Le théorème de Noether ou le premier théorème de Noether stipule que chaque symétrie différentiable de l' action d'un système physique avec des forces conservatrices a une loi de conservation correspondante . [1] Le théorème a été prouvé par le mathématicien Emmy Noether en 1915 et publié en 1918, [2] après qu'un cas particulier a été prouvé par E. Cosserat et F. Cosserat en 1909. [3] L'action d'un système physique est l' intégrale au fil du temps d'un lagrangienfonction, à partir de laquelle le comportement du système peut être déterminé par le principe de moindre action . Ce théorème ne s'applique qu'aux symétries continues et lisses sur l'espace physique .

Première page de l 'article d' Emmy Noether "Invariante Variationsprobleme" (1918), où elle a prouvé son théorème.

Le théorème de Noether est utilisé en physique théorique et dans le calcul des variations . Généralisation des formulations sur les constantes de mouvement en mécanique lagrangienne et hamiltonienne (développées respectivement en 1788 et 1833), elle ne s'applique pas aux systèmes qui ne peuvent pas être modélisés avec un lagrangien seul (par exemple, les systèmes avec une fonction de dissipation de Rayleigh ). En particulier, les systèmes dissipatifs avec des symétries continues n'ont pas besoin d'avoir une loi de conservation correspondante.

## Illustrations de base et arrière-plan

A titre d'illustration, si un système physique se comporte de la même manière quelle que soit son orientation dans l'espace, son lagrangien est symétrique en rotation continue: à partir de cette symétrie, le théorème de Noether dicte que le moment cinétique du système soit conservé, en conséquence de sa lois du mouvement. Le système physique lui-même n'a pas besoin d'être symétrique; un astéroïde dentelé tombant dans l'espace conserve le moment cinétique malgré son asymétrie. Ce sont les lois de son mouvement qui sont symétriques.

Comme autre exemple, si un processus physique présente les mêmes résultats quel que soit le lieu ou le temps, alors son lagrangien est symétrique sous des traductions continues dans l'espace et le temps respectivement: par le théorème de Noether, ces symétries rendent compte des lois de conservation de l'impulsion linéaire et de l' énergie dans ce système, respectivement.

Le théorème de Noether est important, à la fois en raison de la compréhension qu'il donne des lois de conservation, et aussi en tant qu'outil de calcul pratique. Il permet aux chercheurs de déterminer les quantités conservées (invariants) à partir des symétries observées d'un système physique. Inversement, il permet aux chercheurs de considérer des classes entières de Lagrangiens hypothétiques avec des invariants donnés, pour décrire un système physique. A titre d'illustration, supposons qu'une théorie physique est proposée qui conserve une quantité X . Un chercheur peut calculer les types de Lagrangiens qui conservent X grâce à une symétrie continue. En raison du théorème de Noether, les propriétés de ces Lagrangiens fournissent des critères supplémentaires pour comprendre les implications et juger de l'adéquation de la nouvelle théorie.

Il existe de nombreuses versions du théorème de Noether, avec divers degrés de généralité. Il existe des équivalents quantiques naturels de ce théorème, exprimés dans les identités Ward-Takahashi . Il existe également des généralisations du théorème de Noether aux superspaces . [ citation nécessaire ]

## Déclaration informelle du théorème

Tous les beaux points techniques mis à part, le théorème de Noether peut être énoncé de manière informelle

Si un système a une propriété de symétrie continue, il existe des quantités correspondantes dont les valeurs sont conservées dans le temps. [4]

Une version plus sophistiquée du théorème impliquant des champs déclare que:

A chaque symétrie différentiable engendrée par des actions locales correspond un courant conservé .

Le mot «symétrie» dans l'énoncé ci-dessus se réfère plus précisément à la covariance de la forme qu'une loi physique prend par rapport à un groupe de transformations de Lie unidimensionnel satisfaisant à certains critères techniques. La loi de conservation d'une grandeur physique est généralement exprimée sous la forme d'une équation de continuité .

La preuve formelle du théorème utilise la condition d'invariance pour dériver une expression pour un courant associé à une quantité physique conservée. Dans la terminologie moderne (depuis environ 1980 [5] ), la quantité conservée est appelée la charge Noether , tandis que le flux transportant cette charge est appelé le courant Noether . Le courant Noether est défini jusqu'à un champ vectoriel solénoïdal (sans divergence).

Dans le contexte de la gravitation, la déclaration de Felix Klein du théorème de Noether pour l'action I stipule pour les invariants: [6]

Si une intégrale I est invariante sous un groupe continu G ρ avec ρ paramètres, alors ρ les combinaisons linéairement indépendantes des expressions lagrangiennes sont des divergences.

## Brève illustration et aperçu du concept

Graphique illustrant le théorème de Noether pour une symétrie par coordonnées.

L'idée principale derrière le théorème de Noether est plus facilement illustrée par un système avec une coordonnée ${\ displaystyle q}$ et une symétrie continue ${\ displaystyle \ varphi: q \ mapsto q + \ delta q}$(flèches grises sur le diagramme). Considérez n'importe quelle trajectoire${\ displaystyle q (t)}$(en gras sur le diagramme) qui satisfait les lois du mouvement du système . Autrement dit, l' action ${\ displaystyle S}$régissant ce système est stationnaire sur cette trajectoire, c'est-à-dire ne change pas sous aucune variation locale de la trajectoire. En particulier, il ne changerait pas sous une variante qui applique le flux de symétrie${\ displaystyle \ varphi}$sur un segment de temps [ t 0 , t 1 ] et est immobile à l'extérieur de ce segment. Pour garder la trajectoire continue, nous utilisons des périodes de "tamponnage" de petit temps${\ displaystyle \ tau}$ pour faire la transition entre les segments progressivement.

