Noether's theorem: Difference between revisions

Revision as of 19:11, 6 March 2004

Revision as of 04:42, 7 March 2004

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Suppose given [[boundary condition]]s, which are basically a specification of the value of &phi; at the [[boundary]] of M is [[compact]], or some limit on &phi; as x approaches <math>\infty</math>; this will help in doing [[integration by parts]]). We can denote by N the [[subset]] of <math>\mathcal{C}</math> consisting of functions, &phi; such that all [[functional derivative]]s of S at &phi; are zero and &phi; satisfies the given boundary conditions.

Now, suppose we have an [[infinitesimal]] [[TransformationLinear (mathematics)transformation|transformation]] on <math>\mathcal{C}</math>, given by a [[functional derivative]], &delta; such that

:<math>\delta\int_N d^nx\mathcal{L}=\int_{\partial N}ds_\mu f^\mu(\phi(x),\partial\phi,\partial\partial\phi,...)</math>

for all [[compact|Compact space]] submanifolds N. Then, we say &delta; is a generator of a 1-parameter [[symmetry]] [[Lie group]].

Now, for any N, because of the [[Euler-Lagrange]] theorem, we have

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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|Integral]] this current over a spacelike slice, we get a [[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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