## Quantum fields

Follow the links to find a brief summary (meaning very brief) of my lecture notes from professor Sterman’s lectures on quantum field theory during the fall of 2007 and spring of 2008. The **first part of the course** follows:

- Week 01 –
*Introduction:*historical perspective, review of quantum mechanics of point particles, relativistic point particle, Klein-Gordon equation, Lagrange formulation of classical field theory, local field theory, action principles, Lagrange’s equations of motion, examples, transformations of fields - Week 02 –
*More classical field theory:*form invariance, general infinitesimal transformations - Week 03 –
*Symmetries and conservation:*Translations, Noether’s theorem, phase invariance for complex field, Noether currents and charge;*Group theory:*matrices, determinant and identities, representations, Lie groups and algebras, generators of the algebra and structure constants - Week 04 –
*More group theory:*generators for and ;*Spacetime symmetries:*Lorentz group, Lorentz transformations, generators, Lorentz algebra, extended Lorentz group - Week 05 –
*More spacetime symmetries:*Poincaré group, Poincaré algebra, generators;*Quantization:*Heisenberg equation, scalar field and conjugate momentum, equal time commutator relations, Fourier transform and alternate coordinate, creation and annihilation operators, Lagrange and Hamilton densities, Hamiltonian - Week 06 –
*More quantization:*natural units, vacuum state, creation of quanta, Fock space, number operator, normal ordering, completeness relation;*Quantum symmetries:*unitary transformations, conserved charges as quantum numbers and labels for states - Week 07 –
*Poincaré symmetry:*representations with unitary matrices, invariance of matrix elements, causality;*Particles:*eigenstates for position, wavefunction, probability density, Bose symmetry, induced and spontaneous emission - Week 08 –
*Green functions:*for the Klein-Gordon equation, as matrix elements, advanced and retarded functions, Feynman propagator, causal Green function, in and out states, S-matrix - Week 09 –
*More Green functions:*reduction formulas;*Path integrals*: introduction and definition, interaction picture versus Schrödinger or Heisenberg pictures - Week 10 –
*Path integrals*: free particle case, Wick rotations - Week 11 –
*More path integrals:*harmonic oscillator, Gaussian integrals, time-ordered matrix elements - Week 12 –
*Path integrals for field theories:*sources, functional derivatives, free field, generating functionals, Green functions - Week 13 –
*More path integrals for field theories:*interacting fields, Feynman graphs and rules - Week 14 – Feynman rules in momentum space; cross sections
- Week 15 – More on cross sections, decay rates, complex scalar fields, background vector fields

The **second part of this course** can be found among these:

- Week 16 –
- Week 17 –
- Week 18 –
- Week 19 –
- Week 20 –
- Week 21 –
- Week 22 –
- Week 23 –
- Week 24 –
- Week 25 –
- Week 26 –
- Week 27 –
- Week 28 –
- Week 29 –
- Week 30 –

More **advanced topics** were covered in professor Siegel’s advanced field theory course during the spring of 2008.

- Week 01 – SU(N), double-line color diagrams, review of Weyl spinors
- Week 02 –
- Week 03 –
- Week 04 –
- Week 05 –
- Week 06 –
- Week 07 –
- Week 08 –
- Week 09 –
- Week 10 –
- Week 11 –
- Week 12 –
- Week 13 –
- Week 14 –
- Week 15 –

During the Fall semester of 2008 I attended the lectures on advanced quantum field theory by Professor Peter van Nieuwenhuizen. The class contained very little overlap with Siegel’s advanced course and instead concentrated more on extended objects like instantons and solitons. This course can be outlined as follows:

Week 01 – *BRST symmetry*

- Yang-Mills action, gauge-fixing, ghost action
- BRST transformations
- Nilpotency of BRST transformations and BRST charge
- Auxiliary fields

Week 02 – *Renormalization of gauge theories*

- Path integrals and functionals, sources for ghosts and gauge fields
- Jacobian of BRST transformation
- Classical fields
- Zinn-Justin sources, BRST transformations, correlation functions
- Renormalization Z-factors: sources, fields, gauge-fixing parameter, coupling strength
- Effective action: as functional for 1-particle-irreducible graphs, Legendre transformation of functional for connected graphs
- Ward Identities for proper graphs
- Left and right functional derivatives
- Slavnov operator and its nilpotency
- Invariance of quantum action under Slavnov transformation
- Regulated functionals and renormalized functionals
- Relation between renormalization Z-factors and solution to Slavnov identities; proof of renormalizability to all-loops

