Quantum Field Theory Course

Total: 440+ pages covering classical field theory through the Standard Model

Quantum Field Theory

Part VII: Advanced Topics

Beyond perturbative QFT: Exploring non-perturbative phenomena, quantum anomalies, topological structures, thermal field theory, curved spacetime effects, and the foundations of supersymmetry.

Overview

Part VII ventures into the frontier of quantum field theory, exploring phenomena that cannot be understood through perturbation theory alone. These advanced topics reveal the deep mathematical structure underlying QFT and connect to modern research in particle physics, cosmology, and quantum gravity.

We begin with anomaliesβ€”quantum violations of classical symmetries that have profound physical consequences. Then we explore non-perturbative phenomena like instantons and solitons, which arise from the topological structure of field configurations. Moving to finite temperature QFT, we see how thermal effects modify quantum fields. We then study QFT in curved spacetime, discovering phenomena like Hawking radiation. Finally, we introduce supersymmetry, a profound symmetry connecting bosons and fermions.

Why These Topics Matter

  • Anomalies constrain possible gauge theories and explain phenomena like π⁰ β†’ Ξ³Ξ³ decay
  • Instantons mediate quantum tunneling and affect vacuum structure in non-Abelian gauge theories
  • Solitons provide stable, particle-like solutions in field theory (monopoles, vortices, domain walls)
  • Thermal QFT describes the early universe, quark-gluon plasma, and phase transitions
  • QFT in curved spacetime connects to cosmology and black hole physics
  • Supersymmetry is a cornerstone of beyond-Standard-Model physics

Course Structure

1. Anomalies

⚑

Quantum violations of classical symmetries: chiral anomaly, axial anomaly, ABJ anomaly, triangle diagrams, and their physical consequences. Understanding why certain symmetries cannot survive quantization.

Chiral anomaly β€’ Triangle diagrams β€’ π⁰ decay β€’ Anomaly cancellation β€’ 9 pages

2. Instantons & Tunneling

πŸŒ€

Non-perturbative tunneling configurations in Euclidean spacetime. BPST instantons in Yang-Mills theory, theta vacua, and the role of topology in quantum tunneling processes.

Euclidean path integrals β€’ BPST instanton β€’ Theta vacua β€’ Tunneling amplitude β€’ 8 pages

3. Solitons & Topological Objects

🎯

Stable, localized field configurations: kinks, vortices, magnetic monopoles, domain walls. Understanding how topology protects these objects from decay and their role in gauge theories.

Kinks β€’ 't Hooft-Polyakov monopole β€’ Vortices β€’ Topological charge β€’ 8 pages

4. Finite Temperature QFT

πŸ”₯

Thermal quantum field theory: imaginary time formalism, Matsubara frequencies, thermal propagators, and phase transitions. Applications to the early universe and quark-gluon plasma.

Matsubara formalism β€’ Thermal propagators β€’ Phase transitions β€’ Early universe β€’ 10 pages

5. QFT in Curved Spacetime

🌌

Quantum fields in gravitational backgrounds: particle creation in expanding universes, Unruh effect, Hawking radiation from black holes, and the connection between thermodynamics and horizons.

Unruh effect β€’ Hawking radiation β€’ Particle creation β€’ Horizon thermodynamics β€’ 10 pages

6. Introduction to SUSY

✨

Supersymmetry: the symmetry between bosons and fermions. SUSY algebra, superfields, Wess-Zumino model, and why SUSY is essential for many beyond-Standard-Model scenarios.

SUSY algebra β€’ Superspace β€’ Superfields β€’ Wess-Zumino model β€’ 5 pages

Prerequisites

Part VII assumes mastery of earlier QFT material. Before starting, you should be thoroughly comfortable with:

  • Part I-III - Classical fields, canonical quantization, and path integrals
  • Part IV - Gauge theories (QED, Yang-Mills, spontaneous symmetry breaking)
  • Part V - Non-Abelian gauge theories and the Standard Model
  • Part VI - Renormalization and running couplings
  • Differential Geometry - Manifolds, fiber bundles, and topology

Key Concepts You'll Master

Theoretical Frameworks:

  • Quantum anomalies and consistency conditions
  • Topological quantum numbers
  • Non-perturbative methods
  • Euclidean field theory
  • Thermal field theory
  • QFT on curved backgrounds
  • Supersymmetric field theories

Physical Phenomena:

  • Chiral anomaly in the Standard Model
  • Instanton effects in QCD
  • Magnetic monopoles
  • Cosmological phase transitions
  • Hawking radiation from black holes
  • Unruh temperature
  • SUSY partner particles

Mathematical Toolkit

Advanced topics require sophisticated mathematical tools:

Topology & Geometry

  • β€’ Homotopy groups Ο€n
  • β€’ Fiber bundles and connections
  • β€’ Chern classes and index theorems
  • β€’ Pontryagin index

Advanced Analysis

  • β€’ Functional determinants
  • β€’ Zeta function regularization
  • β€’ Matsubara frequency sums
  • β€’ Grassmann calculus

Connection to Modern Research

The topics in Part VII connect directly to active research areas:

Anomalies: Essential for Standard Model consistency, neutrino physics, and searches for new physics beyond the SM.
Instantons: Key to understanding QCD vacuum, CP violation, and the strong CP problem (related to axion dark matter searches).
Solitons: Models for cosmic strings, domain walls in cosmology, and confined magnetic flux in superconductors.
Thermal QFT: Describes quark-gluon plasma at RHIC/LHC, electroweak phase transition, and the early universe.
QFT in Curved Spacetime: Foundation for quantum cosmology, black hole information paradox, and approaches to quantum gravity.
Supersymmetry: Central to string theory, SUSY breaking mechanisms, and searches for superpartners at particle colliders.
← Part VI: RenormalizationStart Part VII: Anomalies β†’