Editing Noether's theorem (section)

==Mathematical expression==
{{see also|Perturbation theory}}

===Simple form using perturbations===

The essence of Noether's theorem is generalizing the notion of ignorable coordinates.

One can assume that the Lagrangian ''L'' defined above is invariant under small perturbations (warpings) of the time variable ''t'' and the [[generalized coordinate]]s '''q'''. One may write

:<math>\begin{align}
           t &\rightarrow t^{\prime} = t + \delta t \\
  \mathbf{q} &\rightarrow \mathbf{q}^{\prime} = \mathbf{q} + \delta \mathbf{q} ~,
\end{align}</math>

where the perturbations ''δt'' and ''δ'''''q''' are both small, but variable. For generality, assume there are (say) ''N'' such [[symmetry transformations]] of the action, i.e. transformations leaving the action unchanged; labelled by an index ''r''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3,&nbsp;...,&nbsp;''N''.

Then the resultant perturbation can be written as a linear sum of the individual types of perturbations,
:<math>\begin{align}
           \delta t &= \sum_r \varepsilon_r T_r \\
  \delta \mathbf{q} &= \sum_r \varepsilon_r \mathbf{Q}_r ~,
\end{align}</math>

where ''ε''<sub>''r''</sub> are [[infinitesimal]] parameter coefficients corresponding to each: 
* [[Lie group#The exponential map|generator]] ''T<sub>r</sub>'' of [[time evolution]], and
* [[Lie group#The exponential map|generator]] '''Q'''<sub>''r''</sub> of the generalized coordinates.
For translations, '''Q'''<sub>''r''</sub> is a constant with units of [[length]]; for rotations, it is an expression linear in the components of '''q''', and the parameters make up an [[angle]].

Using these definitions, [[Emmy Noether|Noether]] showed that the ''N'' quantities

:<math>\left(\frac{\partial L}{\partial \dot{\mathbf{q}}} \cdot \dot{\mathbf{q}} - L \right) T_r - \frac{\partial L}{\partial \dot{\mathbf{q}}} \cdot \mathbf{Q}_r</math>

are conserved ([[constants of motion]]).

==== Examples ====

'''I. Time invariance'''

For illustration, consider a Lagrangian that does not depend on time, i.e., that is invariant (symmetric) under changes ''t'' → ''t'' + δ''t'', without any change in the coordinates '''q'''. In this case, ''N''&nbsp;=&nbsp;1, ''T''&nbsp;=&nbsp;1 and '''Q'''&nbsp;=&nbsp;0; the corresponding conserved quantity is the total [[energy]] ''H''<ref name=Lanczos1970>{{cite book | author-link= Cornelius Lanczos |last=Lanczos |first=C. | year = 1970 | title = The Variational Principles of Mechanics | edition = 4th | publisher = Dover Publications | location = New York | isbn = 0-486-65067-7}}</ref>{{rp|401}}

:<math>H = \frac{\partial L}{\partial \dot{\mathbf{q}}} \cdot \dot{\mathbf{q}} - L. </math>

'''II. Translational invariance'''

Consider a Lagrangian which does not depend on an ("ignorable", as above) coordinate ''q''<sub>''k''</sub>; so  it is invariant (symmetric) under changes ''q''<sub>''k''</sub> → ''q''<sub>''k''</sub> + ''δq''<sub>''k''</sub>. In that case, ''N''&nbsp;=&nbsp;1, ''T''&nbsp;=&nbsp;0, and ''Q''<sub>''k''</sub>&nbsp;=&nbsp;1; the conserved quantity is the corresponding linear [[momentum]] ''p''<sub>''k''</sub><ref name=Lanczos1970/>{{rp|403–404}}

:<math>p_k = \frac{\partial L}{\partial \dot{q_k}}.</math>

In [[special relativity|special]] and [[general relativity]], these two conservation laws can be expressed either ''globally'' (as it is done above), or ''locally'' as a continuity equation. The global versions can be united into a single global conservation law: the conservation of the energy-momentum 4-vector. The local versions of energy and momentum conservation (at any point in space-time) can also be united, into the conservation of a quantity defined ''locally'' at the space-time point: the [[stress–energy tensor]]<ref name="Goldstein1980">{{cite book |last=Goldstein |first=Herbert |author-link=Herbert Goldstein |year=1980 |title= [[Classical Mechanics (Goldstein)|Classical Mechanics]] |edition=2nd |publisher=Addison-Wesley |location=Reading, MA |isbn= 0-201-02918-9 }}</ref>{{rp|592}}(this will be derived in the next section).