Le changement total dans l'action ${\ displaystyle S}$comprend maintenant les changements apportés par chaque intervalle de jeu. Les pièces, où la variation elle-même disparaît, n'apportent${\ displaystyle \ Delta S}$. La partie centrale ne change pas non plus l'action, car sa transformation${\ displaystyle \ varphi}$ est une symétrie et préserve ainsi le lagrangien ${\ displaystyle L}$ et l'action ${\ textstyle S = \ int L}$. Les seules pièces restantes sont les pièces «tampons». En gros, ils contribuent principalement par leur «biais»${\ displaystyle {\ dot {q}} \ rightarrow {\ dot {q}} \ pm \ delta q / \ tau}$.

Cela change le lagrangien en ${\ displaystyle \ Delta L \ approx {\ bigl (} \ partial L / \ partial {\ dot {q}} {\ bigr)} \ Delta {\ dot {q}}}$, qui s'intègre à

${\ displaystyle \ Delta S = \ int \ Delta L \ approx \ int {\ frac {\ partial L} {\ partial {\ dot {q}}}} \ Delta {\ dot {q}} \ approx \ int { \ frac {\ partial L} {\ partial {\ dot {q}}}} \ left (\ pm {\ frac {\ delta q} {\ tau}} \ right) \ approx \ \ pm {\ frac {\ partial L} {\ partial {\ dot {q}}}} \ delta q = \ pm {\ frac {\ partial L} {\ partial {\ dot {q}}}} \ varphi.}$

Ces derniers termes, évalués autour des endpoints ${\ displaystyle t_ {0}}$ et ${\ displaystyle t_ {1}}$, doivent s'annuler pour effectuer le changement total de l'action ${\ displaystyle \ Delta S}$être nul, comme on pourrait s'y attendre si la trajectoire est une solution. C'est-à-dire

${\displaystyle \left({\frac {\partial L}{\partial {\dot {q}}}}\varphi \right)(t_{0})=\left({\frac {\partial L}{\partial {\dot {q}}}}\varphi \right)(t_{1}),}$

meaning the quantity ${\displaystyle \left(\partial L/\partial {\dot {q}}\right)\varphi }$ is conserved, which is the conclusion of Noether's theorem. For instance if pure translations of ${\displaystyle \varphi }$ by a constant are the symmetry, then the conserved quantity becomes just ${\displaystyle \left(\partial L/\partial {\dot {q}}\right)=p}$, the canonical momentum.

More general cases follow the same idea:

• When more coordinates ${\displaystyle q_{r}}$ undergo a symmetry transformation ${\displaystyle q_{r}\mapsto q_{r}+\varphi _{r}}$, their effects add up by linearity to a conserved quantity ${\textstyle \sum _{r}\left(\partial L/\partial {\dot {q}}_{r}\right)\varphi _{r}}$.
• When there are time transformations ${\displaystyle t\mapsto t+T}$, they cause the "buffering" segments to contribute the two following terms to ${\displaystyle \Delta S}$:
${\displaystyle \Delta S\approx \pm \left(TL+\int {\frac {\partial L}{\partial {\dot {q}}_{r}}}\Delta {\dot {q}}_{r}\right)\approx \pm T\left(L-{\frac {\partial L}{\partial {\dot {q}}_{r}}}{\dot {q}}_{r}\right),}$
first term being due to stretching in temporal dimension of the "buffering" segment (that changes the size of the domain of integration), and the second is due to its "slanting" just as in the exemplar case. Together they add a summand ${\textstyle T\left(L-\sum _{r}\left(\partial L/\partial {\dot {q}}_{r}\right){\dot {q}}_{r}\right)}$ to the conserved quantity.
• Finally, when instead of a trajectory ${\displaystyle q(t)}$ entire fields ${\displaystyle \psi (q_{r},t)}$ are considered, the argument replaces
• the interval ${\displaystyle [t_{0},t_{1}]}$ with a bounded region ${\displaystyle U}$ of the ${\displaystyle (q_{r},t)}$-domain,
• the endpoints ${\displaystyle t_{0}}$ and ${\displaystyle t_{1}}$ with the boundary ${\displaystyle \partial U}$ of the region,
• and its contribution to ${\displaystyle \Delta S}$ is interpreted as a flux of a conserved current ${\displaystyle j_{r}}$, that is built in a way analogous to the prior definition of a conserved quantity.
Now, the zero contribution of the "buffering" ${\displaystyle \partial U}$ to ${\displaystyle \Delta S}$ is interpreted as vanishing of the total flux of the current ${\displaystyle j_{r}}$ through the ${\displaystyle \partial U}$. That is the sense in which it is conserved: how much is "flowing" in, just as much is "flowing" out.

## Contexte historique

A conservation law states that some quantity X in the mathematical description of a system's evolution remains constant throughout its motion – it is an invariant. Mathematically, the rate of change of X (its derivative with respect to time) is zero,

${\displaystyle {\frac {dX}{dt}}={\dot {X}}=0~.}$

Such quantities are said to be conserved; they are often called constants of motion (although motion per se need not be involved, just evolution in time). For example, if the energy of a system is conserved, its energy is invariant at all times, which imposes a constraint on the system's motion and may help in solving for it. Aside from insights that such constants of motion give into the nature of a system, they are a useful calculational tool; for example, an approximate solution can be corrected by finding the nearest state that satisfies the suitable conservation laws.

The earliest constants of motion discovered were momentum and energy, which were proposed in the 17th century by René Descartes and Gottfried Leibniz on the basis of collision experiments, and refined by subsequent researchers. Isaac Newton was the first to enunciate the conservation of momentum in its modern form, and showed that it was a consequence of Newton's third law. According to general relativity, the conservation laws of linear momentum, energy and angular momentum are only exactly true globally when expressed in terms of the sum of the stress–energy tensor (non-gravitational stress–energy) and the Landau–Lifshitz stress–energy–momentum pseudotensor (gravitational stress–energy). The local conservation of non-gravitational linear momentum and energy in a free-falling reference frame is expressed by the vanishing of the covariant divergence of the stress–energy tensor. Another important conserved quantity, discovered in studies of the celestial mechanics of astronomical bodies, is the Laplace–Runge–Lenz vector.

