Re: Noether charge & generators of symmetry

ygor.geurts@xxxxxxxxx wrote:

I have the following question. I'm wondering if anyone could explain
to me the mathematics behind the (physics) statement that the noether
charge of an infinitesimal symmetry is the generator of the symmetry?

This statement is relation between dynamic in tangent bundle and
cotangent bundle (phase space). Same Lie group act on tangent bundle
and we want to induce this action on phase space. This is difficult
task in general, but we can take explicitly constructed conserved charge
Q(q,q`,t) and rewrite it in terms o phase space: Q(q,p,t). Now using
Poisson bracket we have actions of our Lie algebra on phase space
via {Q_i,*}. Of course by construction {Q_i , H}=0.
Key question: is it true that the {Q_i,Q_j} obey Lie algebra
relations? In general is not (so called classical anomaly)!
But in concrete cases we can check relations by hands. In many
physical problem {Q_i,Q_j} give Lie algebra of the symmetry group.

.

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