'''III. Rotational invariance'''

The conservation of the [[angular momentum]] '''L''' = '''r''' × '''p''' is  analogous to its linear momentum counterpart.<ref name=Lanczos1970/>{{rp|404–405}} It is assumed that the symmetry of the Lagrangian is rotational, i.e., that the Lagrangian does not depend on the absolute orientation of the physical system in space. For concreteness, assume that the Lagrangian does not change under small rotations of an angle ''δθ'' about an axis '''n'''; such a rotation transforms the [[Cartesian coordinate system|Cartesian coordinates]] by the equation

:<math>\mathbf{r} \rightarrow \mathbf{r} + \delta\theta \, \mathbf{n} \times \mathbf{r}.</math>

Since time is not being transformed, ''T'' = 0, and ''N'' = 1. Taking ''δθ'' as the ''ε'' parameter and the Cartesian coordinates '''r''' as the generalized coordinates '''q''', the corresponding '''Q''' variables are given by

:<math>\mathbf{Q} = \mathbf{n} \times \mathbf{r}.</math>

Then Noether's theorem states that the following quantity is conserved,
:<math>
\frac{\partial L}{\partial \dot{\mathbf{q}}} \cdot \mathbf{Q} = 
\mathbf{p} \cdot \left( \mathbf{n} \times \mathbf{r} \right) = 
\mathbf{n} \cdot \left( \mathbf{r} \times \mathbf{p} \right) = 
\mathbf{n} \cdot \mathbf{L}.
</math>

In other words, the component of the angular momentum '''L''' along the '''n''' axis is conserved. And if '''n''' is arbitrary, i.e., if the system is insensitive to any rotation, then every component of '''L''' is conserved; in short, [[angular momentum]] is conserved.

===Field theory version===
Although useful in its own right, the version of Noether's theorem just given is a special case of the general version derived in 1915. To give the flavor of the general theorem, a version of Noether's theorem for continuous fields in four-dimensional [[space–time]] is now given. Since field theory problems are more common in modern physics than [[mechanics]] problems, this field theory version is the most commonly used (or most often implemented) version of Noether's theorem.

Let there be a set of differentiable [[Field (physics)|fields]] <math>\varphi</math> defined over all space and time; for example, the temperature <math>T(\mathbf{x}, t)</math> would be representative of such a field, being a number defined at every place and time. The [[principle of least action]] can be applied to such fields, but the action is now an integral over space and time

:<math>\mathcal{S} = \int \mathcal{L} \left(\varphi, \partial_\mu \varphi, x^\mu \right) \, d^4 x</math>

(the theorem can be further generalized to the case where the Lagrangian depends on up to the ''n''<sup>th</sup> derivative, and can also be formulated using [[jet bundle]]s).

A continuous transformation of the fields <math>\varphi</math> can be written infinitesimally as

:<math>\varphi \mapsto \varphi + \varepsilon \Psi,</math>

where <math>\Psi</math> is in general a function that may depend on both <math>x^\mu</math> and <math>\varphi</math>. The condition for <math>\Psi</math> to generate a physical symmetry is that the action <math>\mathcal{S}</math> is left invariant. This will certainly be true if the Lagrangian density <math>\mathcal{L}</math> is left invariant, but it will also be true if the Lagrangian changes by a divergence,

:<math>\mathcal{L} \mapsto \mathcal{L} + \varepsilon \partial_\mu \Lambda^\mu,</math>

since the integral of a divergence becomes a boundary term according to the [[divergence theorem]]. A system described by a given action might have multiple independent symmetries of this type, indexed by <math>r = 1, 2, \ldots, N,</math> so the most general symmetry transformation would be written as

:<math>\varphi \mapsto \varphi + \varepsilon_r \Psi_r,</math>

with the consequence

:<math>\mathcal{L} \mapsto \mathcal{L} + \varepsilon_r \partial_\mu \Lambda^\mu_r.</math>

For such systems, Noether's theorem states that there are <math>N</math> conserved [[conserved current|current densities]]

:<math>j^\nu_r = \Lambda^\nu_r - \frac{\partial \mathcal{L}}{\partial \varphi_{,\nu}} \cdot \Psi_r</math>

(where the [[dot product]] is understood to contract the ''field'' indices, not the <math>\nu</math> index or <math>r</math> index).

In such cases, the [[conservation law]] is expressed in a four-dimensional way

:<math>\partial_\nu j^\nu = 0,</math>

which expresses the idea that the amount of a conserved quantity within a sphere cannot change unless some of it flows out of the sphere. For example, [[electric charge]] is conserved; the amount of charge within a sphere cannot change unless some of the charge leaves the sphere.

