The polysymplectic formalism in local field theory, developed by Günther [J. Diff. Geom. 25, 23 (1987)], is revised. A new approach and new results on momentum maps and reduction are given.

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The aim of this paper is to generalize the classical Marsden-Weinstein reduction procedure for symplectic manifolds to polysymplectic manifolds in order to obtain quotient manifolds which inherit the polysymplectic structure. This generalization allows us to reduce polysymplectic Hamiltonian systems with symmetries, such as those appearing in certain kinds of classical field theories. As an application of this technique, an analogous to the Kirillov-Kostant-Souriau theorem for polysymplectic manifolds is obtained and some other mathematical examples are also analyzed.Our procedure corrects some mistakes and inaccuracies in previous papers [29,50] on this subject.