Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mathematician Emmy Noether in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem applies to continuous and smooth symmetries of physical space. Noether's formulation is quite general and has been applied across classical mechanics, high energy physics, and recently statistical mechanics. Noether's theorem is used in theoretical physics and the calculus 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 on constants of motion in Lagrangian and Hamiltonian 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 a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law.
Basic illustrations and background As an illustration, if a physical system behaves the same regardless of how it is oriented in space (that is, it is invariant), its Lagrangian is symmetric under continuous rotation: from this symmetry, Noether's theorem dictates that the angular momentum of the system be conserved, as a consequence of its laws of motion. 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 the conservation laws of linear momentum and energy within this system, respectively. 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. As an illustration, suppose that a physical theory is proposed which conserves a quantity X. A researcher can calculate the types of Lagrangians that conserve X 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 the Ward–Takahashi identities. Generalizations of Noether's theorem to superspaces also exist.
Informal statement of the theorem 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. A more sophisticated version of the theorem involving fields states that:
To every continuous symmetry generated by local actions there corresponds a conserved current and vice versa. The word "symmetry" in the above statement refers more precisely to the covariance of the form that a physical law takes with respect to a one-dimensional Lie group of transformations satisfying certain technical criteria. The conservation law of a physical quantity is usually expressed as a continuity 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 the Noether charge, while the flow carrying that charge is called the Noether current. The Noether current is defined up to a solenoidal (divergenceless) vector field. In the context of gravitation, Felix Klein's statement of Noether's theorem for action I stipulates for the invariants:
If an integral I is invariant under a continuous group Gρ with ρ parameters, then ρ linearly independent combinations of the Lagrangian expressions are divergences.
Brief illustration and overview of the concept
The main idea behind Noether's theorem is most easily illustrated by a system with one coordinate q {\displaystyle q} and a continuous symmetry φ : q ↦ q + δ q {\displaystyle \varphi :q\mapsto q+\delta q} (gray arrows on the diagram). Consider any trajectory q ( t ) {\displaystyle q(t)} (bold on the diagram) that satisfies the system's laws of motion. That is, the action S {\displaystyle S} governing this system is stationary on this trajectory, i.e. does not change under any local variation of the trajectory. In particular it would not change under a variation that applies the symmetry flow φ {\displaystyle \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 τ {\displaystyle \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 ] {\displaystyle [t_{0},t_{1}]} , bring no Δ S {\displaystyle \Delta S} . The middle part does not change the action either, because its transformation φ {\displaystyle \varphi } is a symmetry and thus preserves the Lagrangian L {\displaystyle L} and the action S = ∫ L {\textstyle S=\int L} . The only remaining parts are the "buffering" pieces. In these regions both the coordinate q {\displaystyle q} and velocity q ˙ {\displaystyle {\dot {q}}} change, but q ˙ {\displaystyle {\dot {q}}} changes by δ q / τ {\displaystyle \delta q/\tau } , and the change δ q {\displaystyle \delta q} in the coordinate is negligible by comparison since the time span τ {\displaystyle \tau } of the buffering is small (taken to the limit of 0), so δ q / τ ≫ δ q {\displaystyle \delta q/\tau \gg \delta q} . So the regions contribute mostly through their "slanting" q ˙ → q ˙ ± δ q / τ {\displaystyle {\dot {q}}\rightarrow {\dot {q}}\pm \delta q/\tau } . That changes the Lagrangian by Δ L ≈ ( ∂ L / ∂ q ˙ ) Δ q ˙ {\displaystyle \Delta L\approx {\bigl (}\partial L/\partial {\dot {q}}{\bigr )}\Delta {\dot {q}}} , which integrates to
Δ S = ∫ Δ L ≈ ∫ ∂ L ∂ q ˙ Δ q ˙ ≈ ∫ ∂ L ∂ q ˙ ( ± δ q τ ) ≈ ± ∂ L ∂ q ˙ δ q = ± ∂ L ∂ q ˙ φ . {\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 .}
These last terms, evaluated around the endpoints t 0 {\displaystyle t_{0}} and t 1 {\displaystyle t_{1}} , should cancel each other in order to make the total change in the action Δ S {\displaystyle \Delta S} be zero, as would be expected if the trajectory is a solution. That is
( ∂ L ∂ q ˙ φ ) ( t 0 ) = ( ∂ L ∂ q ˙ φ ) ( t 1 ) , {\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 ( ∂ L / ∂ q ˙ ) φ {\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 q {\displaystyle q} by a constant are the symmetry, then the conserved quantity becomes just ( ∂ L / ∂ q ˙ ) = p {\displaystyle \left(\partial L/\partial {\dot {q}}\right)=p} , the canonical momentum.