Weeks 03 04 05 06 & 07 – *Instantons*

- Yang-Mills action in Euclidean space, definition of zero-modes, theta-terms
- Winding number and topological invariants
- Explicit examples of instantons: SU(2) , SU(N), embedded solutions, multi-instantons
- Collective coordinates: bosonic and fermionic moduli and zero-modes, explicit examples
- (anti) self-dual solutions as minimum of the pure Yang-Mills action
- Quantum operator for zero modes; counting with index theorems
- Path integral measure for zero modes
- Fluctuations around an instanton background
- Exact -function for 1, 2, 4 super Yang-Mills theory
- Tunneling phenomena in Minkowski spacetime as instanton in Euclidean space

Weeks 08 & 09 – *Kinks and domain walls*

- Lagrangian and energy density; solution to classical field equations; mass
- Fluctuations around kink vacuum; renormalization of mass; topological vacuum
- Quantum operator; Modes: bound-state, zero-modes, continuum
- Superkinks: Lagrangian, SUSY transformations, auxiliary fields
- Modified dimensional regularization and additional modes
- Factorization of the quantum operators
- Domain walls: mass renormalization and quantum corrections; counter-terms
- BPS saturation: central charges, supersymmetric currents, anti-commutation relations, interaction picture, regularization, corrections to central charge, breaking of half of supersymmetries

Weeks 10 11 12 & 13 – *Vortices*

- Lagrangian and energy density; winding number; vacua
- BPS conditions and minimization of energy
- Collective coordinates and moduli
- Supervortex: model; superspace formulation
- Quantization of vortex and supervortex, corrections to VEV; gauge-fixing term
- Topological vortices: Bogomol`nyi bound, winding number; vortex and antivortex
- Background splitting of fields; fluctuations around vortex background; quadratic terms
- Renormalization: tadpole cancellation condition
- Index theorem for spectral densities
- Fermionic and bosonic zero-modes: explicit solutions and collective coordinates
- Multiplet shortening: kinks and vortices, supersymmetry algebra, Olive-Witten result

Weeks 13 14 & 15 – *Monopoles*

- Lagrangian and energy density; electric and magnetic charge: dyons
- BPS equations and Bogomol`nyi trick; Weinberg’s transformation of monopole solution to dyon solution
- 2 super Yang-Mills theory in 4 dimensions as dimensional reduction of 1 super Yang-Mills in 6 dimensions and supersymmetric monopoles
- 4 super Yang-Mills theory in 4 dimensions as dimensional reduction of 1 super Yang-Mills in 10 dimensions and supersymmetric monopoles
- Renormalization of composite operators

The spring 2009 term saw the second part of this course:

Weeks 02 and 03 – *Renormalization of Higgs models*

- Spontaneous symmetry breaking
- Goldstone’s theorem and Goldstone bosons
- Tadpole cancellation and path-integral calculations; renormalization procedure
- Higgs model: lagrangian, gauge transformations, ghost lagrangian, gauge fixing term
- Renormalization of Higgs models: -equation and -factors, Slavnov-Taylor identities

Weeks 03 04 05 06 07 08 and 09 – *Anomalies*

- Anomaly cancellation for the Standard Model: triangle graphs and hypercharge, lepton and baryon current
- Chiral anomaly and the chiral basis
- Triangle, box and pentagon graphs
- basis and chiral current
- Dimensional regularization and spinors
- Explicit calculations of triangle graphs: and anomalies
- Dimensional reduction applied to anomaly calculation
- Singlet and non-Abelian anomalies, group theory factors and symbols, safe groups and anomaly cancellation
- Anomalies and grand unified theories
- Calculations in the chiral basis
- Gauge anomalies and non-renormalizability and unitarity
- Consistency conditions; the consistent anomaly versus the covariant anomaly; Bardeen curvatures; Wess-Zumino terms
- Pauli-Villars regularization
- Descent equations and Chern-Simons terms; Chern forms
- Fujikawa method; path-integral measure
- Anomalies in higher dimensions

Weeks 09 and 10 – *Background Field Formalism*

- Definition of the formalism; renormalization simplifications
- Background and quantum fields; gauge transformations; functionals and Ward Identities; sources for fields
- S-matrix and the background field formalism
- Renormalization and background field formalism; Extended BRST method
- One-loop beta-function calculation for pure Yang-Mills theory

Weeks 10 11 12 and 13 – *Unitariy and the cutting rules*

- Feynman propagator
- Largest-time equation
- Unitarity and renormalization
- Sum over states and cancellation of unphysical states

Weeks 13 and 14 – *Dirac formalism and antifields*

- Primary constraints and their descendants
- First class and second class constraints
- Hamiltonian BRST formalism
- Antifield formalism: antifields and antibrackets

Week 15 – **Feynman rules in superspace**

- Wess-Zumino model
- Basics of superspace
- Super Yang-Mills

Roughly one can argue that Sterman’s course serves as QFT I and II, van Nieuwenhuizen’s as QFT III and IV and Siegel’s as QFT V. That means I can brag about having taken five courses on quantum field theory with non-trivial overlap in content.