Editing Noether's theorem (section)

===Field-theoretic derivation===
{{unreferenced|section|date=March 2025}}
Noether's theorem may also be derived for tensor fields <math>\varphi^A</math> where the index ''A'' ranges over the various components of the various tensor fields. These field quantities are functions defined over a four-dimensional space whose points are labeled by coordinates ''x''<sup>μ</sup> where the index ''μ'' ranges over time (''μ''&nbsp;=&nbsp;0) and three spatial dimensions (''μ''&nbsp;=&nbsp;1,&nbsp;2,&nbsp;3). These four coordinates are the independent variables; and the values of the fields at each event are the dependent variables. Under an infinitesimal transformation, the variation in the coordinates is written

:<math>x^\mu \rightarrow \xi^\mu = x^\mu + \delta x^\mu</math>

whereas the transformation of the field variables is expressed as

:<math>\varphi^A \rightarrow \alpha^A \left(\xi^\mu\right) = \varphi^A \left(x^\mu\right) + \delta \varphi^A \left(x^\mu\right)\,.</math>

By this definition, the field variations <math>\delta\varphi^A</math>  result from two factors: intrinsic changes in the field themselves and changes in coordinates, since the transformed field ''α''<sup>''A''</sup> depends on the transformed coordinates ξ<sup>μ</sup>. To isolate the intrinsic changes, the field variation at a single point ''x''<sup>μ</sup> may be defined

:<math>\alpha^A \left(x^\mu\right) = \varphi^A \left(x^\mu\right) + \bar{\delta} \varphi^A \left(x^\mu\right)\,.</math>

If the coordinates are changed, the boundary of the region of space–time over which the Lagrangian is being integrated also changes; the original boundary and its transformed version are denoted as Ω and Ω’, respectively.

Noether's theorem begins with the assumption that a specific transformation of the coordinates and field variables does not change the [[action (physics)|action]], which is defined as the integral of the Lagrangian density over the given region of spacetime. Expressed mathematically, this assumption may be written as

:<math>\int_{\Omega^\prime} L \left( \alpha^A, {\alpha^A}_{,\nu}, \xi^\mu \right) d^4\xi - \int_{\Omega} L \left( \varphi^A, {\varphi^A}_{,\nu}, x^\mu \right) d^{4}x = 0</math>

where the comma subscript indicates a partial derivative with respect to the coordinate(s) that follows the comma, e.g.

:<math>{\varphi^A}_{,\sigma} = \frac{\partial \varphi^A}{\partial x^\sigma}\,.</math>

Since ξ is a dummy variable of integration, and since the change in the boundary Ω is infinitesimal by assumption, the two integrals may be combined using the four-dimensional version of the [[divergence theorem]] into the following form

:<math>
  \int_\Omega \left\{ 
    \left[ L \left( \alpha^A,  {\alpha^A}_{,\nu},  x^\mu \right) - 
           L \left( \varphi^A, {\varphi^A}_{,\nu}, x^\mu \right) \right] +
   \frac{\partial}{\partial x^\sigma} \left[ L \left( \varphi^A, {\varphi^A}_{,\nu}, x^\mu \right) \delta x^\sigma \right]
  \right\} d^4 x = 0
\,.</math>

The difference in Lagrangians can be written to first-order in the infinitesimal variations as

:<math>
  \left[
    L \left( \alpha^A, {\alpha^A}_{,\nu}, x^\mu \right) - 
    L \left( \varphi^A, {\varphi^A}_{,\nu}, x^\mu \right)
  \right] = 
  \frac{\partial L}{\partial \varphi^A} \bar{\delta} \varphi^A + 
    \frac{\partial L}{\partial {\varphi^A}_{,\sigma}} \bar{\delta} {\varphi^A}_{,\sigma}
\,.</math>

However, because the variations are defined at the same point as described above, the variation and the derivative can be done in reverse order; they [[commutativity|commute]]

:<math>
  \bar{\delta} {\varphi^A}_{,\sigma} = 
  \bar{\delta} \frac{\partial \varphi^A}{\partial x^\sigma} = 
  \frac{\partial}{\partial x^\sigma} \left(\bar{\delta} \varphi^A\right)
\,.</math>

Using the Euler–Lagrange field equations

:<math>
  \frac{\partial}{\partial x^\sigma} \left( \frac{\partial L}{\partial {\varphi^A}_{,\sigma}} \right) =
  \frac{\partial L}{\partial\varphi^A}
</math>

the difference in Lagrangians can be written neatly as

:<math>\begin{align}
      &\left[ L \left( \alpha^A, {\alpha^A}_{,\nu}, x^\mu \right) - L \left( \varphi^A, {\varphi^A}_{,\nu}, x^\mu \right) \right] \\[4pt]
  ={} &\frac{\partial}{\partial x^\sigma} \left( \frac{\partial L}{\partial {\varphi^A}_{,\sigma}} \right) \bar{\delta} \varphi^A + \frac{\partial L}{\partial {\varphi^A}_{,\sigma}} \bar{\delta} {\varphi^A}_{,\sigma}
  =    \frac{\partial}{\partial x^\sigma} \left( \frac{\partial L}{\partial {\varphi^A}_{,\sigma}} \bar{\delta} \varphi^A \right).
\end{align}</math>

Thus, the change in the action can be written as

:<math>
  \int_\Omega \frac{\partial}{\partial x^\sigma} \left\{
    \frac{\partial L}{\partial {\varphi^A}_{,\sigma}} \bar{\delta} \varphi^A + 
    L \left( \varphi^A, {\varphi^A}_{,\nu}, x^\mu \right) \delta x^\sigma
  \right\} d^{4}x = 0
\,.</math>

Since this holds for any region Ω, the integrand must be zero

:<math>
  \frac{\partial}{\partial x^\sigma} \left\{
    \frac{\partial L}{\partial {\varphi^A}_{,\sigma}} \bar{\delta} \varphi^A + 
    L \left( \varphi^A, {\varphi^A}_{,\nu}, x^\mu \right) \delta x^\sigma
  \right\} = 0
\,.</math>

For any combination of the various [[symmetry in physics|symmetry]] transformations, the perturbation can be written

:<math>\begin{align}
    \delta x^{\mu} &= \varepsilon X^\mu \\
  \delta \varphi^A &= \varepsilon \Psi^A = \bar{\delta} \varphi^A + \varepsilon \mathcal{L}_X \varphi^A
\end{align}</math>

where <math>\mathcal{L}_X \varphi^A</math> is the [[Lie derivative]] of 
<math>\varphi^A</math> in the ''X''<sup>''μ''</sup> direction. When <math>\varphi^A</math> is a scalar or <math>{X^\mu}_{,\nu} = 0 </math>,

:<math>\mathcal{L}_X \varphi^A = \frac{\partial \varphi^A}{\partial x^\mu} X^\mu\,.</math>

These equations imply that the field variation taken at one point equals

:<math>\bar{\delta} \varphi^A = \varepsilon \Psi^A - \varepsilon \mathcal{L}_X \varphi^A\,.</math>

Differentiating the above divergence with respect to ''ε'' at ''ε''&nbsp;=&nbsp;0 and changing the sign yields the conservation law

:<math>\frac{\partial}{\partial x^\sigma} j^\sigma = 0</math>

where the conserved current equals

:<math>
    j^\sigma = 
    \left[\frac{\partial L}{\partial {\varphi^A}_{,\sigma}} \mathcal{L}_X \varphi^A - L \, X^\sigma\right]
  - \left(\frac{\partial L}{\partial {\varphi^A}_{,\sigma}} \right) \Psi^A\,.
</math>