In the late 18th and early 19th centuries, physicists developed more systematic methods for discovering invariants. A major advance came in 1788 with the development of Lagrangian mechanics, which is related to the principle of least action. In this approach, the state of the system can be described by any type of generalized coordinates q; the laws of motion need not be expressed in a Cartesian coordinate system, as was customary in Newtonian mechanics. The action is defined as the time integral I of a function known as the Lagrangian L

${\displaystyle I=\int L(\mathbf {q} ,{\dot {\mathbf {q} }},t)\,dt~,}$

where the dot over q signifies the rate of change of the coordinates q,

${\displaystyle {\dot {\mathbf {q} }}={\frac {d\mathbf {q} }{dt}}~.}$

Hamilton's principle states that the physical path q(t)—the one actually taken by the system—is a path for which infinitesimal variations in that path cause no change in I, at least up to first order. This principle results in the Euler–Lagrange equations,

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right)={\frac {\partial L}{\partial \mathbf {q} }}~.}$

Thus, if one of the coordinates, say qk, does not appear in the Lagrangian, the right-hand side of the equation is zero, and the left-hand side requires that

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}_{k}}}\right)={\frac {dp_{k}}{dt}}=0~,}$

where the momentum

${\displaystyle p_{k}={\frac {\partial L}{\partial {\dot {q}}_{k}}}}$

is conserved throughout the motion (on the physical path).

Thus, the absence of the ignorable coordinate qk from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations of qk; the Lagrangian is invariant, and is said to exhibit a symmetry under such transformations. This is the seed idea generalized in Noether's theorem.

Several alternative methods for finding conserved quantities were developed in the 19th century, especially by William Rowan Hamilton. For example, he developed a theory of canonical transformations which allowed changing coordinates so that some coordinates disappeared from the Lagrangian, as above, resulting in conserved canonical momenta. Another approach, and perhaps the most efficient for finding conserved quantities, is the Hamilton–Jacobi equation.

## Expression mathématique

### Simple form using perturbations

The essence of Noether's theorem is generalizing the ignorable coordinates outlined.[clarification needed]

One can assume that the Lagrangian L defined above is invariant under small perturbations (warpings) of the time variable t and the generalized coordinates q. One may write

{\displaystyle {\begin{aligned}t&\rightarrow t^{\prime }=t+\delta t\\\mathbf {q} &\rightarrow \mathbf {q} ^{\prime }=\mathbf {q} +\delta \mathbf {q} ~,\end{aligned}}}

where the perturbations δt and δq 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 = 1, 2, 3, ..., N.

Then the resultant perturbation can be written as a linear sum of the individual types of perturbations,

{\displaystyle {\begin{aligned}\delta t&=\sum _{r}\varepsilon _{r}T_{r}\\\delta \mathbf {q} &=\sum _{r}\varepsilon _{r}\mathbf {Q} _{r}~,\end{aligned}}}

where εr are infinitesimal parameter coefficients corresponding to each:

• generator Tr of time evolution, and
• generator Qr of the generalized coordinates.

For translations, Qr 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, Noether showed that the N quantities

${\displaystyle \left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot {\dot {\mathbf {q} }}-L\right)T_{r}-{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot \mathbf {Q} _{r}}$

(which have the dimensions of [energy]·[time] + [momentum]·[length] = [action]) are conserved (constants of motion).

#### Examples

##### Time invariance

For illustration, consider a Lagrangian that does not depend on time, i.e., that is invariant (symmetric) under changes tt + δt, without any change in the coordinates q. In this case, N = 1, T = 1 and Q = 0; the corresponding conserved quantity is the total energy H[7]

${\displaystyle H={\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot {\dot {\mathbf {q} }}-L.}$
##### Translational invariance

Consider a Lagrangian which does not depend on an ("ignorable", as above) coordinate qk; so it is invariant (symmetric) under changes qkqk + δqk. In that case, N = 1, T = 0, and Qk = 1; the conserved quantity is the corresponding linear momentum pk[8]

${\displaystyle p_{k}={\frac {\partial L}{\partial {\dot {q_{k}}}}}.}$

In special and general relativity, these apparently separate conservation laws are aspects of a single conservation law, that of the stress–energy tensor,[9] that is derived in the next section.

##### Rotational invariance

The conservation of the angular momentum L = r × p is analogous to its linear momentum counterpart.[10] 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 coordinates by the equation

${\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\delta \theta \,\mathbf {n} \times \mathbf {r} .}$

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

${\displaystyle \mathbf {Q} =\mathbf {n} \times \mathbf {r} .}$

Then Noether's theorem states that the following quantity is conserved,

${\displaystyle {\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot \mathbf {Q} _{r}=\mathbf {p} \cdot \left(\mathbf {n} \times \mathbf {r} \right)=\mathbf {n} \cdot \left(\mathbf {r} \times \mathbf {p} \right)=\mathbf {n} \cdot \mathbf {L} .}$

In other words, the component of the angular momentum L along the n axis is conserved.

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 fields ${\displaystyle \varphi }$ defined over all space and time; for example, the temperature ${\displaystyle T(\mathbf {x} ,t)}$ 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

${\displaystyle {\mathcal {S}}=\int {\mathcal {L}}\left(\varphi ,\partial _{\mu }\varphi ,x^{\mu }\right)\,d^{4}x}$

(the theorem can be further generalized to the case where the Lagrangian depends on up to the nth derivative, and can also be formulated using jet bundles).

