Editing Noether's theorem (section)

== Derivations ==

===One independent variable===
{{unreferenced|section|date=March 2025}}
Consider the simplest case, a system with one independent variable, time. Suppose the dependent variables '''q''' are such that the action integral

<math display="block">I = \int_{t_1}^{t_2} L [\mathbf{q} [t], \dot{\mathbf{q}} [t], t] \, dt </math>

is invariant under brief infinitesimal variations in the dependent variables. In other words, they satisfy the [[Euler–Lagrange equation]]s

:<math>\frac{d}{dt} \frac{\partial L}{\partial \dot{\mathbf{q}}} [t] = \frac{\partial L}{\partial \mathbf{q}} [t].</math>

And suppose that the integral is invariant under a continuous symmetry. Mathematically such a symmetry is represented as a [[flow (mathematics)|flow]], '''φ''', which acts on the variables as follows

:<math>\begin{align}
               t &\rightarrow t' = t + \varepsilon T \\
  \mathbf{q} [t] &\rightarrow \mathbf{q}' [t'] = \varphi [\mathbf{q} [t], \varepsilon] = \varphi [\mathbf{q} [t' - \varepsilon T], \varepsilon]
\end{align}</math>

where ''ε'' is a real variable indicating the amount of flow, and ''T'' is a real constant (which could be zero) indicating how much the flow shifts time.

:<math>
\dot{\mathbf{q}} [t] \rightarrow \dot{\mathbf{q}}' [t'] = \frac{d}{dt} \varphi [\mathbf{q} [t], \varepsilon] = \frac{\partial \varphi}{\partial \mathbf{q}} [\mathbf{q} [t' - \varepsilon T], \varepsilon] \dot{\mathbf{q}} [t' - \varepsilon T]
.</math>

The action integral flows to

:<math>
\begin{align}
I' [\varepsilon] & = \int_{t_1 + \varepsilon T}^{t_2 + \varepsilon T} L [\mathbf{q}'[t'], \dot{\mathbf{q}}' [t'], t'] \, dt' \\[6pt]
& = \int_{t_1 + \varepsilon T}^{t_2 + \varepsilon T} L [\varphi [\mathbf{q} [t' - \varepsilon T], \varepsilon], \frac{\partial \varphi}{\partial \mathbf{q}} [\mathbf{q} [t' - \varepsilon T], \varepsilon] \dot{\mathbf{q}} [t' - \varepsilon T], t'] \, dt'
\end{align}
</math>

which may be regarded as a function of ''ε''. Calculating the derivative at ''ε'' = 0 and using [[Leibniz's rule (derivatives and integrals)|Leibniz's rule]], we get

:<math>
\begin{align}
0 = \frac{d I'}{d \varepsilon} [0] = {} & L [\mathbf{q} [t_2], \dot{\mathbf{q}} [t_2], t_2] T - L [\mathbf{q} [t_1], \dot{\mathbf{q}} [t_1], t_1] T \\[6pt]
& {} + \int_{t_1}^{t_2} \frac{\partial L}{\partial \mathbf{q}} \left( - \frac{\partial \varphi}{\partial \mathbf{q}} \dot{\mathbf{q}} T + \frac{\partial \varphi}{\partial \varepsilon} \right) + \frac{\partial L}{\partial \dot{\mathbf{q}}} \left( - \frac{\partial^2 \varphi}{(\partial \mathbf{q})^2} {\dot{\mathbf{q}}}^2 T + \frac{\partial^2 \varphi}{\partial \varepsilon \partial \mathbf{q}} \dot{\mathbf{q}} -
\frac{\partial \varphi}{\partial \mathbf{q}} \ddot{\mathbf{q}} T \right) \, dt.
\end{align}
</math>

Notice that the Euler–Lagrange equations imply

:<math>
\begin{align}
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \mathbf{q}} \dot{\mathbf{q}} T \right) 
& = \left( \frac{d}{dt} \frac{\partial L}{\partial \dot{\mathbf{q}}} \right) \frac{\partial \varphi}{\partial \mathbf{q}} \dot{\mathbf{q}} T + \frac{\partial L}{\partial \dot{\mathbf{q}}} \left( \frac{d}{dt} \frac{\partial \varphi}{\partial \mathbf{q}} \right) \dot{\mathbf{q}} T + \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \mathbf{q}} \ddot{\mathbf{q}} \, T \\[6pt]
& = \frac{\partial L}{\partial \mathbf{q}} \frac{\partial \varphi}{\partial \mathbf{q}} \dot{\mathbf{q}} T + \frac{\partial L}{\partial \dot{\mathbf{q}}} \left( \frac{\partial^2 \varphi}{(\partial \mathbf{q})^2} \dot{\mathbf{q}} \right) \dot{\mathbf{q}} T + \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \mathbf{q}} \ddot{\mathbf{q}} \, T.
\end{align}
</math>

