Content deleted Content added
grammar |
Added a proof of one version of Noether's theorem |
||
Line 3:
The formal statement of the theorem derives an expression for the physical quantity that is conserved (and hence also defines it), from the condition of invariance alone. For example, the invariance of physical systems with respect to translation (when simply stated, it is just that the laws of physics don't vary with ___location in space) translates into the law of conservation of [[linear momentum]]. Invariance with respect to rotation gives law of conservation of [[angular momentum]], invariance with respect to time gives the well known [[law of conservation of energy]], et cetera. When it comes to [[quantum field theory]], the invariance with respect to general [[gauge transformation]]s gives the law of conservation of [[electric charge]] and so on. Thus, the result is a very important contribution to physics in general, as it helps to provide powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved, invariant.
== Proof ==
Suppose we have an n-dimensional [[manifold]], M and a target [[manifold]] T. Let <math>\mathcal{C}</math> be the configuration space of [[Smooth (mathematics)|smooth]] [[Function (mathematics)|maps]] from M to T.
Before we go on, let's give some examples:
* In [[classical mechanics]], M is the one dimensional [[manifold]] <math>\mathbb{R}</math>, representing time and the target space is the [[tangent bundle]] of [[space]] of generalized positions.
* 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, φ<sub>1</sub>,...,φ<sub>m</sub>, 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.
To get to the usual version of Noether's theorom, we need additional restrictions on the [[Action (physics)|action]]. If <math>\phi\in\mathcal{C}</math>, we assume S(φ) is the [[integral]] over M of a function of φ, its [[derivative]] and the position called the [[Lagrangian]], <math>\mathcal{L}(\phi,\partial_\mu\phi,x)</math>. In other words,
<math>\forall\phi\in\mathcal{C}\, S[\phi]\equiv\int_M d^nx \mathcal{L}(\phi(x),\partial_\mu\phi(x),x)</math>.
Given [[boundary]] conditions, which is basically a specification of the value of φ at the [[boundary]] of M is [[compact]] or some limit on φ as x approaches <math>\infty</math> (this will help in doing [[integration by parts]]), we can denote the [[subset]] of <math>\mathcal{C}</math> consisting of functions, φ such that all [[functional derivative]]s of S at φ are zero and φ satisfies the given boundary conditions.
Now, suppose we have an [[infinitesimal]] [[Transformation (mathematics)|transformation]] on <math>\mathcal{C}</math>, given by a [[functional derivative]], δ 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]] submanifolds N. Then, we say δ is a generator of a 1-parameter [[symmetry]] [[Lie group]].
Now, for any N, because of the [[Euler-Lagrange]] theorem, we have
<math>
\delta\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)})\delta\phi+
\int_{\partial N}ds_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}=
\int_{\partial_N}ds_\mu\frac{\partial\mathcal{L}}{\partial(\partial_\mu\phi)}
</math>.
Since this is true for any N, we have
<math>
\partial_\mu(\frac{\partial\mathcal{L}}{\partial(\partial_\mu\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)}-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 [[conserved]] quantity called the Noether charge (provided, of course, if M is [[compact|noncompact]], the currents fall off sufficiently fast at infinity).
== External links ==
| |||