A Mathematical Primer on Relativistic Classical Field Theory

- Lecturer: Pedro Lauridsen Ribeiro, Universidade Federal do ABC - Brasil - Venue: Polo Ferrari 1 – Via Sommarive 5 – Povo - Room A102 - Period:  15-22-29 giugno 2018 - 14:30-16:30 p.m.  6-13-20 luglio 2018 - 14:30-16:30 p.m. - Abstract: Since their inception through the experimental findings of Michael Faraday on electrodynamics, relativistic fields have been a fundamental concept in our description of Nature. Roughly, (classical) fields describe any quantity that is prescribed locally through space-time – mathematically, they correspond to sections of some fiber bundle over the space-time manifold whose typical fiber encodes the (pointwise) internal degrees of freedom of the field. Relativistic fields are distinguished for having a time evolution whose local perturbations propagate with finite speed, meaning that the field dynamics is described locally by a system of hyperbolic partial differential equations. As such, relativistic field theory provides an extremely rich ground where several different mathematical disciplines meet, such as geometry, partial differential equations, analysis, topology and algebra. The goal of this minicourse is to browse through the main elements of relativistic classical field theory in a concise but mathematically and conceptually sound fashion, without pretense to completeness. Emphasis will be put on modern methods which put the theory in a form suitable to quantization at a later stage (which shall not be discussed here).

- Approximate Programme: Adjustments will be made as the minicourse progresses. Lecture 1: What is a field? Kinematical underpinnings – fields as sections of bundles over space-time, manifold structure of the space of field configurations Lecture 2: Local point symmetries of fields – G-structures and local gauge transformations. G-equivariant connections as (gauge) fields Lecture 3: Field dynamics and the variational principle – jet bundles, action functionals and the Euler-Lagrange operator. Symmetries of the action and Noether’s theorems, the stress-energy tensor and currents Lecture 4: What is a relativistic field? Local relativistic field dynamics = hyperbolic PDE’s – globally hyperbolic space-times, a priori energy estimates and basic well-posedness theory for the Cauchy problem in the linear case, retarded and advanced Green operators, remarks about non-linear dynamics Lecture 5: Singularities of Green operators. A brief introduction to microlocal analysis – wave front sets and its basic calculus, propagation of singularities Lecture 6: Field observables as function(al)s on the space of field configurations – smooth, additive, microlocal and microcausal functionals, the Peierls bracket, the Poisson algebra of microcausal functionals

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