Editing Noether's theorem (section)

===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.

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]].)

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.