A continuous transformation of the fields ${\displaystyle \varphi }$ can be written infinitesimally as

${\displaystyle \varphi \mapsto \varphi +\varepsilon \Psi ,}$

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

${\displaystyle {\mathcal {L}}\mapsto {\mathcal {L}}+\varepsilon \partial _{\mu }\Lambda ^{\mu },}$

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 ${\displaystyle r=1,2,\ldots ,N,}$ so the most general symmetry transformation would be written as

${\displaystyle \varphi \mapsto \varphi +\varepsilon _{r}\Psi _{r},}$

with the consequence

${\displaystyle {\mathcal {L}}\mapsto {\mathcal {L}}+\varepsilon _{r}\partial _{\mu }\Lambda _{r}^{\mu }.}$

For such systems, Noether's theorem states that there are ${\displaystyle N}$ conserved current densities

${\displaystyle j_{r}^{\nu }=\Lambda _{r}^{\nu }-{\frac {\partial {\mathcal {L}}}{\partial \varphi _{,\nu }}}\cdot \Psi _{r}}$

(where the dot product is understood to contract the field indices, not the ${\displaystyle \nu }$ index or ${\displaystyle r}$ index).

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

${\displaystyle \partial _{\nu }j^{\nu }=0,}$

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, ${\displaystyle L\left({\boldsymbol {\varphi }},\partial _{\mu }{\boldsymbol {\varphi }},x^{\mu }\right)}$ is constant in its third argument. In that case, N = 4, one for each dimension of space and time. An infinitesimal translation in space, ${\displaystyle x^{\mu }\mapsto x^{\mu }+\varepsilon _{r}\delta _{r}^{\mu }}$ (with ${\displaystyle \delta }$ denoting the Kronecker delta), affects the fields as ${\displaystyle \varphi (x^{\mu })\mapsto \varphi \left(x^{\mu }-\varepsilon _{r}\delta _{r}^{\mu }\right)}$: 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 ${\displaystyle x^{\mu }}$ with the value at the point ${\displaystyle x^{\mu }-\varepsilon X^{\mu }}$ "behind" it which would be mapped onto ${\displaystyle x^{\mu }}$ by the infinitesimal displacement under consideration. Since this is infinitesimal, we may write this transformation as

${\displaystyle \Psi _{r}=-\delta _{r}^{\mu }\partial _{\mu }\varphi .}$

The Lagrangian density transforms in the same way, ${\displaystyle {\mathcal {L}}\left(x^{\mu }\right)\mapsto {\mathcal {L}}\left(x^{\mu }-\varepsilon _{r}\delta _{r}^{\mu }\right)}$, so

${\displaystyle \Lambda _{r}^{\mu }=-\delta _{r}^{\mu }{\mathcal {L}}}$

and thus Noether's theorem corresponds to the conservation law for the stress–energy tensor Tμν,[9] where we have used ${\displaystyle \mu }$ in place of ${\displaystyle r}$. To wit, by using the expression given earlier, and collecting the four conserved currents (one for each ${\displaystyle \mu }$) into a tensor ${\displaystyle T}$, Noether's theorem gives

${\displaystyle T_{\mu }{}^{\nu }=-\delta _{\mu }^{\nu }{\mathcal {L}}+\delta _{\mu }^{\sigma }\partial _{\sigma }\varphi {\frac {\partial {\mathcal {L}}}{\partial \varphi _{,\nu }}}=\left({\frac {\partial {\mathcal {L}}}{\partial \varphi _{,\nu }}}\right)\cdot \varphi _{,\mu }-\delta _{\mu }^{\nu }{\mathcal {L}}}$

with

${\displaystyle T_{\mu }{}^{\nu }{}_{,\nu }=0}$

(we relabelled ${\displaystyle \mu }$ as ${\displaystyle \sigma }$ at an intermediate step to avoid conflict). (However, the ${\displaystyle T}$ obtained in this way may differ from the symmetric tensor used as the source term in general relativity; see 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.[11] 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|2 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* that leave |ψ|2 unchanged, such as

${\displaystyle \psi \rightarrow e^{i\theta }\psi \ ,\ \psi ^{*}\rightarrow e^{-i\theta }\psi ^{*}~,}$

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 and −*, respectively. A specific example is the Klein–Gordon equation, the relativistically correct version of the Schrödinger equation for spinless particles, which has the Lagrangian density

${\displaystyle L=\psi _{,\nu }\psi _{,\mu }^{*}\eta ^{\nu \mu }+m^{2}\psi \psi ^{*}.}$

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

${\displaystyle j^{\nu }=i\left({\frac {\partial \psi }{\partial x^{\mu }}}\psi ^{*}-{\frac {\partial \psi ^{*}}{\partial x^{\mu }}}\psi \right)\eta ^{\nu \mu }~,}$

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 symmetries of physics.

## Dérivations

### One independent variable

Consider the simplest case, a system with one independent variable, time. Suppose the dependent variables q are such that the action integral

${\displaystyle I=\int _{t_{1}}^{t_{2}}L[\mathbf {q} [t],{\dot {\mathbf {q} }}[t],t]\,dt}$

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

${\displaystyle {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}[t]={\frac {\partial L}{\partial \mathbf {q} }}[t].}$

And suppose that the integral is invariant under a continuous symmetry. Mathematically such a symmetry is represented as a flow, φ, which acts on the variables as follows

{\displaystyle {\begin{aligned}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{aligned}}}

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.

${\displaystyle {\dot {\mathbf {q} }}[t]\rightarrow {\dot {\mathbf {q} }}'[t']={\frac {d}{dt}}\varphi [\mathbf {q} [t],\varepsilon ]={\frac {\partial \varphi }{\partial \mathbf {q} }}[\mathbf {q} [t'-\varepsilon T],\varepsilon ]{\dot {\mathbf {q} }}[t'-\varepsilon T].}$

The action integral flows to

{\displaystyle {\begin{aligned}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{aligned}}}

which may be regarded as a function of ε. Calculating the derivative at ε' = 0 and using Leibniz's rule, we get

{\displaystyle {\begin{aligned}0={\frac {dI'}{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{aligned}}}

Notice that the Euler–Lagrange equations imply

{\displaystyle {\begin{aligned}{\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{aligned}}}

Substituting this into the previous equation, one gets

{\displaystyle {\begin{aligned}0={\frac {dI'}{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{aligned}}}