==== Examples ====
'''I. The [[stress–energy tensor]]'''

For illustration, consider a physical system of fields that behaves the same under translations in time and space, as considered above; in other words, <math>L \left(\boldsymbol\varphi, \partial_\mu{\boldsymbol\varphi}, x^\mu \right)</math> is constant in its third argument. In that case, ''N''&nbsp;=&nbsp;4, one for each dimension of space and time. An infinitesimal translation in space, <math>x^\mu \mapsto x^\mu + \varepsilon_r \delta^\mu_r</math> (with <math>\delta</math> denoting the [[Kronecker delta]]), affects the fields as <math>\varphi(x^\mu) \mapsto \varphi\left(x^\mu - \varepsilon_r \delta^\mu_r\right)</math>: that is, relabelling the coordinates is equivalent to leaving the coordinates in place while translating the field itself, which in turn is equivalent to transforming the field by replacing its value at each point <math>x^\mu</math> with the value at the point <math>x^\mu - \varepsilon X^\mu</math> "behind" it which would be mapped onto <math>x^\mu</math> by the infinitesimal displacement under consideration. Since this is infinitesimal, we may write this transformation as

:<math>\Psi_r = -\delta^\mu_r \partial_\mu \varphi.</math>

The Lagrangian density transforms in the same way, <math>\mathcal{L}\left(x^\mu\right) \mapsto \mathcal{L}\left(x^\mu - \varepsilon_r \delta^\mu_r\right)</math>, so

:<math>\Lambda^\mu_r = -\delta^\mu_r \mathcal{L}</math>

and thus Noether's theorem corresponds<ref name="Goldstein1980" />{{rp|592}} to the conservation law for the [[stress–energy tensor]] ''T''<sub>''μ''</sub><sup>''ν''</sup>, where we have used <math>\mu</math> in place of <math>r</math>. To wit, by using the expression given earlier, and collecting the four conserved currents (one for each <math>\mu</math>) into a tensor <math>T</math>, Noether's theorem gives

:<math>
  T_\mu{}^\nu =
  -\delta^\nu_\mu \mathcal{L} + \delta^\sigma_\mu \partial_\sigma \varphi \frac{\partial \mathcal{L}}{\partial \varphi_{,\nu}} =
  \left(\frac{\partial \mathcal{L}}{\partial \varphi_{,\nu}}\right) \cdot \varphi_{,\mu} - \delta^\nu_\mu \mathcal{L}
</math>

with

:<math>T_\mu{}^\nu{}_{,\nu} = 0</math>

(we relabelled <math>\mu</math> as <math>\sigma</math> at an intermediate step to avoid conflict). (However, the <math>T</math> obtained in this way may differ from the symmetric tensor used as the source term in general relativity; see [[Stress–energy tensor#Canonical stress.E2.80.93energy tensor|Canonical stress–energy tensor]].)

'''I. The [[electric charge]]'''

The conservation of [[electric charge]], by contrast, can be derived by considering ''Ψ'' linear in the fields ''φ'' rather than in the derivatives.<ref name="Goldstein1980"/>{{rp|593–594}} In [[quantum mechanics]], the [[probability amplitude]] ''ψ''('''x''') of finding a particle at a point '''x''' is a complex field ''φ'', because it ascribes a [[complex number]] to every point in space and time. The probability amplitude itself is physically unmeasurable; only the probability ''p'' = |''ψ''|<sup>2</sup> can be inferred  from a set of  measurements. Therefore, the system is invariant under transformations of the ''ψ'' field and its [[complex conjugate]] field ''ψ''<sup>*</sup> that leave |''ψ''|<sup>2</sup> unchanged, such as

:<math>\psi \rightarrow e^{i\theta} \psi\ ,\ \psi^{*} \rightarrow e^{-i\theta} \psi^{*}~,</math>

a complex rotation. In the limit when the phase ''θ'' becomes infinitesimally small, ''δθ'', it may be taken as the parameter ''ε'', while the ''Ψ'' are equal to ''iψ'' and −''iψ''*, respectively. A specific example is the [[Klein–Gordon equation]], the [[special relativity|relativistically correct]] version of the [[Schrödinger equation]] for [[Spin (physics)|spinless]] particles, which has the Lagrangian density

:<math>L = \partial_{\nu}\psi \partial_{\mu}\psi^{*} \eta^{\nu \mu} + m^2 \psi \psi^{*}.</math>

In this case, Noether's theorem states that the conserved (∂&nbsp;⋅&nbsp;''j''&nbsp;=&nbsp;0) current equals

:<math>j^\nu = i \left( \frac{\partial \psi}{\partial x^\mu} \psi^{*} - \frac{\partial \psi^{*}}{\partial x^\mu} \psi \right) \eta^{\nu \mu}~,</math>

which, when multiplied by the charge on that species of particle, equals the electric current density due to that type of particle. This "gauge invariance" was first noted by [[Hermann Weyl]], and is one of the prototype [[gauge symmetry|gauge symmetries]] of physics.