Poisson brackets and symmetries on the space of solutions via coisotropic embeddings: Electrodynamics
Ponente: Luca Schiavone (UC3M & Università di Napoli Federico II)Fecha: martes 25 de enero de 2022 - 13:00Lugar: Room 2.3.B03, UC3M
The search for a solid classical analogue of the unequal time commutation relations of Quantum Field Theory has been a task that repeatedly received attention within theoretical and mathematical physical community in the last decades starting from the seminal paper of R. E. Peierls of 1952.
From the mathematical point of view, following the ideas of Souriau, the problem can be formulated as the search for a Poisson structure on the space of solutions of a classical field theory.
That the space of solutions of some non-singular first order field theories can be equipped with a symplectic (and, thus, a Poisson) structure is well known. On the other hand, it happens that within gauge theories such a structure turns out to be only pre-symplectic, in the sense that it presents a non-trivial kernel.
In this talk we show how to induce, in some circumstances, a Poisson bracket on the space of solutions even in this pre-symplectic case, i.e, we show how to construct a Poisson bracket on the space of solutions of a class of gauge theories, by using a construction related with the so called coisotropic embedding theorem.
Within this approach, we also construct a consistent theory of symmetries that deals with 1st and 2nd Noether’s theorem on an equal footing.
Free Classical Electrodynamics will be our guiding example throughout all our constructions.