Movatterモバイル変換

[0]ホーム

Jump to content

Noether's theorem

Edit links

From Wikipedia, the free encyclopedia

(Redirected fromNoether charge)

Statement relating differentiable symmetries to conserved quantities

This article is about Emmy Noether's first theorem, which derives conserved quantities from symmetries. For other uses, seeNoether's theorem (disambiguation).

First page ofEmmy Noether's article "Invariante Variationsprobleme" (1918), where she proved her theorem

Part of a series of articles about

Calculus

\int _{a}^{b}f'(t)\,dt=f(b)-f(a)

Fundamental theorem

Differential

Definitions
Derivative (generalizations) Differential infinitesimal of a function total
Concepts
Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem
Rules and identities
Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds

Integral

Definitions
Lists of integrals Integral transform Leibniz integral rule
Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions
Integration by
Parts Discs Cylindrical shells Substitution (trigonometric,tangent half-angle,Euler) Euler's formula Partial fractions (Heaviside's method) Changing order Reduction formulae Differentiating under the integral sign Risch algorithm

Series

Convergence tests
Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor
Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel

Vector

Theorems
Gradient Divergence Curl Laplacian Directional derivative Identities
Gradient Green's Stokes' Divergence Generalized Stokes Helmholtz decomposition

Multivariable

Formalisms
Matrix Tensor Exterior Geometric
Definitions
Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian

Advanced

Specialized

Miscellanea

Noether's theorem states that everycontinuous symmetry of theaction of a physical system withconservative forces has a correspondingconservation law. This is the first of two theorems (seeNoether's second theorem) published by the mathematicianEmmy Noether in 1918.^[1] The action of a physical system is theintegral over time of aLagrangian function, from which the system's behavior can be determined by theprinciple of least action. This theorem applies to continuous and smoothsymmetries of physical space. Noether's formulation is quite general and has been applied across classical mechanics, high energy physics, and recentlystatistical mechanics.^[2]

Noether's theorem is used intheoretical physics and thecalculus of variations. It reveals the fundamental relation between the symmetries of a physical system and the conservation laws. It also made modern theoretical physicists much more focused on symmetries of physical systems. A generalization of the formulations onconstants of motion in Lagrangian andHamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g., systems with aRayleigh dissipation function). In particular,dissipative systems withcontinuous symmetries need not have a corresponding conservation law.^[3]

Basic illustrations and background

[edit]

As an illustration, if a physical system behaves the same regardless of how it is oriented in space (that is, it isinvariant), itsLagrangian is symmetric under continuous rotation: from this symmetry, Noether's theorem dictates that theangular momentum of the system be conserved, as a consequence of its laws of motion.^[4]^: 126 The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. It is the laws of its motion that are symmetric.

As another example, if a physical process exhibits the same outcomes regardless of place or time, then its Lagrangian is symmetric under continuous translations in space and time respectively: by Noether's theorem, these symmetries account for theconservation laws oflinear momentum andenergy within this system, respectively.^[5]^: 23^[6]^: 261

Noether's theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows investigators to determine the conserved quantities (invariants) from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians with given invariants, to describe a physical system.^[4]^: 127 As an illustration, suppose that a physical theory is proposed which conserves a quantityX. A researcher can calculate the types of Lagrangians that conserveX through a continuous symmetry. Due to Noether's theorem, the properties of these Lagrangians provide further criteria to understand the implications and judge the fitness of the new theory.

There are numerous versions of Noether's theorem, with varying degrees of generality. There are natural quantum counterparts of this theorem, expressed in theWard–Takahashi identities. Generalizations of Noether's theorem tosuperspaces also exist.^[7]

Informal statement of the theorem

[edit]

All fine technical points aside, Noether's theorem can be stated informally as:

If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.^[8]

A more sophisticated version of the theorem involving fields states that:

To every continuoussymmetry generated by local actions there corresponds aconserved current and vice versa.

The word "symmetry" in the above statement refers more precisely to thecovariance of the form that a physical law takes with respect to a one-dimensionalLie group of transformations satisfying certain technical criteria. Theconservation law of aphysical quantity is usually expressed as acontinuity equation.

The formal proof of the theorem utilizes the condition of invariance to derive an expression for a current associated with a conserved physical quantity. In modern terminology, the conserved quantity is called theNoether charge, while the flow carrying that charge is called theNoether current. The Noether current is definedup to asolenoidal (divergenceless) vector field.

