Noether's theorem: Difference between revisions

Revision as of 11:31, 27 October 2003

Revision as of 11:59, 27 October 2003

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* In [[field theory]], M is the [[spacetime]] [[manifold]] and the target space is the set of values the fields can take at any given point. For example, if there are m [[real]]-valued [[scalar]] fields, &phi;₁,...,&phi;_m, then the target [[manifold]] is <math>\mathbb{R}^m</math>. If the field is a [[real]] vector field, then the target [[manifold]] is [[isomorphic]] to <math>\mathbb{R}^n</math>. There's actually a much more elegant way using [[tangent bundle]]s over M, but for the purposes of this proof, we'd just stick to this version.

Now suppose there's a [[Functional analysis|functional]], <math>S:\mathcal{C}\rightarrow \mathbb{R}</math>, called the [[Action (physics)|action]]. Note it's a [[mapping]] to <math>\mathbb{R}</math>, not <math>\mathbb{C}</math>. This has got to do with physical reasons and does not really matter for this proof.