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We establish a link between the multisymplectic and the covariant phase spaceapproach to geometric field theory by showing how to derive the symplectic formon the latter, as introduced by Crnkovic-Witten and Zuckerman, from themultisymplectic form. The main result is that the Poisson bracket associatedwith this symplectic structure, according to the standard rules, is preciselythe covariant bracket due to Peierls and DeWitt.Comment: 42 page

Covariant Poisson Brackets in Geometric Field Theory