In the context of gravitation,Felix Klein's statement of Noether's theorem for actionI stipulates for the invariants:^[9]

If an integral I is invariant under a continuous groupG_ρ withρ parameters, thenρ linearly independent combinations of the Lagrangian expressions are divergences.

Brief illustration and overview of the concept

[edit]

Plot illustrating Noether's theorem for a coordinate-wise symmetry

The main idea behind Noether's theorem is most easily illustrated by a system with one coordinate $q {\displaystyle q}$ and a continuous symmetry $\varphi :q\mapsto q+\delta q$ (gray arrows on the diagram).

Consider any trajectory $q(t)$ (bold on the diagram) that satisfies the system'slaws of motion. That is, theaction $S {\displaystyle S}$ governing this system isstationary on this trajectory, i.e. does not change under any localvariation of the trajectory. In particular it would not change under a variation that applies the symmetry flow $\varphi$ on a time segment[t₀,t₁] and is motionless outside that segment. To keep the trajectory continuous, we use "buffering" periods of small time $\tau$ to transition between the segments gradually.

The total change in the action $S {\displaystyle S}$ now comprises changes brought by every interval in play. Parts where variation itself vanishes, i.e outside $[t_{0},t_{1}]$ , bring no $\Delta S$ . The middle part does not change the action either, because its transformation $\varphi$ is a symmetry and thus preserves the Lagrangian $L {\displaystyle L}$ and the action ${\textstyle S=\int L}$ . The only remaining parts are the "buffering" pieces. In these regions both the coordinate $q {\displaystyle q}$ and velocity ${\dot {q}}$ change, but ${\dot {q}}$ changes by $\delta q/\tau$ , and the change $\delta q$ in the coordinate is negligible by comparison since the time span $\tau$ of the buffering is small (taken to the limit of 0), so $\delta q/\tau \gg \delta q$ . So the regions contribute mostly through their "slanting" ${\dot {q}}\rightarrow {\dot {q}}\pm \delta q/\tau$ .

That changes the Lagrangian by $\Delta L\approx {\bigl (}\partial L/\partial {\dot {q}}{\bigr )}\Delta {\dot {q}}$ , which integrates to $\Delta S=\int \Delta L\approx \int {\frac {\partial L}{\partial {\dot {q}}}}\Delta {\dot {q}}\approx \int {\frac {\partial L}{\partial {\dot {q}}}}\left(\pm {\frac {\delta q}{\tau }}\right)\approx \ \pm {\frac {\partial L}{\partial {\dot {q}}}}\delta q=\pm {\frac {\partial L}{\partial {\dot {q}}}}\varphi .$

These last terms, evaluated around the endpoints $t_{0}$ and $t_{1}$ , should cancel each other in order to make the total change in the action $\Delta S$ be zero, as would be expected if the trajectory is a solution. That is $\left({\frac {\partial L}{\partial {\dot {q}}}}\varphi \right)(t_{0})=\left({\frac {\partial L}{\partial {\dot {q}}}}\varphi \right)(t_{1}),$ meaning the quantity $\left(\partial L/\partial {\dot {q}}\right)\varphi$ is conserved, which is the conclusion of Noether's theorem. For instance if pure translations of $q {\displaystyle q}$ by a constant are the symmetry, then the conserved quantity becomes just $\left(\partial L/\partial {\dot {q}}\right)=p$ , the canonical momentum.

More general cases follow the same idea:

When more coordinates $q_{r}$ undergo a symmetry transformation $q_{r}\mapsto q_{r}+\varphi _{r}$ , their effects add up by linearity to a conserved quantity ${\textstyle \sum _{r}\left(\partial L/\partial {\dot {q}}_{r}\right)\varphi _{r}}$ .
Time invariance implies conservation of energy: Suppose the Lagrangian is invariant to time transformations, $t\mapsto t+T$ . We effect such a transformation with a very small time shift $T\ll \tau$ in the time between $t_{0}+\tau$ and $t_{1}-\tau$ , by stretching the first buffering segment $(t_{0},t_{0}+\tau )$ to $(t_{0},t_{0}+\tau +T)$ and compressing the second buffering segment $(t_{1}-\tau ,t_{1})$ to $(t_{1}-\tau +T,t_{1})$ . Again, the action outside the interval $(t_{0},t_{1})$ and between the buffering segments remains the same. However, the buffering segments each contribute two terms to the change of the action:
$\Delta S\approx \pm \left(TL+\int \sum _{r}{\frac {\partial L}{\partial {\dot {q}}_{r}}}\Delta {\dot {q}}_{r}\right)\approx \pm T\left(L-\sum _{r}{\frac {\partial L}{\partial {\dot {q}}_{r}}}{\dot {q}}_{r}\right).$
The first term $T L {\displaystyle TL}$ is due to the changing sizes of the "buffering" segments. The first segment changes its size from $\tau$ to $\tau +T$ , and the second segment form $\tau$ to $\tau -T$ . Therefore, the integral over the first segment changes by $+TL(t_{0})$ and the integral over the second segment changes by $-TL(t_{1})$ . The second term is due to the time dilation by a factor $(\tau +T)/\tau$ in the first segment and by $(\tau -T)/\tau$ in the second segment, which changes all time derivatives by the dilation factor. These time dilations change ${\dot {q}}_{r}$ to ${\dot {q}}_{r}\mp (T/\tau ){\dot {q}}_{r}$ (to first order in $T/\tau$ ) in the first (-) and second (+) segment. Together they add to the conserved action S a term ${\textstyle \pm T\left(L-\sum _{r}\left(\partial L/\partial {\dot {q}}_{r}\right){\dot {q}}_{r}\right)}$ for the first (+) and second (-) segment. Since the change of action must be zero, $\Delta S=0$ , we conclude that the total energy $\sum _{r}{\frac {\partial L}{\partial {\dot {q}}_{r}}}{\dot {q}}_{r}-L$ must be equal at times $t_{0}$ and $t_{1}$ , so total energy is conserved.
Finally, when instead of a trajectory $q(t)$ entire fields $\psi (q_{r},t)$ are considered, the argument replaces
- the interval $[t_{0},t_{1}]$ with a bounded region $U {\displaystyle U}$ of the $(q_{r},t)$ -domain,
- the endpoints $t_{0}$ and $t_{1}$ with the boundary $\partial U$ of the region,
- and its contribution to $\Delta S$ is interpreted as a flux of aconserved current $j_{r}$ , that is built in a way analogous to the prior definition of a conserved quantity.
Now, the zero contribution of the "buffering" $\partial U$ to $\Delta S$ is interpreted as vanishing of the total flux of the current $j_{r}$ through the $\partial U$ . That is the sense in which it is conserved: how much is "flowing" in, just as much is "flowing" out.

Historical context

[edit]

Main articles:Constant of motion,conservation law, andconserved current

Aconservation law states that some quantityX in the mathematical description of a system's evolution remains constant throughout its motion – it is aninvariant. Mathematically, the rate of change ofX (itsderivative with respect totime) is zero,

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

Such quantities are said to be conserved; they are often calledconstants of motion (although motionper 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 weremomentum andkinetic energy, which were proposed in the 17th century byRené Descartes andGottfried Leibniz on the basis ofcollision 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 ofNewton's laws of motion. According togeneral relativity, the conservation laws of linear momentum, energy and angular momentum are only exactly true globally when expressed in terms of the sum of thestress–energy tensor (non-gravitational stress–energy) and theLandau–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 covariantdivergence of thestress–energy tensor. Another important conserved quantity, discovered in studies of thecelestial mechanics of astronomical bodies, is theLaplace–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 ofLagrangian mechanics, which is related to theprinciple of least action. In this approach, the state of the system can be described by any type ofgeneralized coordinatesq; the laws of motion need not be expressed in aCartesian coordinate system, as was customary in Newtonian mechanics. Theaction is defined as the time integralI of a function known as theLagrangian L

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

where the dot overq signifies the rate of change of the coordinatesq,

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

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

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

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

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

where the momentum

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

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

Thus, the absence of theignorable coordinateq_k from the Lagrangian implies that the Lagrangian is unaffected by changes or transformations ofq_k; the Lagrangian is invariant, and is said to exhibit asymmetry 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 byWilliam Rowan Hamilton. For example, he developed a theory ofcanonical 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 theHamilton–Jacobi equation.

