Relativistic Quantum Field Theory I

What this course covers

• Quantum Field Theory as a Discipline
• QFT as Quantum Mechanics with Infinite Degrees of Freedom
• Building Intuition Through Examples and Practice
• Connecting Formalism to Concrete Physical Problems
• Principle of Locality
• Fields as Mathematical Realization of Locality
• Local Equations and Maxwell's Equations
• Minkowski metric convention
• Fields as infinite degrees of freedom
• Lagrangian density and local action
• Canonical momentum density and Hamiltonian density
• Free scalar field theory
• Definition of symmetry in field theory
• Spacetime symmetries: translations and Lorentz invariance
• Continuous vs. discrete symmetries
• Infinitesimal form of continuous symmetries
• Noether's theorem: symmetry implies conservation law
• Global vs. local (gauge) symmetries
• Noether's Theorem and Conserved Currents
• Klein-Gordon Equation as Relativistic Wave Equation
• Klein-Gordon Equation as Classical Field Equation
• Difficulties of Relativistic Quantum Mechanics
• Quantum Field Theory as the Resolution
• Field Theory from Discrete Systems: Continuum Limit
• Quantization of the Harmonic Oscillator
• Klein-Gordon field quantization review
• Canonical commutation relations
• Commutation relations for mode operators
• QFT as infinite harmonic oscillators
• Hamiltonian in terms of mode operators
• Zero-point energy divergence
• Time independence of the Hamiltonian
• Conserved momentum from spatial translation
• Conserved currents and charges in classical field theory
• Conserved charges as quantum operators
• Conserved charge as a symmetry generator
• Operator ordering ambiguity in quantum field theory
• Proof of the generator relation for the restricted case
• Time translation and the Hamiltonian as a generator
• Finite symmetry transformations via exponentiation of charges
• Propagator in non-relativistic quantum mechanics
• No position operator in QFT
• Lorentz covariance obstruction to localized states
• Approximate position states via the field operator
• Wave function analog in QFT
• Two-point correlation functions G⁺ and G⁻
• Retarded and advanced Green's functions
• Feynman propagator G_F
• Commutativity of fields at spacelike separation
• Propagator and its two interpretations
• Feynman propagator as time-ordered correlator
• Poles of the propagator in momentum space
• Contour prescriptions for retarded, advanced, and Feynman propagators
• Iε prescription as an equivalent contour shift
• General N-point vacuum correlation functions
• Composite operators and ultraviolet divergences
• S-matrix and scattering amplitude
• LSZ theorem and correlation functions
• Perturbation theory in the coupling constant
• Naive perturbative expansion of correlation functions
• Interaction picture
• Path integral formulation of quantum mechanics
• Path integral formulation of quantum mechanics
• Mathematical definition of the path integral
• Free particle as a Gaussian path integral
• Generalization to multiple degrees of freedom
• Path integral in quantum field theory
• Field theory path integral measure D[φ]
• Free scalar field theory as Gaussian path integral
• Time-ordered correlation functions in the path integral
• Path integral formula for correlation functions
• Motivation for the generating functional
• Generating function in one dimension
• Recovering moments by differentiation
• Generating functional Z[J] for path integrals
• Functional derivative and its delta-function rule
• n-point functions from Z[J] via functional derivatives
• Z[J]/Z[0] as a time-ordered exponential
• Power-series expansion of the time-ordered exponential
• Harmonic oscillator as a worked example
• Path integral master formula
• Splitting action into free and interacting parts
• Interacting correlators from free-theory correlators
• Perturbative expansion in the interaction
• Applying Wick's theorem to evaluate terms
• Feynman diagrams as diagrammatic bookkeeping
• Combinatorial (symmetry) factors
• Disconnected and vacuum bubble diagrams
• Feynman rules for λφ⁴ theory
• Perturbative expansion of the interacting n-point function
• Wick's theorem and Feynman diagrams as computational tools
• Structure of diagrams for the numerator (star)
• Vacuum diagrams and the denominator (star star)
• Cancellation of vacuum bubbles in the ratio
• Connected vs. disconnected Feynman diagrams
• Momentum-space correlation functions and on-shell limit
• Generalization to multiple interaction terms and multiple fields
• LSZ reduction formula
• Sign convention for initial vs. final state momenta
• Truncation of external propagators
• Field-strength renormalization constant Z
• Motivation for the Dirac equation
• Dirac's minimal requirement: plane-wave solutions
• Clifford algebra conditions on Dirac matrices
• Four-dimensional matrix representations of Dirac matrices
• Dirac Equation and Spinor Space
• Gamma Matrix Algebra (Clifford Algebra)
• Hermiticity Properties of Gamma Matrices
• Lorentz Covariance of Field Equations
• Lorentz Transformation of the Vector Field
• Spinor Field Transformation Law and the Matrix S(Λ)
• Covariance Condition: S γ^μ S⁻¹ = Λ^μ_ν γ^ν
• Infinitesimal Lorentz Transformations and the Spinor Generator σ^{μν}
• Dirac equation plane wave solutions
• Rest-frame method for U and V
• Lorentz-boost method for general momentum
• Algebraic shortcut using (K̸)² = K²
• Normalization and orthogonality of spinors
• Explicit spinor components in Dirac representation
• Non-relativistic limit of Dirac spinors
• Ultra-relativistic limit and linear dependence of U and V
• Spinor projection operators
• Failure of canonical commutator quantization for Dirac field
• Canonical anti-commutation relations for Dirac field
• Self-consistency of the anti-commutation relation
• Mode expansion of the Dirac field
• Anti-commutation relations for mode operators
• Positive-definite Hamiltonian from anti-commutation
• Conserved charge and particle/anti-particle interpretation
• Fermionic harmonic oscillator
• Pauli exclusion principle from anti-commutation
• Spin-statistics theorem
• Dirac spinor Lorentz transformation
• Reducing spinor components under Lorentz symmetry
• Chiral (Weyl) representation of gamma matrices
• Left- and right-handed chiral spinors
• Mass as the reason for four-component spinors
• Gamma-5 matrix and its properties
• Chiral projection operators P_L and P_R
• Lorentz invariance of chiral subspaces via γ^5
• Discrete symmetries: P, T, and C
• Quantum implementation via unitary operators
• Time reversal requires an anti-unitary operator
• Anti-linear operators and their properties
• Anti-unitary operators
• Unitary implementation of P, T, C in scalar QFT
• Intrinsic parity phases: scalars vs. pseudo-scalars
• Action of parity on creation/annihilation operators
• Grassmann variables and anti-commutativity
• Taylor expansion of Grassmann functions
• Differentiation of Grassmann variables
• Integration rules for Grassmann variables
• Gaussian integrals over real Grassmann variables
• Complex Grassmann variables
• Gaussian integrals over complex Grassmann variables
• Generating functional with linear source terms
• Two-point function from Grassmann generating functional
• Dirac field as Grassmann-valued field
• Path integral for the Dirac field
• Classical Maxwell Lagrangian
• Maxwell Equations from Euler-Lagrange
• Current Conservation from Antisymmetry
• Gauge Symmetry of the Maxwell Action
• Lorenz gauge action and equation of motion
• Polarization vectors for the gauge field
• Canonical quantization of the gauge field
• Negative-norm states from timelike polarization
• Too many degrees of freedom in naive quantization
• Path integral formulation and the propagator kernel
• Non-invertibility of the Maxwell kernel due to gauge symmetry
• Gauge fixing in the path integral via delta functional
• Physical Hilbert space: eliminating unphysical states
• QED Lagrangian construction
• U(1) global symmetry and Noether current
• Coupling fermions to photons via J_mu
• Electric charge as coupling constant
• Covariant derivative D_mu
• Local (gauge) symmetry of QED
• Gauge principle as a dynamical principle
• Gauge symmetry in all fundamental interactions
• Relativistic cross section: setup and requirements
• Cross section defined via transition probability and incident flux
• S-matrix element and scattering amplitude M_{αβ}
• Squaring the delta function: spacetime volume V·T
• Single-particle state normalisation with plane waves
• Lorentz-invariant phase space (LIPS) measure dμ
• Final relativistic cross section formula σ_{αβ} = |M|²/4Σ · dμ
• Mandelstam variable s and the kinematic factor Σ
• Centre-of-mass frame and momentum p_{CM}
• Differential cross section for 2→2 scattering in the CM frame
• Particle decay rate Γ and lifetime
• Spin sums: averaging over initial and summing over final polarisations
• E⁺E⁻ annihilation as a particle discovery tool
• Discovery of the J/ψ and charm quark evidence
• R-ratio and evidence for quark color
• Feynman amplitude for e⁺e⁻ → B B̄
• Spin averaging and summing for unpolarized cross section
• Squaring the amplitude and taking the complex conjugate
• Trace trick for evaluating spin sums
• Spin-sum completeness relations for Dirac spinors
• Cross section divergence at threshold velocity
• Crossing symmetry between related processes
• Momentum relabeling between crossed processes
• Wave functions for particles vs. antiparticles (u vs. v spinors)
• Spin-sum identity restoring crossing for unpolarized amplitudes
• Field-theoretic origin of crossing symmetry
• Tree-level diagrams in QED
• Loop diagrams and free momentum integration
• UV divergences from loop integrals
• Divergences as signals of unknown short-distance physics
• Vacuum fluctuations from loop diagrams
• Bare vs. physical parameters
• Electron self-energy and mass renormalization
• Field renormalization
• Charge renormalization and gauge symmetry preservation
• Anomalous magnetic moment of the electron