Again using the Euler–Lagrange equations we get

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \varepsilon }}\right)=\left({\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right){\frac {\partial \varphi }{\partial \varepsilon }}+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial ^{2}\varphi }{\partial \varepsilon \partial \mathbf {q} }}{\dot {\mathbf {q} }}={\frac {\partial L}{\partial \mathbf {q} }}{\frac {\partial \varphi }{\partial \varepsilon }}+{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial ^{2}\varphi }{\partial \varepsilon \partial \mathbf {q} }}{\dot {\mathbf {q} }}.}$

Substituting this into the previous equation, one gets

{\displaystyle {\begin{aligned}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{aligned}}}

From which one can see that

${\displaystyle \left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \mathbf {q} }}{\dot {\mathbf {q} }}-L\right)T-{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \varepsilon }}}$

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

${\displaystyle \left({\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\dot {\mathbf {q} }}-L\right)T-{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}{\frac {\partial \varphi }{\partial \varepsilon }}.}$

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.

### Field-theoretic derivation

Noether's theorem may also be derived for tensor fields φA 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 xμ where the index μ ranges over time (μ = 0) and three spatial dimensions (μ = 1, 2, 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

${\displaystyle x^{\mu }\rightarrow \xi ^{\mu }=x^{\mu }+\delta x^{\mu }}$

whereas the transformation of the field variables is expressed as

${\displaystyle \varphi ^{A}\rightarrow \alpha ^{A}\left(\xi ^{\mu }\right)=\varphi ^{A}\left(x^{\mu }\right)+\delta \varphi ^{A}\left(x^{\mu }\right)\,.}$

By this definition, the field variations δφA result from two factors: intrinsic changes in the field themselves and changes in coordinates, since the transformed field αA depends on the transformed coordinates ξμ. To isolate the intrinsic changes, the field variation at a single point xμ may be defined

${\displaystyle \alpha ^{A}\left(x^{\mu }\right)=\varphi ^{A}\left(x^{\mu }\right)+{\bar {\delta }}\varphi ^{A}\left(x^{\mu }\right)\,.}$

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, which is defined as the integral of the Lagrangian density over the given region of spacetime. Expressed mathematically, this assumption may be written as

${\displaystyle \int _{\Omega ^{\prime }}L\left(\alpha ^{A},{\alpha ^{A}}_{,\nu },\xi ^{\mu }\right)d^{4}\xi -\int _{\Omega }L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)d^{4}x=0}$

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

${\displaystyle {\varphi ^{A}}_{,\sigma }={\frac {\partial \varphi ^{A}}{\partial x^{\sigma }}}\,.}$

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

${\displaystyle \int _{\Omega }\left\{\left[L\left(\alpha ^{A},{\alpha ^{A}}_{,\nu },x^{\mu }\right)-L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)\right]+{\frac {\partial }{\partial x^{\sigma }}}\left[L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right]\right\}d^{4}x=0\,.}$

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

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

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 commute

${\displaystyle {\bar {\delta }}{\varphi ^{A}}_{,\sigma }={\bar {\delta }}{\frac {\partial \varphi ^{A}}{\partial x^{\sigma }}}={\frac {\partial }{\partial x^{\sigma }}}\left({\bar {\delta }}\varphi ^{A}\right)\,.}$

Using the Euler–Lagrange field equations

${\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}\right)={\frac {\partial L}{\partial \varphi ^{A}}}}$

the difference in Lagrangians can be written neatly as

{\displaystyle {\begin{aligned}&\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{aligned}}}

Thus, the change in the action can be written as

${\displaystyle \int _{\Omega }{\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}\varphi ^{A}+L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}d^{4}x=0\,.}$

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

${\displaystyle {\frac {\partial }{\partial x^{\sigma }}}\left\{{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\bar {\delta }}\varphi ^{A}+L\left(\varphi ^{A},{\varphi ^{A}}_{,\nu },x^{\mu }\right)\delta x^{\sigma }\right\}=0\,.}$

For any combination of the various symmetry transformations, the perturbation can be written

{\displaystyle {\begin{aligned}\delta x^{\mu }&=\varepsilon X^{\mu }\\\delta \varphi ^{A}&=\varepsilon \Psi ^{A}={\bar {\delta }}\varphi ^{A}+\varepsilon {\mathcal {L}}_{X}\varphi ^{A}\end{aligned}}}

where ${\displaystyle {\mathcal {L}}_{X}\varphi ^{A}}$ is the Lie derivative of φA in the Xμ direction. When φA is a scalar or ${\displaystyle {X^{\mu }}_{,\nu }=0}$,

${\displaystyle {\mathcal {L}}_{X}\varphi ^{A}={\frac {\partial \varphi ^{A}}{\partial x^{\mu }}}X^{\mu }\,.}$

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

${\displaystyle {\bar {\delta }}\varphi ^{A}=\varepsilon \Psi ^{A}-\varepsilon {\mathcal {L}}_{X}\varphi ^{A}\,.}$

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

${\displaystyle {\frac {\partial }{\partial x^{\sigma }}}j^{\sigma }=0}$

where the conserved current equals

${\displaystyle j^{\sigma }=\left[{\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}{\mathcal {L}}_{X}\varphi ^{A}-L\,X^{\sigma }\right]-\left({\frac {\partial L}{\partial {\varphi ^{A}}_{,\sigma }}}\right)\Psi ^{A}\,.}$

### Manifold/fiber bundle derivation

Suppose we have an n-dimensional oriented Riemannian manifold, M and a target manifold T. Let ${\displaystyle {\mathcal {C}}}$ be the configuration space of smooth functions from M to T. (More generally, we can have smooth sections of a fiber bundle over M.)

Examples of this M in physics include:

• In classical mechanics, in the Hamiltonian formulation, M is the one-dimensional manifold ${\displaystyle \mathbb {R} }$, representing time and the target space is the cotangent bundle of space of generalized positions.
• In 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-valued scalar fields, ${\displaystyle \varphi _{1},\ldots ,\varphi _{m}}$, then the target manifold is ${\displaystyle \mathbb {R} ^{m}}$. If the field is a real vector field, then the target manifold is isomorphic to ${\displaystyle \mathbb {R} ^{3}}$.

