Caminos de investigaciónApplied MathematicsLección disponible
Gauge theory and the geometry of physics
Optional research direction: forces as curvature. Principal bundles and connections, curvature as field strength, Maxwell's equations and electromagnetism, Yang–Mills theory and instantons, the Aharonov–Bohm effect, and how gauge theory reshaped four-dimensional topology — with the Yang–Mills mass gap as an open problem.
math.applied.gauge-theory
Se basa en
Vector bundles and characteristic classesLie groups and Lie algebrasMathematical foundations of quantum mechanics
Requisitos previos obligatorios, hasta el fondo: 56
Aprenderás a
- Describe connections and curvature on principal bundles, and write Maxwell's equations as dF = 0, d⋆F = J.
- Formulate the Yang–Mills equations, define instantons and explain the topological charge as a characteristic number.
- Explain the Aharonov–Bohm effect as holonomy of a flat connection, and state the Yang–Mills existence and mass gap problem as open.
Definición
- A principal G-bundle is a fibre bundle P → M with a free right G-action that preserves the fibres and acts simply transitively on each of them, and is locally trivial: locally P ≅ U × G, compatibly with the action.
- A connection on a principal bundle is a G-equivariant choice of horizontal subspaces; its curvature F is a Lie-algebra-valued 2-form.
- An instanton is a solution of the anti-self-duality equation F = −⋆F on a Riemannian 4-manifold.
Teorema
- Electromagnetism is a U(1) gauge theory: the electromagnetic field is the curvature of a connection on a U(1)-bundle.
- Instantons minimise the Yang–Mills energy in their topological class, with energy proportional to |c₂|.
- Donaldson (1983): a definite intersection form of a smooth closed simply connected 4-manifold is diagonalisable over ℤ.
Errores frecuentes
- Thinking a gauge transformation changes the physics; physical quantities are gauge-invariant by definition.
- Believing a flat connection has no observable effects; its holonomy around non-contractible loops can be measured (Aharonov–Bohm).
Descubrimientos opcionales
- Low-dimensional topology: 3- and 4-manifolds Donaldson's theorem on definite intersection forms, proved with instantons, showed that many topological 4-manifolds admit no smooth structure.
- Knot theory Chern–Simons gauge theory gives a physical construction of the Jones polynomial (Witten).
- ← Vector bundles and characteristic classes
- ← The geometry of spacetime: special and general relativity
- ← Low-dimensional topology: 3- and 4-manifolds
Itinerarios de aprendizaje
Lecturas complementarias
- John Baez and Javier P. Muniain, Gauge Fields, Knots and Gravity (1994)
- Mikio Nakahara, Geometry, Topology and Physics (1990)