the algorithm
  • Survey of renormalizable theories for spin 0 and spin ½
• Problems 14
• Solutions 14
• Vector fields
  • The free real vector field
  • The Proca equation and its solutions
  • Canonical quantization of the Proca field
  • The limit μ → 0: a simple physical consequence
  • Feynman rules for a real massive vector field
• Electromagnetic interactions and minimal coupling
  • Gauge invariance and conserved currents
  • The minimal coupling prescription
  • Technical problems
• Problems 15
• Solutions 15
• Functional integration and Feynman rules
  • First steps with functional integrals
  • Functional integrals in field theory
  • The Euclidean Z₀[J] for a free theory
  • The Euclidean Z[J] for an interacting field theory
  • Feynman rules from functional integrals
  • The functional integral for massive vector mesons
• Extending the methods of functional integrals
  • Functional integration for Fermi fields
  • Derivative interactions via functional integrals
  • Ghost fields
  • The Hamiltonian form of the generating functional
  • How to eliminate constrained variables
  • Functional integrals for QED with massive photons
• Problems 16
• Solutions 16
• Electrodynamics with a massive photon
  • Obtaining the Feynman rules for scalar electrodynamics
  • The Feynman rules for massive photon electrodynamics
  • Some low order computations in spinor electrodynamics
  • Quantizing massless electrodynamics with functional integrals
• The Faddeev–Popov prescription
  • The prescription in a finite number of dimensions
  • Extending the prescription to a gauge field theory
  • Applying the prescription to QED
  • Equivalence of the Faddeev-Popov prescription and canonical quantization
  • Revisiting the massive vector theory
  • A first look at renormalization in QED
• Problems 17
• Solutions 17
• Generating functionals and Green’s functions
  • The loop expansion
  • The generating functional for 1PI Green’s functions
  • Connecting statistical mechanics with quantum field theory
  • Quantum electrodynamics in a covariant gauge
• The renormalization of QED
  • Counterterms and gauge invariance
  • Counterterms in QED with a massive photon
  • Gauge-invariant cutoffs
  • The Ward identity and Green’s functions
  • The Ward identity and counterterms
• Problems 18
• Solutions 18
• Two famous results in QED
  • Coulomb’s Law
  • The electron’s anomalous magnetic moment in quantum mechanics
  • The electron’s anomalous magnetic moment in QED
• Confronting experiment with QED
  • Higher order contributions to the electron’s magnetic moment
  • The anomalous magnetic moment of the muon
  • A low-energy theorem
  • Photon-induced corrections to strong interaction processes (via symmetries)
• Problems 19
• Solutions 19
• Introducing SU(3)
  • Decays of the η
  • An informal historical introduction to SU(3)
  • Tensor methods for SU(n)
  • Applying tensor methods in SU(2)
  • Tensor representations of SU(3)
• Irreducible multiplets in SU(3)
  • The irreducible representations q and $\overline{q}$
  • Matrix tricks with SU(3)
  • Isospin and hypercharge decomposition
  • Direct products in SU(3)
  • Symmetry and antisymmetry in the Clebsch–Gordan coefficients
• Problems 20
• Solutions 20
• SU(3): Proofs and applications
  • Irreducibility, inequivalence, and completeness of the IR’s
  • The operators I, Y and Q in SU(3)
  • Electromagnetic form factors of the baryon octet
  • Electromagnetic mass splittings of the baryon octet
• Broken SU(3) and the naive quark model
  • The Gell-Mann-Okubo mass formula derived
  • The GMO formula applied
  • The GMO formula challenged
  • The naive quark model (and how it grew)
  • What can you build out of three quarks?
  • A sketch of quantum chromodynamics
• Problems 21
• Solutions 21
• Weak interactions and their currents
  • The weak interactions circa 1965
  • The conserved vector current hypothesis
  • The Cabibbo angle
  • The Goldberger–Treiman relation
• Current algebra and PCAC
  • The PCAC hypothesis and its interpretation
  • Two isotriplet currents
  • The gradient-coupling model
  • Adler’s Rule for the emission of a soft pion
  • Equal-time current commutators
• Problems 22
• Solutions 22
• Current algebra and pion scattering
  • Pion–hadron scattering without current algebra
  • Pion–hadron scattering and current algebra
  • Pion–pion scattering
  • Some operators and their eigenvalues
• A first look at spontaneous symmetry breaking
  • The man in a ferromagnet
  • Spontaneous symmetry breaking in field theory: Examples
  • Spontaneous symmetry breaking in field theory: The general case
  • Goldstone’s Theorem
• Problems 23
• Solutions 23
• Perturbative spontaneous symmetry breaking
  • One vacuum or many?
  • Perturbative spontaneous symmetry breaking in the general case
  • Calculating the effective potential
  • The physical meaning of the effective potential
• Topics in spontaneous symmetry breaking
  • Three heuristic aspects of the effective potential
  • Fermions and the effective potential
  • Spontaneous symmetry breaking and soft pions: the sigma model
  • The physics of the sigma model
• Problems 24
• Solutions 24
• The Higgs mechanism and non-Abelian gauge fields
  • The Abelian Higgs model
  • Non-Abelian gauge field theories
  • Yang-Mills fields and spontaneous symmetry breaking
• Quantizing non-Abelian gauge fields
  • Quantization of gauge fields by the Faddeev–Popov method
  • Feynman rules for a non-Abelian gauge theory
  • Renormalization of pure gauge field theories
  • The effective potential for a gauge theory
• Problems 25
• Solutions 25
• The Glashow–Salam–Weinberg Model I. A theory of leptons
  • Putting the pieces together
  • The electron-neutrino weak interactions
  • Electromagnetic interactions of the electron and neutrino
  • Adding the other leptons
  • Summary and outlook
• The Glashow–Salam–Weinberg Model II. Adding quarks
  • A simplified quark model
  • Charm and the GIM mechanism
  • Lower bounds on scalar boson masses
• The Renormalization Group
  • The renormalization group for ø⁴ theory
  • The renormalization group equation
  • The solution to the renormalization group equation
  • Applications of the renormalization group equation

The following sections are included:

• Concordance of videos and chapters
• Index