Colchete de Poisson covariante na teoria geométrica dos campos

Abstract

Este exemp]ai corresponde à redução final da tese devidamerlte corrigida e defendida por Sandro Vieira Romero e aprovada pela comissão julgadora.

Hamiltonian multivector fields and Poisson forms in multisymplectic field theory

“…Briefly, covariant phase space is defined as the space S of solutions of the equations of motion and, formally viewed as an infinite-dimensional manifold, carries a naturally defined symplectic form Ω [12][13][14]. A systematic general investigation of the Peierls -DeWitt bracket in the multisymplectic framework, including a proof of the fact that it is precisely the canonical Poisson bracket for functionals on S derived from the symplectic form Ω on S, has been carried out recently [10,11]. In order to establish the desired relation, we must restrict this bracket to a certain class of functionals, namely functionals F obtained by using fields to pull Hamiltonian forms or Poisson forms f on extended multiphase space back to space-time and then integrate over submanifolds Σ of the corresponding dimension.…”

Covariant Poisson Brackets in Geometric Field Theory

“…The present paper, based on the PhD thesis of the second author [21], is intended to revitalize this tradition by systematizing and further developing the link between the two approaches, thus contributing to integrate them into one common picture. It is organized into two main sections.…”

The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory

“…[4] -except for the direct construction of the Legendre transformation FH associated with a Hamiltonian H, which was first derived in Ref. [19]; see also Ref. [20].…”

“…Finally, we would like to point out that there exists another construction of a covariant Poisson bracket in classical field theory, based on the same functional approach that underlies the construction of "covariant phase space" of Crnkovic-Witten [13,14] and Zuckerman [15]. This bracket, originally due to Peierls [16] and further elaborated by de Witt [17,18], has recently been adapted to the multiphase space approach by Romero [19] and shown to be precisely the Poisson bracket associated with the sym-plectic form on covariant phase space introduced in Refs [13,14] and [15]; these results will be presented elsewhere [20]. It would be interesting to identify the relation between that bracket and the one introduced here; this question is presently under investigation.…”