Editing Noether's theorem (section)

==== 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.