Noether's theorem: Difference between revisions

Revision as of 01:18, 30 July 2004

Revision as of 19:22, 30 July 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 transformation]] on <math>\mathcal{C}</math>, given by a [[functional]] derivative[[derivation]], &delta;Q such that

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

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

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

:<math>

\deltaQ[\int_N d^nx\mathcal{L}]=

\int_Nd^nx(\frac{\partial\mathcal{L}}{\partial\phi}-

\partial_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)})\deltaQ[\phi]+

\int_{\partial N}ds_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}Q[\phi]=

\int_{\partial_Npartial N}ds_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}Q[\phi].

</math>

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:<math>

\partial_\mu(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}Q[\phi]-f^\mu)=0.

</math>

You might immediately recognize this as the [[continuity equation]] for the current

:<math>

J^\mu\equiv\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}Q[\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 [[space-like]] 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).