Emmy Noether's work on the invariance theorem began in 1915 when she was helpingFelix Klein and David Hilbert with their work related toAlbert Einstein's theory of general relativity^[10]^: 31 By March 1918 she had most of the key ideas for the paper which would be published later in the year.^[11]^: 81

Mathematical expression

[edit]

Simple form using perturbations

[edit]

The essence of Noether's theorem is generalizing the notion of ignorable coordinates.

One can assume that the LagrangianL defined above is invariant under small perturbations (warpings) of the time variablet and thegeneralized coordinatesq. One may write

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

where the perturbationsδt andδq are both small, but variable. For generality, assume there are (say)N suchsymmetry transformations of the action, i.e. transformations leaving the action unchanged; labelled by an indexr = 1, 2, 3, ..., N.

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

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

whereε_r areinfinitesimal parameter coefficients corresponding to each:

generatorT_r oftime evolution, and
generatorQ_r of the generalized coordinates.

For translations,Q_r is a constant with units oflength; for rotations, it is an expression linear in the components ofq, and the parameters make up anangle.

Using these definitions,Noether showed that theN quantities

\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}

are conserved (constants of motion).

Examples

[edit]

I. Time invariance

For illustration, consider a Lagrangian that does not depend on time, i.e., that is invariant (symmetric) under changest →t + δt, without any change in the coordinatesq. In this case,N = 1,T = 1 andQ = 0; the corresponding conserved quantity is the totalenergyH^[12]^: 401

H={\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot {\dot {\mathbf {q} }}-L.

II. Translational invariance

Consider a Lagrangian which does not depend on an ("ignorable", as above) coordinateq_k; so it is invariant (symmetric) under changesq_k →q_k +δq_k. In that case,N = 1,T = 0, andQ_k = 1; the conserved quantity is the corresponding linearmomentump_k^[12]^{: 403–404}

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

Inspecial andgeneral relativity, these two conservation laws can be expressed eitherglobally (as it is done above), orlocally as a continuity equation. The global versions can be united into a single global conservation law: the conservation of the energy-momentum 4-vector. The local versions of energy and momentum conservation (at any point in space-time) can also be united, into the conservation of a quantity definedlocally at the space-time point: thestress–energy tensor^[13]^: 592(this will be derived in the next section).

III. Rotational invariance

The conservation of theangular momentumL =r ×p is analogous to its linear momentum counterpart.^[12]^{: 404–405} 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 angleδθ about an axisn; such a rotation transforms theCartesian coordinates by the equation

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

Since time is not being transformed,T = 0, andN = 1. Takingδθ as theε parameter and the Cartesian coordinatesr as the generalized coordinatesq, the correspondingQ variables are given by

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

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

{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\cdot \mathbf {Q} =\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 momentumL along then axis is conserved. And ifn is arbitrary, i.e., if the system is insensitive to any rotation, then every component ofL is conserved; in short,angular momentum is conserved.

Field theory version

[edit]

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-dimensionalspace–time is now given. Since field theory problems are more common in modern physics thanmechanics 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 differentiablefields $\varphi$ defined over all space and time; for example, the temperature $T(\mathbf {x} ,t)$ would be representative of such a field, being a number defined at every place and time. Theprinciple of least action can be applied to such fields, but the action is now an integral over space and time

{\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 then^th derivative, and can also be formulated usingjet bundles).

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

\varphi \mapsto \varphi +\varepsilon \Psi ,

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

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

since the integral of a divergence becomes a boundary term according to thedivergence theorem. A system described by a given action might have multiple independent symmetries of this type, indexed by $r=1,2,\ldots ,N,$ so the most general symmetry transformation would be written as

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

with the consequence

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

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

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

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

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

\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.

Examples

[edit]

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

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

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

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

and thus Noether's theorem corresponds^[13]^: 592 to the conservation law for thestress–energy tensorT_μ^ν, where we have used $\mu$ in place of $r {\displaystyle r}$ . To wit, by using the expression given earlier, and collecting the four conserved currents (one for each $\mu$ ) into a tensor $T {\displaystyle T}$ , Noether's theorem gives

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

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

(we relabelled $\mu$ as $\sigma$ at an intermediate step to avoid conflict). (However, the $T {\displaystyle T}$ obtained in this way may differ from the symmetric tensor used as the source term in general relativity; seeCanonical stress–energy tensor.)