Now suppose there is a functional

${\displaystyle {\mathcal {S}}:{\mathcal {C}}\rightarrow \mathbb {R} ,}$

called the action. (It takes values into ${\displaystyle \mathbb {R} }$, rather than ${\displaystyle \mathbb {C} }$; 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. We assume ${\displaystyle {\mathcal {S}}[\varphi ]}$ is the integral over M of a function

${\displaystyle {\mathcal {L}}(\varphi ,\partial _{\mu }\varphi ,x)}$

called the Lagrangian density, depending on φ, its derivative and the position. In other words, for φ in ${\displaystyle {\mathcal {C}}}$

${\displaystyle {\mathcal {S}}[\varphi ]\,=\,\int _{M}{\mathcal {L}}[\varphi (x),\partial _{\mu }\varphi (x),x]\,d^{n}x.}$

Suppose we are given boundary conditions, i.e., a specification of the value of φ at the boundary if M is compact, or some limit on φ as x approaches ∞. Then the subspace of ${\displaystyle {\mathcal {C}}}$ consisting of functions φ such that all functional derivatives of ${\displaystyle {\mathcal {S}}}$ at φ are zero, that is:

${\displaystyle {\frac {\delta {\mathcal {S}}[\varphi ]}{\delta \varphi (x)}}\approx 0}$

and that φ 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 ${\displaystyle {\mathcal {C}}}$, generated by a functional derivation, Q such that

${\displaystyle Q\left[\int _{N}{\mathcal {L}}\,\mathrm {d} ^{n}x\right]\approx \int _{\partial N}f^{\mu }[\varphi (x),\partial \varphi ,\partial \partial \varphi ,\ldots ]\,ds_{\mu }}$

for all compact submanifolds N or in other words,

${\displaystyle Q[{\mathcal {L}}(x)]\approx \partial _{\mu }f^{\mu }(x)}$

for all x, where we set

${\displaystyle {\mathcal {L}}(x)={\mathcal {L}}[\varphi (x),\partial _{\mu }\varphi (x),x].}$

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 symmetry Lie group.

Now, for any N, because of the Euler–Lagrange theorem, on shell (and only on-shell), we have

{\displaystyle {\begin{aligned}Q\left[\int _{N}{\mathcal {L}}\,\mathrm {d} ^{n}x\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} ^{n}x+\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{aligned}}}

Since this is true for any N, we have

${\displaystyle \partial _{\mu }\left[{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}Q[\varphi ]-f^{\mu }\right]\approx 0.}$

But this is the continuity equation for the current ${\displaystyle J^{\mu }}$ defined by:[12]

${\displaystyle J^{\mu }\,=\,{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\varphi )}}Q[\varphi ]-f^{\mu },}$

which is called the Noether current associated with the symmetry. The continuity equation tells us that if we integrate this current over a space-like slice, we get a conserved quantity called the Noether charge (provided, of course, if M is noncompact, the currents fall off sufficiently fast at infinity).

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

${\displaystyle \int _{\partial N}J^{\mu }ds_{\mu }\approx 0.}$

The quantum analogs of Noether's theorem involving expectation values (e.g., ${\textstyle \left\langle \int d^{4}x~\partial \cdot {\textbf {J}}\right\rangle =0}$) probing off shell quantities as well are the Ward–Takahashi identities.

### Generalization to Lie algebras

Suppose we have two symmetry derivations Q1 and Q2. Then, [Q1Q2] is also a symmetry derivation. Let's see this explicitly. Let's say

${\displaystyle Q_{1}[{\mathcal {L}}]\approx \partial _{\mu }f_{1}^{\mu }}$

and

${\displaystyle Q_{2}[{\mathcal {L}}]\approx \partial _{\mu }f_{2}^{\mu }}$

Then,

${\displaystyle [Q_{1},Q_{2}][{\mathcal {L}}]=Q_{1}[Q_{2}[{\mathcal {L}}]]-Q_{2}[Q_{1}[{\mathcal {L}}]]\approx \partial _{\mu }f_{12}^{\mu }}$

where f12 = Q1[f2μ] − Q2[f1μ]. So,

${\displaystyle j_{12}^{\mu }=\left({\frac {\partial }{\partial (\partial _{\mu }\varphi )}}{\mathcal {L}}\right)(Q_{1}[Q_{2}[\varphi ]]-Q_{2}[Q_{1}[\varphi ]])-f_{12}^{\mu }.}$

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 ≈ 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 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] ≈ 0 for every ε, or more compactly, q(x)[S] ≈ 0 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 φ and its first derivatives. Also, assume

${\displaystyle Q[{\mathcal {L}}]\approx \partial _{\mu }f^{\mu }}$

Then,

{\displaystyle {\begin{aligned}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{aligned}}}

for all ${\displaystyle \varepsilon }$.

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

${\displaystyle \partial _{\mu }\left[f^{\mu }-\left[{\frac {\partial }{\partial (\partial _{\mu }\varphi )}}{\mathcal {L}}\right]Q[\varphi ]-2\left[{\frac {\partial }{\partial (\partial _{\mu }\partial _{\nu }\varphi )}}{\mathcal {L}}\right]\partial _{\nu }Q[\varphi ]+\partial _{\nu }\left[\left[{\frac {\partial }{\partial (\partial _{\mu }\partial _{\nu }\varphi )}}{\mathcal {L}}\right]Q[\varphi ]\right]-\,\dotsm \right]\approx 0.}$

## Exemples

### Example 1: Conservation of energy

Looking at the specific case of a Newtonian particle of mass m, coordinate x, moving under the influence of a potential V, coordinatized by time t. The action, S, is:

