Editing Noether's theorem (section)

===Manifold/fiber bundle derivation===
Suppose we have an ''n''-dimensional oriented [[Riemannian manifold]], ''M'' and a target manifold ''T''. Let <math>\mathcal{C}</math> be the [[Configuration space (physics)|configuration space]] of [[smooth function]]s from ''M'' to ''T''. (More generally, we can have smooth sections of a [[fiber bundle]] ''T'' over ''M''.)

Examples of this ''M'' in physics include:
* In [[classical mechanics]], in the [[Hamiltonian mechanics|Hamiltonian]] formulation, ''M'' is the one-dimensional manifold <math>\mathbb{R}</math>, representing time and the target space is the [[cotangent bundle]] of [[space]] of generalized positions.
* In [[field (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 field]]s, <math>\varphi_1,\ldots,\varphi_m</math>, 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}^{3}</math>.

Now suppose there is a [[functional (mathematics)|functional]]

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

called the [[Action (physics)|action]]. (It takes values into <math>\mathbb{R}</math>, rather than <math>\mathbb{C}</math>; this is for physical reasons, and is unimportant for this proof.)

To get to the usual version of Noether's theorem, we need additional restrictions on the [[Action (physics)|action]]. We assume <math>\mathcal{S}[\varphi]</math> is the [[integral]] over ''M'' of a function

:<math>\mathcal{L}(\varphi,\partial_\mu\varphi,x)</math>

called the [[Lagrangian (field theory)|Lagrangian density]], depending on <math>\varphi</math>, its [[derivative]] and the position. In other words, for <math>\varphi</math> in <math>\mathcal{C}</math>

:<math> \mathcal{S}[\varphi]\,=\,\int_M \mathcal{L}[\varphi(x),\partial_\mu\varphi(x),x] \, d^{n}x.</math>

Suppose we are given [[boundary condition]]s, i.e., a specification of the value of <math>\varphi</math> at the [[Boundary (topology)|boundary]] if ''M'' is [[Compact space|compact]], or some limit on <math>\varphi</math> as ''x'' approaches ∞. Then the [[subspace topology|subspace]] of <math>\mathcal{C}</math> consisting of functions <math>\varphi</math> such that all [[functional derivative]]s of <math>\mathcal{S}</math> at <math>\varphi</math> are zero, that is:

:<math>\frac{\delta \mathcal{S}[\varphi]}{\delta \varphi(x)}\approx 0</math>

and that <math>\varphi</math> satisfies the given boundary conditions, is the subspace of [[on shell]] solutions. (See [[principle of stationary action]])

Now, suppose we have an [[infinitesimal transformation]] on <math>\mathcal{C}</math>, generated by a [[functional (mathematics)|functional]] [[derivation (abstract algebra)|derivation]], ''Q'' such that

:<math>Q \left[ \int_N \mathcal{L} \, \mathrm{d}^n x \right] \approx \int_{\partial N} f^\mu [\varphi(x),\partial\varphi,\partial\partial\varphi,\ldots] \, ds_\mu </math>

for all compact submanifolds ''N'' or in other words,

:<math>Q[\mathcal{L}(x)]\approx\partial_\mu f^\mu(x)</math>

for all ''x'', where we set

:<math>\mathcal{L}(x)=\mathcal{L}[\varphi(x), \partial_\mu \varphi(x),x].</math>

If this holds [[on shell]] and [[off shell]], we say ''Q'' generates an off-shell symmetry. If this only holds [[on shell]], we say ''Q'' generates an on-shell symmetry. Then, we say ''Q'' is a generator of a [[one-parameter group|one parameter]] [[symmetry]] [[Lie group]].

Now, for any ''N'', because of the [[Euler–Lagrange]] theorem, [[on shell]] (and only on-shell), we have

:<math>
\begin{align}
Q\left[\int_N \mathcal{L} \, \mathrm{d}^nx \right]
& =\int_N \left[\frac{\partial\mathcal{L}}{\partial\varphi} - \partial_\mu \frac{\partial\mathcal{L}}{\partial(\partial_\mu\varphi)} \right]Q[\varphi] \, \mathrm{d}^nx + \int_{\partial N} \frac{\partial\mathcal{L}}{\partial(\partial_\mu\varphi)}Q[\varphi] \, \mathrm{d}s_\mu \\
& \approx\int_{\partial N} f^\mu \, \mathrm{d}s_\mu.
\end{align}
</math>

Since this is true for any ''N'', we have

:<math>\partial_\mu\left[\frac{\partial\mathcal{L}}{\partial(\partial_\mu\varphi)}Q[\varphi]-f^\mu\right]\approx 0.</math>

But this is the [[continuity equation]] for the current <math>J^\mu</math> defined by:<ref name=Peskin>{{cite book |title=An Introduction to Quantum Field Theory |url=https://books.google.com/books?id=i35LALN0GosC&q=weinberg+%22symmetry+%22&pg=PA689 |page=18 |author1=Michael E. Peskin |author2=Daniel V. Schroeder |publisher=Basic Books |isbn=0-201-50397-2 |year=1995 }}</ref>

:<math>J^\mu\,=\,\frac{\partial\mathcal{L}}{\partial(\partial_\mu\varphi)}Q[\varphi]-f^\mu,</math>

which is called the '''Noether current''' associated with the [[symmetry]]. The continuity equation tells us that if we [[Integral|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 noncompact, the currents fall off sufficiently fast at infinity).