How to give a natural geometric definition of a covariant Poisson bracket inclassical field theory has for a long time been an open problem - as testifiedby the extensive literature on "multisymplectic Poisson brackets", togetherwith the fact that all these proposals suffer from serious defects. On theother hand, the functional approach does provide a good candidate which hascome to be known as the Peierls - De Witt bracket and whose construction in ageometrical setting is now well understood. Here, we show how the basic"multisymplectic Poisson bracket" already proposed in the 1970s can be derivedfrom the Peierls - De Witt bracket, applied to a special class of functionals.This relation allows to trace back most (if not all) of the problemsencountered in the past to ambiguities (the relation between differential formson multiphase space and the functionals they define is not one-to-one) and alsoto the fact that this class of functionals does not form a Poisson subalgebra.
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