Substituting this into the previous equation, one gets

:<math>
\begin{align}
0 = \frac{d I'}{d \varepsilon} [0] = {} & L [\mathbf{q} [t_2], \dot{\mathbf{q}} [t_2], t_2] T - L [\mathbf{q} [t_1], \dot{\mathbf{q}} [t_1], t_1] T - \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \mathbf{q}} \dot{\mathbf{q}} [t_2] T + \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \mathbf{q}} \dot{\mathbf{q}} [t_1] T \\[6pt]
& {} + \int_{t_1}^{t_2} \frac{\partial L}{\partial \mathbf{q}} \frac{\partial \varphi}{\partial \varepsilon} + \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial^2 \varphi}{\partial \varepsilon \partial \mathbf{q}} \dot{\mathbf{q}} \, dt.
\end{align}
</math>

Again using the Euler–Lagrange equations we get

:<math>
\frac{d}{d t} \left( \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \varepsilon} \right) 
= \left( \frac{d}{d t} \frac{\partial L}{\partial \dot{\mathbf{q}}} \right) \frac{\partial \varphi}{\partial \varepsilon} + \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial^2 \varphi}{\partial \varepsilon \partial \mathbf{q}} \dot{\mathbf{q}}
= \frac{\partial L}{\partial \mathbf{q}} \frac{\partial \varphi}{\partial \varepsilon} + \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial^2 \varphi}{\partial \varepsilon \partial \mathbf{q}} \dot{\mathbf{q}}.
</math>

Substituting this into the previous equation, one gets

:<math>
\begin{align}
0 = {} & L [\mathbf{q} [t_2], \dot{\mathbf{q}} [t_2], t_2] T - L [\mathbf{q} [t_1], \dot{\mathbf{q}} [t_1], t_1] T - \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \mathbf{q}} \dot{\mathbf{q}} [t_2] T + \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \mathbf{q}} \dot{\mathbf{q}} [t_1] T \\[6pt]
& {} + \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \varepsilon} [t_2] - \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \varepsilon} [t_1].
\end{align}
</math>

From which one can see that

:<math>\left( \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \mathbf{q}} \dot{\mathbf{q}} - L \right) T - \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \varepsilon}</math>

is a constant of the motion, i.e., it is a conserved quantity. Since φ['''q''', 0] = '''q''', we get <math>\frac{\partial \varphi}{\partial \mathbf{q}} = 1</math> and so the conserved quantity simplifies to

:<math>\left( \frac{\partial L}{\partial \dot{\mathbf{q}}} \dot{\mathbf{q}} - L \right) T - \frac{\partial L}{\partial \dot{\mathbf{q}}} \frac{\partial \varphi}{\partial \varepsilon}.</math>

To avoid excessive complication of the formulas, this derivation assumed that the flow does not change as time passes. The same result can be obtained in the more general case.

=== Geometric derivation ===

The Noether’s theorem can be seen as a consequence of the [[fundamental theorem of Calculus#Generalizations|fundamental theorem of calculus]] (known by various names in physics such as the [[Generalized Stokes theorem]] or the [[Gradient theorem]]):<ref>{{cite journal | last= Houchmandzadeh |first=B. | year = 2025 | title = A geometric derivation of Noether's theorem | journal = European Journal of Physics | volume = 46 | pages = 025003 |url= https://hal.science/hal-04682603v3/document}}
</ref> 
for a function 
<math display="inline">S</math> 
analytical in a domain 
<math display="inline">{\cal {D}}</math>,
<math display="block">\int_{{\cal {\cal P}}}dS=0</math> 
[[File:Geometric derivation of Noether's theorem.svg|thumb|Integration path that leads to Noether's theorem]]
where <math display="inline">{\cal P}</math> is a closed path in <math display="inline">{\cal D}</math>. Here, the ''function'' <math display="inline">S(\mathbf{q},t)</math> is the action ''function'' that is computed by the integration of the Lagrangian over optimal trajectories or equivalently obtained through the [[Hamilton-Jacobi equation]]. As <math display="inline">\partial S/\partial\mathbf{q}=\mathbf{p}</math> (where <math display="inline">\mathbf{p}</math>is the momentum) and <math display="inline">\partial S/\partial t=-H</math> (where <math display="inline">H</math> is the Hamiltonian), the differential of this function is given by <math display="inline">dS=\mathbf{p}d\mathbf{q}-Hdt</math>.

Using the geometrical approach, the conserved quantity for a symmetry in Noether’s sense can be derived. The symmetry is expressed as an infinitesimal transformation:<math display="block">\begin{aligned}
\mathbf{q'} & = & \mathbf{q}+\epsilon\phi_{\mathbf{q}}(\mathbf{q},t)\\
t' & = & t+\epsilon\phi_{t}(\mathbf{q},t)
\end{aligned}</math>  Let <math display="inline">{\cal C}</math> be an optimal trajectory and <math display="inline">{\cal C}'</math> its image under the above transformation <math display="inline">(\phi_{\mathbf{q}},\phi_{t})^{T}</math> (which is also an optimal trajectory). The closed path <math display="inline">{\cal P}</math> of integration is chosen as <math display="inline">ABB'A'</math>, where the branches <math display="inline">AB</math> and <math display="inline">A'B'</math> are given <math display="inline">{\cal C}</math> and <math display="inline">{\cal C}'</math> . By the hypothesis of Noether theorem, to the first order in <math display="inline">\epsilon</math>, <math display="block">\int_{{\cal C}}dS=\int_{{\cal C}'}dS</math> therefore, <math display="block">\int_{A}^{A'}dS=\int_{B}^{B'}dS</math> By definition, on the <math display="inline">AA'</math> branch we have <math display="inline">d\mathbf{q}=\epsilon\phi_{\mathbf{q}}(\mathbf{q},t)</math> and <math display="inline">dt=\epsilon\phi_{t}(\mathbf{q},t)</math>. Therefore, to the first order in <math display="inline">\epsilon</math>, the quantity <math display="block">I=\mathbf{p}\phi_{\mathbf{q}}-H\phi_{t}</math> is conserved along the trajectory.

