Noether's theorem: Difference between revisions

Revision as of 15:27, 31 May 2004

Revision as of 08:39, 13 June 2004

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* In [[Field_theory_(physics)|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 number|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 is a [[functional]]

:<math>S:\mathcal{C}\rightarrow \mathbb{R}</math>,

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J^\mu\equiv\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}-f^\mu

</math>

which is called the [[Noether current]] associated with the [[symmetry]]. The continuity equation tells us if we [[integrate]] this current over a spacelike slice, we get a [[conservation law|conserved]] quantity called the [[Noether charge]] (provided, of course, if M is [[compact|noncompact]], the currents fall off sufficiently fast at infinity).

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