II. Theelectric charge

The conservation ofelectric charge, by contrast, can be derived by consideringΨ linear in the fieldsφ rather than in the derivatives.^[13]^{: 593–594} Inquantum mechanics, theprobability amplitudeψ(x) of finding a particle at a pointx is a complex fieldφ, because it ascribes acomplex number to every point in space and time. The probability amplitude itself is physically unmeasurable; only the probabilityp = |ψ|² can be inferred from a set of measurements. Therefore, the system is invariant under transformations of theψ field and itscomplex conjugate fieldψ^* that leave |ψ|² unchanged, such as

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

a complex rotation. In the limit when the phaseθ becomes infinitesimally small,δθ, it may be taken as the parameterε, while theΨ are equal toiψ and −iψ*, respectively. A specific example is theKlein–Gordon equation, therelativistically correct version of theSchrödinger equation forspinless particles, which has the Lagrangian density

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

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

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 byHermann Weyl, and is one of the prototypegauge symmetries of physics.

Derivations

[edit]

One independent variable

[edit]

This sectiondoes notcite anysources. Please helpimprove this section byadding citations to reliable sources. Unsourced material may be challenged andremoved.
Find sources: "Noether's theorem" – news ·newspapers ·books ·scholar ·JSTOR(March 2025) (Learn how and when to remove this message)

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

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

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

{\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 aflow,φ, which acts on the variables as follows

{\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}}

whereε is a real variable indicating the amount of flow, andT is a real constant (which could be zero) indicating how much the flow shifts time.

{\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

{\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 ofε. Calculating the derivative atε = 0 and usingLeibniz's rule, we get

{\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

{\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

{\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

{\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

{\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

\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 ${\frac {\partial \varphi }{\partial \mathbf {q} }}=1$ and so the conserved quantity simplifies to

\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.

Geometric derivation

[edit]

The Noether's theorem can be seen as a consequence of thefundamental theorem of calculus (known by various names in physics such as theGeneralized Stokes theorem or theGradient theorem):^[14] for a function ${\textstyle S}$ analytical in a domain ${\textstyle {\cal {D}}}$ , $\int _{\cal {\cal {P}}}dS=0$

Integration path that leads to Noether's theorem

where ${\textstyle {\cal {P}}}$ is a closed path in ${\textstyle {\cal {D}}}$ . Here, thefunction ${\textstyle S(\mathbf {q} ,t)}$ is the actionfunction that is computed by the integration of the Lagrangian over optimal trajectories or equivalently obtained through theHamilton-Jacobi equation. As ${\textstyle \partial S/\partial \mathbf {q} =\mathbf {p} }$ (where ${\textstyle \mathbf {p} }$ is the momentum) and ${\textstyle \partial S/\partial t=-H}$ (where ${\textstyle H}$ is the Hamiltonian), the differential of this function is given by ${\textstyle dS=\mathbf {p} d\mathbf {q} -Hdt}$ .

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: ${\begin{aligned}\mathbf {q'} &=&\mathbf {q} +\epsilon \phi _{\mathbf {q} }(\mathbf {q} ,t)\\t'&=&t+\epsilon \phi _{t}(\mathbf {q} ,t)\end{aligned}}$ Let ${\textstyle {\cal {C}}}$ be an optimal trajectory and ${\textstyle {\cal {C}}'}$ its image under the above transformation ${\textstyle (\phi _{\mathbf {q} },\phi _{t})^{T}}$ (which is also an optimal trajectory). The closed path ${\textstyle {\cal {P}}}$ of integration is chosen as ${\textstyle ABB'A'}$ , where the branches ${\textstyle AB}$ and ${\textstyle A'B'}$ are given ${\textstyle {\cal {C}}}$ and ${\textstyle {\cal {C}}'}$ . By the hypothesis of Noether theorem, to the first order in ${\textstyle \epsilon }$ , $\int _{\cal {C}}dS=\int _{{\cal {C}}'}dS$ therefore, $\int _{A}^{A'}dS=\int _{B}^{B'}dS$ By definition, on the ${\textstyle AA'}$ branch we have ${\textstyle d\mathbf {q} =\epsilon \phi _{\mathbf {q} }(\mathbf {q} ,t)}$ and ${\textstyle dt=\epsilon \phi _{t}(\mathbf {q} ,t)}$ . Therefore, to the first order in ${\textstyle \epsilon }$ , the quantity $I=\mathbf {p} \phi _{\mathbf {q} }-H\phi _{t}$ is conserved along the trajectory.