===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>

===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).

=== Comments ===
Noether's theorem is an [[on shell]] theorem: it relies on use of the equations of motion—the classical path. It reflects the relation between the boundary conditions and the variational principle. Assuming no boundary terms in the action, Noether's theorem implies that

: <math>\int_{\partial N} J^\mu ds_{\mu} \approx 0.</math>

The quantum analogs of Noether's theorem involving expectation values (e.g., <math display="inline">\left\langle\int d^{4}x~\partial \cdot \textbf{J} \right\rangle = 0</math>) probing [[off shell]] quantities as well are the [[Ward–Takahashi identity|Ward–Takahashi identities]].

=== Generalization to Lie algebras ===
Suppose we have two symmetry derivations ''Q''<sub>1</sub> and ''Q''<sub>2</sub>. Then, [''Q''<sub>1</sub>,&nbsp;''Q''<sub>2</sub>] is also a symmetry derivation. Let us see this explicitly. Let us say
<math display="block">Q_1[\mathcal{L}]\approx \partial_\mu f_1^\mu</math>
and
<math display="block">Q_2[\mathcal{L}]\approx \partial_\mu f_2^\mu</math>

Then,
<math display="block">[Q_1,Q_2][\mathcal{L}] = Q_1[Q_2[\mathcal{L}]]-Q_2[Q_1[\mathcal{L}]]\approx\partial_\mu f_{12}^\mu</math>
where ''f''<sub>12</sub>&nbsp;=&nbsp;''Q''<sub>1</sub>[''f''<sub>2</sub><sup>''μ''</sup>]&nbsp;−&nbsp;''Q''<sub>2</sub>[''f''<sub>1</sub><sup>''μ''</sup>]. So,
<math display="block">j_{12}^\mu = \left(\frac{\partial}{\partial (\partial_\mu\varphi)} \mathcal{L}\right)(Q_1[Q_2[\varphi]] - Q_2[Q_1[\varphi]])-f_{12}^\mu.</math>

This shows we can extend Noether's theorem to larger Lie algebras in a natural way.

=== Generalization of the proof ===
This applies to ''any'' local symmetry derivation ''Q'' satisfying ''QS''&nbsp;≈&nbsp;0, and also to more general local functional differentiable actions, including ones where the Lagrangian depends on higher derivatives of the fields. Let ''ε'' be any arbitrary smooth function of the spacetime (or time) manifold such that the closure of its support is disjoint from the boundary. ''ε''&nbsp;is a [[test function]]. Then, because of the variational principle (which does ''not'' apply to the boundary, by the way), the derivation distribution q generated by ''q''[''ε''][Φ(''x'')] = ''ε''(''x'')''Q''[Φ(''x'')] satisfies ''q''[''ε''][''S'']&nbsp;≈&nbsp;0 for every&nbsp;''ε'', or more compactly, ''q''(''x'')[''S'']&nbsp;≈ 0&nbsp;for all ''x'' not on the boundary (but remember that ''q''(''x'') is a shorthand for a derivation ''distribution'', not a derivation parametrized by ''x'' in general). This is the generalization of Noether's theorem.

To see how the generalization is related to the version given above, assume that the action is the spacetime integral of a Lagrangian that only depends on <math>\varphi</math> and its first derivatives. Also, assume

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

Then,

:<math>
\begin{align}
q[\varepsilon][\mathcal{S}] & = \int q[\varepsilon][\mathcal{L}] d^{n} x  \\[6pt]
& = \int \left\{ \left(\frac{\partial}{\partial \varphi}\mathcal{L}\right) \varepsilon Q[\varphi]+ \left[\frac{\partial}{\partial (\partial_\mu \varphi)}\mathcal{L}\right]\partial_\mu(\varepsilon Q[\varphi]) \right\} d^{n} x \\[6pt]
& = \int \left\{ \varepsilon Q[\mathcal{L}] + \partial_{\mu}\varepsilon \left[\frac{\partial}{\partial \left( \partial_\mu \varphi\right)} \mathcal{L} \right] Q[\varphi] \right\} \, d^{n} x \\[6pt]
& \approx \int \varepsilon \partial_\mu \left\{f^\mu-\left[\frac{\partial}{\partial (\partial_\mu\varphi)}\mathcal{L}\right]Q[\varphi]\right\} \, d^{n} x
\end{align}
</math>

for all <math>\varepsilon</math>.

More generally, if the Lagrangian depends on higher derivatives, then

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