{\displaystyle {\begin{aligned}{\mathcal {S}}[x]&=\int L\left[x(t),{\dot {x}}(t)\right]\,dt\\&=\int \left({\frac {m}{2}}\sum _{i=1}^{3}{\dot {x}}_{i}^{2}-V(x(t))\right)\,dt.\end{aligned}}}

The first term in the brackets is the kinetic energy of the particle, whilst the second is its potential energy. Consider the generator of time translations Q = d/dt. In other words, ${\displaystyle Q[x(t)]={\dot {x}}(t)}$. The coordinate x has an explicit dependence on time, whilst V does not; consequently:

${\displaystyle Q[L]={\frac {d}{dt}}\left[{\frac {m}{2}}\sum _{i}{\dot {x}}_{i}^{2}-V(x)\right]=m\sum _{i}{\dot {x}}_{i}{\ddot {x}}_{i}-\sum _{i}{\frac {\partial V(x)}{\partial x_{i}}}{\dot {x}}_{i}}$

so we can set

${\displaystyle L={\frac {m}{2}}\sum _{i}{\dot {x}}_{i}^{2}-V(x).}$

Then,

{\displaystyle {\begin{aligned}j&=\sum _{i=1}^{3}{\frac {\partial L}{\partial {\dot {x}}_{i}}}Q[x_{i}]-L\\&=m\sum _{i}{\dot {x}}_{i}^{2}-\left[{\frac {m}{2}}\sum _{i}{\dot {x}}_{i}^{2}-V(x)\right]\\[3pt]&={\frac {m}{2}}\sum _{i}{\dot {x}}_{i}^{2}+V(x).\end{aligned}}}

The right hand side is the energy, and Noether's theorem states that ${\displaystyle dj/dt=0}$ (i.e. the principle of conservation of energy is a consequence of invariance under time translations).

More generally, if the Lagrangian does not depend explicitly on time, the quantity

${\displaystyle \sum _{i=1}^{3}{\frac {\partial L}{\partial {\dot {x}}_{i}}}{\dot {x_{i}}}-L}$

(called the Hamiltonian) is conserved.

### Example 2: Conservation of center of momentum

Still considering 1-dimensional time, let

{\displaystyle {\begin{aligned}{\mathcal {S}}\left[{\vec {x}}\right]&=\int {\mathcal {L}}\left[{\vec {x}}(t),{\dot {\vec {x}}}(t)\right]dt\\[3pt]&=\int \left[\sum _{\alpha =1}^{N}{\frac {m_{\alpha }}{2}}\left({\dot {\vec {x}}}_{\alpha }\right)^{2}-\sum _{\alpha <\beta }V_{\alpha \beta }\left({\vec {x}}_{\beta }-{\vec {x}}_{\alpha }\right)\right]dt,\end{aligned}}}

or ${\displaystyle N}$ Newtonian particles where the potential only depends pairwise upon the relative displacement.

For ${\displaystyle {\vec {Q}}}$, consider the generator of Galilean transformations (i.e. a change in the frame of reference). In other words,

${\displaystyle Q_{i}\left[x_{\alpha }^{j}(t)\right]=t\delta _{i}^{j}.}$

And

{\displaystyle {\begin{aligned}Q_{i}[{\mathcal {L}}]&=\sum _{\alpha }m_{\alpha }{\dot {x}}_{\alpha }^{i}-\sum _{\alpha <\beta }\partial _{i}V_{\alpha \beta }\left({\vec {x}}_{\beta }-{\vec {x}}_{\alpha }\right)(t-t)\\&=\sum _{\alpha }m_{\alpha }{\dot {x}}_{\alpha }^{i}.\end{aligned}}}

This has the form of ${\textstyle {\frac {d}{dt}}\sum _{\alpha }m_{\alpha }x_{\alpha }^{i}}$ so we can set

${\displaystyle {\vec {f}}=\sum _{\alpha }m_{\alpha }{\vec {x}}_{\alpha }.}$

Then,

{\displaystyle {\begin{aligned}{\vec {j}}&=\sum _{\alpha }\left({\frac {\partial }{\partial {\dot {\vec {x}}}_{\alpha }}}{\mathcal {L}}\right)\cdot {\vec {Q}}\left[{\vec {x}}_{\alpha }\right]-{\vec {f}}\\[6pt]&=\sum _{\alpha }\left(m_{\alpha }{\dot {\vec {x}}}_{\alpha }t-m_{\alpha }{\vec {x}}_{\alpha }\right)\\[3pt]&={\vec {P}}t-M{\vec {x}}_{CM}\end{aligned}}}

where ${\displaystyle {\vec {P}}}$ is the total momentum, M is the total mass and ${\displaystyle {\vec {x}}_{CM}}$ is the center of mass. Noether's theorem states:

${\displaystyle {d{\vec {j}}}/dt=0\Rightarrow {\vec {P}}-M{\dot {\vec {x}}}_{CM}=0.}$

### Example 3: Conformal transformation

Both examples 1 and 2 are over a 1-dimensional manifold (time). An example involving spacetime is a conformal transformation of a massless real scalar field with a quartic potential in (3 + 1)-Minkowski spacetime.

