Gauge Theories & Standard Model

Contact: A. Bharucha & S. Lazzarini

Prerequisite: QFT

1st part: Gauge Theories


  • Notion of symmetry: within the framework of geometry or differential algebra.
  • There is a strong link between geometry and physics.
  • Symmetries confer a structure to spaces, give conserved quantities (Noether)
  • In Quantum Mechanics: gauge transformation of the 4-potential vs unitary transformation
  • Gauge principle in field theory: covariant derivative, coupling matter to Maxwell gauge potential (→ Yang-Mills theories.)
  • Analytical Mechanics and symplectic geometry (geometric quantization)
  • General Relativity : Riemann geometry, diffeomorphisms

Covered issues:
- Why the notion of fiber bundles appears in Physics? The very important role of Lie groups
- Which geometric interpretation to give to:

  •  the Yang-Mills potential? / to the Christoffel symbols?  Notion of connection.
  •  the Yang-Mills field? / to the Riemann tensor? Notion of  curvature.

- How to view the notion of connection? get an image of a gauge choice  (local description)
- What about representations of Lie groups? Associated fiber bundles.
- What geometrical meaning for the matter fields?  Notion of smooth sections.
- What is a gauge group? (gauge transformations? gauge fixing?)
- How to formulate a gauge theory in terms of geometrical concepts?
- How to translate an infinitesimal symmetry in algebraic terms? Notion of differential algebra
- Differential geometry & differential algebra: an elegant unifying language for all theories? (Yang-Mills, gravitation, ...)


2nd part: The Standard Model of Particle Physics: Emphasis on symmetries and applications to the standard model

Successes of the standard model

  • Consistent (unitary and renormalizable) theory
  • Weak neutral currents and gauge bosons
  • High energy chromodynamics
  • P and CP violation
  • Cancellation of chiral anomalies
  • Higgs boson

Challenges to the Standard model

  • Derivation of confinement and hadronic physics
  • Hierarchy of masses (electron mass = 0.5 MeV; top mass = 173 GeV)
  • Status of massive neutrinos
  • Number of free parameters ≥ 18
  • Fine tuning of the Higgs mass


• One loop β function, asymptotic freedom
• Construction of the standard model (including massive neutrinos)
• Physical consequences (CP violation, low and high energy behavior in QCD, ....)
• Physics beyond the standard model