Field-theoretic derivation

[edit]

Noether's theorem may also be derived for tensor fields $\varphi ^{A}$ where the indexA 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 coordinatesx^μ where the indexμ 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

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

whereas the transformation of the field variables is expressed as

\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 $\delta \varphi ^{A}$ result from two factors: intrinsic changes in the field themselves and changes in coordinates, since the transformed fieldα^A depends on the transformed coordinates ξ^μ. To isolate the intrinsic changes, the field variation at a single pointx^μ may be defined

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

\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.

{\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 thedivergence theorem into the following form

\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

\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; theycommute

{\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

{\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

{\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

\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

{\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 varioussymmetry transformations, the perturbation can be written

{\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 ${\mathcal {L}}_{X}\varphi ^{A}$ is theLie derivative of $\varphi ^{A}$ in theX^μ direction. When $\varphi ^{A}$ is a scalar or ${X^{\mu }}_{,\nu }=0$ ,

{\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

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

Differentiating the above divergence with respect toε atε = 0 and changing the sign yields the conservation law

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

where the conserved current equals

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

[edit]

Suppose we have ann-dimensional orientedRiemannian manifold,M and a target manifoldT. Let ${\mathcal {C}}$ be theconfiguration space ofsmooth functions fromM toT. (More generally, we can have smooth sections of afiber bundleT overM.)

Examples of thisM in physics include:

Inclassical mechanics, in theHamiltonian formulation,M is the one-dimensional manifold $\mathbb {R}$ , representing time and the target space is thecotangent bundle ofspace of generalized positions.
Infield theory,M is thespacetime manifold and the target space is the set of values the fields can take at any given point. For example, if there aremreal-valuedscalar fields, $\varphi _{1},\ldots ,\varphi _{m}$ , then the target manifold is $\mathbb {R} ^{m}$ . If the field is a real vector field, then the target manifold isisomorphic to $\mathbb {R} ^{3}$ .

Now suppose there is afunctional

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

called theaction. (It takes values into $\mathbb {R}$ , rather than $\mathbb {C}$ ; 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 theaction. We assume ${\mathcal {S}}[\varphi ]$ is theintegral overM of a function

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

called theLagrangian density, depending on $\varphi$ , itsderivative and the position. In other words, for $\varphi$ in ${\mathcal {C}}$

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

Suppose we are givenboundary conditions, i.e., a specification of the value of $\varphi$ at theboundary ifM iscompact, or some limit on $\varphi$ asx approaches ∞. Then thesubspace of ${\mathcal {C}}$ consisting of functions $\varphi$ such that allfunctional derivatives of ${\mathcal {S}}$ at $\varphi$ are zero, that is:

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

and that $\varphi$ satisfies the given boundary conditions, is the subspace ofon shell solutions. (Seeprinciple of stationary action)

Now, suppose we have aninfinitesimal transformation on ${\mathcal {C}}$ , generated by afunctional derivation,Q such that

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 submanifoldsN or in other words,

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

for allx, where we set

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

If this holdson shell andoff shell, we sayQ generates an off-shell symmetry. If this only holdson shell, we sayQ generates an on-shell symmetry. Then, we sayQ is a generator of aone parameter symmetry Lie group.

Now, for anyN, because of theEuler–Lagrange theorem,on shell (and only on-shell), we have

{\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 anyN, we have

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

But this is thecontinuity equation for the current $J^{\mu }$ defined by:^[15]

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

which is called theNoether current associated with thesymmetry. The continuity equation tells us that if weintegrate this current over aspace-like slice, we get aconserved quantity called the Noether charge (provided, of course, ifM is noncompact, the currents fall off sufficiently fast at infinity).