{\displaystyle {\begin{aligned}{\mathcal {S}}[\varphi ]&=\int {\mathcal {L}}\left[\varphi (x),\partial _{\mu }\varphi (x)\right]d^{4}x\\[4pt]&=\int \left({\frac {1}{2}}\partial ^{\mu }\varphi \partial _{\mu }\varphi -\lambda \varphi ^{4}\right)d^{4}x\end{aligned}}}

For Q, consider the generator of a spacetime rescaling. In other words,

${\displaystyle Q[\varphi (x)]=x^{\mu }\partial _{\mu }\varphi (x)+\varphi (x).}$

The second term on the right hand side is due to the "conformal weight" of ${\displaystyle \varphi }$. And

${\displaystyle Q[{\mathcal {L}}]=\partial ^{\mu }\varphi \left(\partial _{\mu }\varphi +x^{\nu }\partial _{\mu }\partial _{\nu }\varphi +\partial _{\mu }\varphi \right)-4\lambda \varphi ^{3}\left(x^{\mu }\partial _{\mu }\varphi +\varphi \right).}$

This has the form of

${\displaystyle \partial _{\mu }\left[{\frac {1}{2}}x^{\mu }\partial ^{\nu }\varphi \partial _{\nu }\varphi -\lambda x^{\mu }\varphi ^{4}\right]=\partial _{\mu }\left(x^{\mu }{\mathcal {L}}\right)}$

(where we have performed a change of dummy indices) so set

${\displaystyle f^{\mu }=x^{\mu }{\mathcal {L}}.}$

Then

{\displaystyle {\begin{aligned}j^{\mu }&=\left[{\frac {\partial }{\partial (\partial _{\mu }\varphi )}}{\mathcal {L}}\right]Q[\varphi ]-f^{\mu }\\[6pt]&=\partial ^{\mu }\varphi \left(x^{\nu }\partial _{\nu }\varphi +\varphi \right)-x^{\mu }\left({\frac {1}{2}}\partial ^{\nu }\varphi \partial _{\nu }\varphi -\lambda \varphi ^{4}\right).\end{aligned}}}

Noether's theorem states that ${\displaystyle \partial _{\mu }j^{\mu }=0}$ (as one may explicitly check by substituting the Euler–Lagrange equations into the left hand side).

If one tries to find the Ward–Takahashi analog of this equation, one runs into a problem because of anomalies.

## Applications

Application of Noether's theorem allows physicists to gain powerful insights into any general theory in physics, by just analyzing the various transformations that would make the form of the laws involved invariant. For example:

• the invariance of physical systems with respect to spatial translation (in other words, that the laws of physics do not vary with locations in space) gives the law of conservation of linear momentum;
• invariance with respect to rotation gives the law of conservation of angular momentum;
• invariance with respect to time translation gives the well-known law of conservation of energy

In quantum field theory, the analog to Noether's theorem, the Ward–Takahashi identity, yields further conservation laws, such as the conservation of electric charge from the invariance with respect to a change in the phase factor of the complex field of the charged particle and the associated gauge of the electric potential and vector potential.

The Noether charge is also used in calculating the entropy of stationary black holes.[13]

## Voir également

• Conservation law
• Charge (physics)
• Gauge symmetry
• Gauge symmetry (mathematics)
• Invariant (physics)
• Goldstone boson
• Symmetry in physics

## Remarques

1. ^ This is sometimes referred to as Noether's first theorem, see Noether's second theorem.
2. ^ Noether, E. (1918). "Invariante Variationsprobleme". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse. 1918: 235–257.
3. ^ Cosserat, E.; Cosserat, F. (1909). Théorie des corps déformables. Paris: Hermann.
4. ^ Thompson, W.J. (1994). Angular Momentum: an illustrated guide to rotational symmetries for physical systems. 1. Wiley. p. 5. ISBN 0-471-55264-X.
5. ^ The term "Noether charge" occurs in Seligman, Group theory and its applications in physics, 1980: Latin American School of Physics, Mexico City, American Institute of Physics, 1981. It comes enters wider use during the 1980s, e.g. by G. Takeda in: Errol Gotsman, Gerald Tauber (eds.) From SU(3) to Gravity: Festschrift in Honor of Yuval Ne'eman, 1985, p. 196.
6. ^ Nina Byers (1998) "E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws". In Proceedings of a Symposium on the Heritage of Emmy Noether, held on 2–4 December 1996, at the Bar-Ilan University, Israel, Appendix B.
7. ^ Lanczos 1970, pp. 401–403
8. ^ Lanczos 1970, pp. 403–404
9. ^ a b Goldstein 1980, pp. 592–593
10. ^ Lanczos 1970, pp. 404–405
11. ^ Goldstein 1980, pp. 593–594
12. ^ Michael E. Peskin; Daniel V. Schroeder (1995). An Introduction to Quantum Field Theory. Basic Books. p. 18. ISBN 0-201-50397-2.
13. ^ Vivek Iyer; Wald (1995). "A comparison of Noether charge and Euclidean methods for Computing the Entropy of Stationary Black Holes". Physical Review D. 52 (8): 4430–9. arXiv:gr-qc/9503052. Bibcode:1995PhRvD..52.4430I. doi:10.1103/PhysRevD.52.4430. PMID 10019667. S2CID 2588285.

## Les références

• Badin, Gualtiero; Crisciani, Fulvio (2018). Variational Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws -. Springer. p. 218. doi:10.1007/978-3-319-59695-2. ISBN 978-3-319-59694-5. S2CID 125902566.
• Goldstein, Herbert (1980). Classical Mechanics (2nd ed.). Reading, MA: Addison-Wesley. pp. 588–596. ISBN 0-201-02918-9.
• Johnson, Tristan (2016). "Noether's Theorem: Symmetry and Conservation". Honors Theses. Union College. Retrieved 28 August 2020.
• Kosmann-Schwarzbach, Yvette (2010). The Noether theorems: Invariance and conservation laws in the twentieth century. Sources and Studies in the History of Mathematics and Physical Sciences. Springer-Verlag. ISBN 978-0-387-87867-6. Online copy.
• Lanczos, C. (1970). The Variational Principles of Mechanics (4th ed.). New York: Dover Publications. pp. 401–5. ISBN 0-486-65067-7.
• Moser, Seth (21 April 2020). "Understanding Noether's Theorem by Visualizing the Lagrangian". Physics Capstone Projects: 1–12. Retrieved 28 August 2020.
• Olver, Peter (1993). Applications of Lie groups to differential equations. Graduate Texts in Mathematics. 107 (2nd ed.). Springer-Verlag. ISBN 0-387-95000-1.
• Sardanashvily, G. (2016). Noether's Theorems. Applications in Mechanics and Field Theory. Springer-Verlag. ISBN 978-94-6239-171-0.