Syllabus

1. Lectures

   1. Lecture 1: Classical Field Theories and Principle of Locality
   2. Lecture 2: Symmetries and Conservation Laws
   3. Lecture 3: Why Quantum Field Theory
   4. Lecture 4: Canonical Quantization of a Free Scalar Field Theory
   5. Lecture 5: Complex Scalar Field Theory and Anti-Particle
   6. Lecture 6: Propagators and Green Functions
   7. Lecture 7: Interacting Theories and S-Matrix
   8. Lecture 8: Path Integral Formalism for Non-Relativistic Quantum Mechanics
   9. Lecture 9: Path Integral Formalism for QFT; Computation of Time-Ordered Correlation Functions
   10. Lecture 10: Time-Ordered Correlation Functions in Field Theory
   11. Lecture 11: Computation of Correlation Functions in Perturbation Theory and Feynman Diagrams
   12. Lecture 12: More on Perturbation Theory and Feynman Diagrams
   13. Lecture 13: Introducing the Dirac Equation
   14. Lecture 14: Lorentz Covariance of the Dirac Equation
   15. Lecture 15: Classical Solutions of Dirac Equations
   16. Lecture 16: Quantization of the Dirac Theory
   17. Lecture 17: Chiral and Majorana Spinors
   18. Lecture 18: Discrete Symmetries
   19. Lecture 19: Path Integrals of Fermions
   20. Lecture 20: Maxwell Theory and its Canonical Quantization
   21. Lecture 21: Quantum Maxwell Theory (continued)
   22. Lecture 22: Quantum Electrodynamics
   23. Lecture 23: Cross Section and Decay Rate
   24. Lecture 24: Elementary Processes in QED (I)
   25. Lecture 25: Elementary Processes in QED (II)
   26. Lecture 26: Quantum Fluctuations and Renormalization