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I'm currently looking for the mathematical foundation behind the

claim, often found in field theory/string theory books that the

noether charge associated to a symmetry of the lagrangian is the

generator of that symmetry, ie. its poisson bracket with a field from

the lagrangian, generates the change in the field.

Recently I came across the article 'Covariant Momentum maps & field

theory part I/II' by Gotay and Marsden. In there I've found that the

concept of a Noether current is in fact given by the covariant

momentum map in the multisymplectic classical field theory

formulation. In particular, this map satisfies the mathematical analog

of noether's theorem as explained in the article.

In part II of the article they define an instantaneous momentum map,

loosely defined by the integral of the covariant momentum map over a

cauchy hypersurface. My question is now, whether this is the

mathematical object that corresponds to the Noether charge (which in

physics is defined as the integral of the time component of the

current over a spacelike hypersurface)? Moreover, if that is so,

somehow I still can't find an explicit demonstration (in the above

article or any other mathematical article) that this charge then

generates the transformation, ie is somehow related to the generators

of the Lie algebra of the Lie group that encodes the symmetry.

Could anyone help me out on clarifying these matters?

Thanks in advance,

Ygor

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# Noether charge in multisymplectic geometry

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