More general cases follow the same idea:
Historical context
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,
d X d t = X ˙ = 0 . {\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 kinetic 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 laws of motion. 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
I = ∫ L ( q , q ˙ , t ) d t , {\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,
q ˙ = d q d t . {\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,
d d t ( ∂ L ∂ q ˙ ) = ∂ L ∂ q . {\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
d d t ( ∂ L ∂ q ˙ k ) = d p k d t = 0 , {\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial {\dot {q}}_{k}}}\right)={\frac {dp_{k}}{dt}}=0~,}
where the momentum
p k = ∂ L ∂ q ˙ k {\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. Emmy Noether's work on the invariance theorem began in 1915 when she was helping Felix Klein and David Hilbert with their work related to Albert Einstein's theory of general relativity By March 1918 she had most of the key ideas for the paper which would be published later in the year.
Mathematical expression
Simple form using perturbations The essence of Noether's theorem is generalizing the notion of ignorable coordinates. 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
t → t ′ = t + δ t q → q ′ = q + δ q , {\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,
δ t = ∑ r ε r T r δ q = ∑ r ε r Q r , {\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
( ∂ L ∂ q ˙ ⋅ q ˙ − L ) T r − ∂ L ∂ q ˙ ⋅ Q r {\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}}
are conserved (constants of motion).
Examples I. Time invariance For illustration, consider a Lagrangian that does not depend on time, i.e., that is invariant (symmetric) under changes t → t + 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
H = ∂ L ∂ q ˙ ⋅ q ˙ − L . {\displaystyle 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) coordinate qk; so it is invariant (symmetric) under changes qk → qk + δqk. In that case, N = 1, T = 0, and Qk = 1; the conserved quantity is the corresponding linear momentum pk
p k = ∂ L ∂ q k ˙ . {\displaystyle p_{k}={\frac {\partial L}{\partial {\dot {q_{k}}}}}.}
In special and general relativity, these two conservation laws can be expressed either globally (as it is done above), or locally 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 defined locally at the space-time point: the stress–energy tensor(this will be derived in the next section). III. Rotational invariance The conservation of the angular momentum L = r × p is analogous to its linear momentum counterpart. 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
r → r + δ θ n × r . {\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\delta \theta \,\mathbf {n} \times \mathbf {r} .}
Since time is not being transformed, T = 0, and N = 1. Taking δθ as the ε parameter and the Cartesian coordinates r as the generalized coordinates q, the corresponding Q variables are given by
Q = n × r . {\displaystyle \mathbf {Q} =\mathbf {n} \times \mathbf {r} .}
Then Noether's theorem states that the following quantity is conserved,
∂ L ∂ q ˙ ⋅ Q = p ⋅ ( n × r ) = n ⋅ ( r × p ) = n ⋅ L . {\displaystyle {\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 momentum L along the n axis is conserved. And 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 given above 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 version of Noether's theorem. Let there be a set of differentiable fields φ {\displaystyle \varphi } defined over all space and time; for example, the temperature T ( x , t ) {\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
S = ∫ L ( φ , ∂ μ φ , x μ ) d 4 x {\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 φ {\displaystyle \varphi } can be written infinitesimally as
φ ↦ φ + ε Ψ , {\displaystyle \varphi \mapsto \varphi +\varepsilon \Psi ,}
where Ψ {\displaystyle \Psi } is in general a function that may depend on both x μ {\displaystyle x^{\mu }} and φ {\displaystyle \varphi } . The condition for Ψ {\displaystyle \Psi } to generate a physical symmetry is that the action S {\displaystyle {\mathcal {S}}} is left invariant. This will certainly be true if the Lagrangian density L {\displaystyle {\mathcal {L}}} is left invariant, but it will also be true if the Lagrangian changes by a divergence,
L ↦ L + ε ∂ μ Λ μ , {\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 r = 1 , 2 , … , N , {\displaystyle r=1,2,\ldots ,N,} so the most general symmetry transformation would be written as
φ ↦ φ + ε r Ψ r , {\displaystyle \varphi \mapsto \varphi +\varepsilon _{r}\Psi _{r},}
with the consequence
L ↦ L + ε r ∂ μ Λ r μ . {\displaystyle {\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} conserved current densities
j r ν = Λ r ν − ∂ L ∂ ( ∂ ν φ ) ⋅ Ψ r {\displaystyle j_{r}^{\nu }=\Lambda _{r}^{\nu }-{\frac {\partial {\mathcal {L}}}{\partial (\partial _{\nu }\varphi )}}\cdot \Psi _{r}}
(where the dot product is understood to contract the field indices, not the ν {\displaystyle \nu } index or r {\displaystyle r} index). In such cases, the conservation law is expressed in a four-dimensional way
∂ ν j ν = 0 , {\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 charge enters or leaves the sphere.
Examples I. The stress–energy tensor For illustration, consider a physical system of fields that behaves the same under translations in time and space, as considered above; in other words, L ( φ , ∂ μ φ , x μ ) {\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, x μ ↦ x μ + ε r δ r μ {\displaystyle x^{\mu }\mapsto x^{\mu }+\varepsilon _{r}\delta _{r}^{\mu }} (with δ {\displaystyle \delta } denoting the Kronecker delta), affects the fields as φ ( x μ ) ↦ φ ( x μ − ε r δ r μ