Mathematical Methods of Theoretical Physics

Course:
Laurea magistrale in Scienze fisiche
Teachers:
Paolo Perinotti
Academic year:
2017/2018 (Altri: 2016/2017 2015/2016 2014/2015 2013/2014)
Semester:
II
Language:
Italiano / english friendly
Code:
500639
SSD:
FIS/02
Credits:
6
Hours:
48

Educational goals

Introduction to the mathematical methods of theoretical Physics with case studies applications to relativity, gauge and quantum field theories.

Prerequisites

Undergraduate Mathematical Methods, Mechanics, Electrodynamics and Quantum Mechanics.

Programme

Differentiable manifolds, vector bundles, principal bundles. Vector fields on a manifold, flow of a vector field, Lie derivative. Linear connection on a vector bundle. Connections as 1-forms with value in the Lie algebra of a Lie group. Parallel transport. The Exponential map. Curvature and holonomy of a connection. Bianchi identities and their geometrical meaning. Curvature tensor. Metrics on a manifold, compatibility with a linear connection. Connections and Yang-Mills fields. Gauge invariance. Yang-Mills equations and their variational derivation. The example of the electromagnetic field. Linear connections on the tangent bundle and Levi-Civita connections. Geodetics and their properties. The Ricci tensor. Isometries and Killing vectors. Geometrical analysis. Differential operators on manifolds and partial differential equations. Banach spaces of sections. L^p and Sobolev Spaces. Elliptic, parabolic, and hyperbolic operators. Examples and applications. The spectral theorem. The spectrum of the Laplace-Beltrami operator on a compact manifold. Applications. Functional determinants and partition functions in quantum field theory and statistical mechanics.

Bibliography

Manfredo do Carmo, Riemannian Geometry, Birkhauser Boston
Y. Choquet- Bruhat, C. DeWitt-Morette, Analysis, Manifolds and Physics, North-Holland
J. Jost, Riemannian Geometry and Geometric analysis, Springer

Exam

Oral examination.