Comments

[edit]

Noether's theorem is anon 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

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

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

Generalization to Lie algebras

[edit]

Suppose we have two symmetry derivationsQ₁ andQ₂. Then, [Q₁, Q₂] is also a symmetry derivation. Let us see this explicitly. Let us say $Q_{1}[{\mathcal {L}}]\approx \partial _{\mu }f_{1}^{\mu }$ and $Q_{2}[{\mathcal {L}}]\approx \partial _{\mu }f_{2}^{\mu }$

Then, $[Q_{1},Q_{2}][{\mathcal {L}}]=Q_{1}[Q_{2}[{\mathcal {L}}]]-Q_{2}[Q_{1}[{\mathcal {L}}]]\approx \partial _{\mu }f_{12}^{\mu }$ wheref₁₂ = Q₁[f₂^μ] − Q₂[f₁^μ]. So, $j_{12}^{\mu }=\left({\frac {\partial }{\partial (\partial _{\mu }\varphi )}}{\mathcal {L}}\right)(Q_{1}[Q_{2}[\varphi ]]-Q_{2}[Q_{1}[\varphi ]])-f_{12}^{\mu }.$

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

Generalization of the proof

[edit]

This applies toany local symmetry derivationQ satisfyingQS ≈ 0, and also to more general local functional differentiable actions, including ones where the Lagrangian depends on higher derivatives of the fields. Letε 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 atest function. Then, because of the variational principle (which doesnot apply to the boundary, by the way), the derivation distribution q generated byq[ε][Φ(x)] =ε(x)Q[Φ(x)] satisfiesq[ε][S] ≈ 0 for every ε, or more compactly,q(x)[S] ≈ 0 for allx not on the boundary (but remember thatq(x) is a shorthand for a derivationdistribution, not a derivation parametrized byx 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 $\varphi$ and its first derivatives. Also, assume

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

Then,

{\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 $\varepsilon$ .

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

\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.

Examples

[edit]

Example 1: Conservation of energy

[edit]

Looking at the specific case of a Newtonian particle of massm, coordinatex, moving under the influence of a potentialV, coordinatized by timet. Theaction,S, is:

{\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 thekinetic energy of the particle, while the second is itspotential energy. Consider the generator oftime translations $Q={\frac {d}{dt}}$ . In other words, $Q[x(t)]={\dot {x}}(t)$ . The coordinatex has an explicit dependence on time, whilstV does not; consequently:

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

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

Then,

{\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 $dj/dt=0$ (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

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

(called theHamiltonian) is conserved.

Example 2: Conservation of center of momentum

[edit]

Still considering 1-dimensional time, let

{\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}}

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

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

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

And

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

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

Then,

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

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

Example 3: Conformal transformation

[edit]

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

{\begin{aligned}{\mathcal {S}}[\varphi ]&=\int {\mathcal {L}}\left[\varphi (x),\partial _{\mu }\varphi (x)\right]d^{4}x\\[3pt]&=\int \left({\frac {1}{2}}\partial ^{\mu }\varphi \partial _{\mu }\varphi -\lambda \varphi ^{4}\right)d^{4}x\end{aligned}}

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

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 $\varphi$ . And

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

\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

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

Then

{\begin{aligned}j^{\mu }&=\left[{\frac {\partial }{\partial (\partial _{\mu }\varphi )}}{\mathcal {L}}\right]Q[\varphi ]-f^{\mu }\\&=\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 $\partial _{\mu }j^{\mu }=0$ (as one may explicitly check by substituting the Euler–Lagrange equations into the left hand side).

If one tries to find theWard–Takahashi analog of this equation, one runs into a problem because ofanomalies.

Applications

[edit]

Application of Noether's theorem allows physicists to gain 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:

Invariance of an isolated system with respect to spatialtranslation (in other words, that the laws of physics are the same at all locations in space) gives the law of conservation oflinear momentum (which states that the total linear momentum of an isolated system is constant)
Invariance of an isolated system with respect totime translation (i.e. that the laws of physics are the same at all points in time) gives thelaw of conservation of energy (which states that the total energy of an isolated system is constant)
Invariance of an isolated system with respect torotation (i.e., that the laws of physics are the same with respect to all angular orientations in space) gives the law of conservation ofangular momentum (which states that the total angular momentum of an isolated system is constant)
Invariance of an isolated system with respect to Lorentz boosts (i.e., that the laws of physics are the same with respect to all inertial reference frames) gives the center-of-mass theorem (which states that the center-of-mass of an isolated system moves at a constant velocity).

Inquantum field theory, the analog to Noether's theorem, theWard–Takahashi identity, yields further conservation laws, such as the conservation ofelectric charge from the invariance with respect to a change in thephase factor of thecomplex field of the charged particle and the associatedgauge of theelectric potential andvector potential.

The Noether charge is also used in calculating theentropy ofstationary black holes.^[16]