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Física Matemática

Infinite-Dimensional (Quasi-)Poisson Structures from Quasi-Linear Hyperbolic PDE Systems in Fiber Bundles: Challenges Off the Shell
Pedro Lauridsen Ribeiro

Terça-feira, 27 de junho de 2017, 17:00
Sala 236.

Quasi-linear hyperbolic systems of partial differential operators acting on smooth sections of a fiber bundle constitute the basic equations of motion of relativistic field theories whenever constraints and/or local gauge symmetries are absent or have been resolved and/or gauge-fixed. Usually such equations come from a variational principle - this essentially amounts to the linearized system being formally self-adjoint, as shown by Helmholtz long ago. In this case, it is possible to formally write an antisymmetric bilinear form in the space of (compactly supported, smooth) linearized initial data at hypersurface. If the linearized system is hyperbolic in the sense of having retarded and advanced fundamental solutions, one may extend this form to a quasi-Poisson bracket - discovered by Peierls in the 50's and thus called the Peierls bracket - on suitable algebras of functionals on smooth sections which may be thought of as observables, without the need for imposing equations of motion. In physics' parlance, we say that the Peierls bracket is defined "off shell". In this talk, we will focus on which kind of quasi-Poisson structure emerges from such a construction: when the Jacobi identity holds, in which sense such structures can be linearized in the sense of Darboux and Weinstein, and so on. Possible connections to twisted Dirac structures shall also be touched upon, if time allows. (joint work with Romeo Brunetti and Klaus Fredenhagen)