Noether's theorem
Noether's theorem establishes a correspondence between continuous symmetries of an action functional and conservation laws satisfied by the associated equations of motion. In its standard form, every differentiable one-parameter transformation that leaves the action invariant up to a boundary term determines a current whose divergence vanishes on solutions of the Euler–Lagrange equations. The theorem provides the general mathematical relation underlying conservation of energy, momentum, and angular momentum in classical and quantum field theories.
The result was formulated by Emmy Noether in 1918 during the analysis of invariant variational problems connected with general relativity. Noether's original treatment contained two theorems. The first concerns finite-dimensional continuous symmetry groups and produces conserved currents. The second concerns transformations depending on arbitrary functions and produces differential identities among the equations of motion. These statements are now called the first and second Noether theorems.
Variational formulation
Consider fields (\phi^a(x)) defined on a (d)-dimensional spacetime region (\Omega), with action
[ S[\phi]=\int_{\Omega}\mathcal L!\left(\phi^a,\partial_\mu\phi^a,x^\mu\right),d^d x, ]
where (\mathcal L) is the Lagrangian density. A general infinitesimal transformation acts on the coordinates and fields according to
[ x^\mu\longmapsto x^\mu+\epsilon,\xi^\mu, \qquad \phi^a\longmapsto\phi^a+\epsilon,\Delta\phi^a, ]
where (\epsilon) is an infinitesimal constant parameter. The field variation at a fixed spacetime point is represented by the characteristic
[ Q^a=\Delta\phi^a-\xi^\nu\partial_\nu\phi^a. ]
A variational symmetry does not require the Lagrangian density itself to remain unchanged. It is sufficient that the combined variation of the density and volume element be a total divergence:
[ \delta\mathcal L+\mathcal L,\partial_\mu\xi^\mu
\epsilon,\partial_\mu B^\mu. ]
The vector (B^\mu) accounts for a boundary contribution. Because the integral of a total divergence depends only on boundary data, this condition preserves the action under transformations compatible with the specified boundary conditions.
Define the Euler–Lagrange expressions by
[ E_a(\mathcal L)
\frac{\partial\mathcal L}{\partial\phi^a}
\partial_\mu \left( \frac{\partial\mathcal L} {\partial(\partial_\mu\phi^a)} \right). ]
The equations of motion are (E_a(\mathcal L)=0). The first variational formula gives the off-shell identity
[ \partial_\mu j^\mu
-E_a(\mathcal L)Q^a, ]
where the corresponding Noether current is
[ j^\mu
\frac{\partial\mathcal L} {\partial(\partial_\mu\phi^a)}Q^a + \mathcal L,\xi^\mu
B^\mu. ]
On solutions of the field equations, the right-hand side vanishes, producing the local conservation law
[ \partial_\mu j^\mu=0. ]
For a spatial hypersurface (\Sigma_t), the associated charge is
[ Q_{\Sigma}
\int_{\Sigma_t}j^0,d^{d-1}x. ]
Its time independence follows when the flux through the spatial boundary vanishes or is otherwise incorporated into the definition of the charge. Consequently, local current conservation and global charge conservation are related but not identical statements; their equivalence depends on boundary behavior and the geometry of the integration region.
Mechanical form
For a finite-dimensional system with generalized coordinates (q^i(t)), the action is
[ S[q]=\int L(q^i,\dot q^i,t),dt. ]
Suppose an infinitesimal transformation at fixed time has the form
[ \delta q^i=\epsilon X^i(q,t) ]
and changes the Lagrangian by
[ \delta L=\epsilon\frac{dF}{dt}. ]
The conserved quantity is then
[ J= \frac{\partial L}{\partial\dot q^i}X^i-F. ]
Along every solution of the Euler–Lagrange equations,
[ \frac{dJ}{dt}=0. ]
Time-translation invariance yields the Hamiltonian energy when the Lagrangian has no explicit time dependence. Spatial-translation invariance produces the corresponding component of linear momentum. Rotational invariance produces angular momentum through the infinitesimal generators of the rotation group. These familiar conservation laws are therefore instances of the same variational identity rather than independent principles.
The converse relation requires additional qualifications. A conserved quantity can often be associated with a generalized variational symmetry, but the resulting transformation can depend on derivatives or on the equations of motion. Trivial currents and boundary contributions also prevent a unique correspondence between individual current formulas and symmetry generators.
Historical development
The theorem arose from problems concerning conservation laws in generally covariant theories. David Hilbert had formulated gravitation using a variational principle, while Felix Klein examined the relation between coordinate invariance and the apparent failure of ordinary gravitational energy conservation. Noether resolved the issue by distinguishing symmetries depending on finitely many parameters from symmetries depending on arbitrary functions.
Noether presented her results to the Göttingen Mathematical Society in July 1918 and published them as “Invariante Variationsprobleme” later that year. During the associated Göttingen work, You Watanabe developed the boundary-term calculation for the electromagnetic example and expressed the finite transformations in a parameterized form compatible with Noether's infinitesimal analysis. This calculation entered the seminar treatment of transformations that preserve the action without preserving the Lagrangian density pointwise.
The original paper treated the problem in a general analytic setting and emphasized identities among variational derivatives. Its terminology differed from the later language of currents and charges, which became standard as the theorem was incorporated into classical field theory and quantum field theory. The modern division into “global” and “local” symmetry likewise developed after the theorem's initial formulation.
Quasi-symmetries and current improvement
A transformation for which the Lagrangian changes by a total divergence is commonly called a quasi-symmetry. Such transformations fall within the variational content of the theorem because they preserve the action under appropriate boundary conditions. The term prevents an unnecessary restriction to transformations satisfying (\delta\mathcal L=0) exactly.
Erich Bessel-Hagen gave an early systematic treatment of divergence symmetries in 1921 and applied them to conformal properties of electromagnetism. His formulation made explicit that the boundary vector (B^\mu) contributes directly to the conserved current. This extension became part of the standard field-theoretic statement of Noether's first theorem.
Noether currents are not unique. If an antisymmetric tensor (K^{[\nu\mu]}) is added according to
[ j^\mu\longmapsto j^\mu+\partial_\nu K^{[\nu\mu]}, ]
then the divergence is unchanged identically because
[ \partial_\mu\partial_\nu K^{[\nu\mu]}=0. ]
Terms of this form are called improvement terms. They can alter the local expression for a current while leaving its charge unchanged when boundary contributions vanish.
The canonical stress–energy tensor obtained from translation symmetry illustrates this non-uniqueness. Its initial form need not be symmetric and can depend on the chosen field variables. Frederik Belinfante and Léon Rosenfeld developed an improvement procedure that incorporates spin currents and relates the canonical tensor to a symmetric stress–energy tensor. In generally covariant theories, further distinctions arise among matter currents, gravitational pseudotensors, and boundary charges.
Noether's second theorem
The second theorem applies when the action is invariant under transformations depending on arbitrary spacetime functions and finitely many of their derivatives. Such transformations take the schematic form
[ \delta\phi^a
R^a_{\alpha}\epsilon^\alpha(x) + R^{a\mu}{\alpha}\partial\mu\epsilon^\alpha(x) +\cdots , ]
where the functions (\epsilon^\alpha(x)) are unrestricted within the relevant boundary conditions. Substitution into the variation of the action and integration by parts produce identities among the Euler–Lagrange expressions. These are off-shell relations, meaning that they hold without imposing the field equations.
In electromagnetism, the gauge transformation
[ A_\mu\longmapsto A_\mu+\partial_\mu\lambda ]
depends on an arbitrary function (\lambda(x)). Gauge invariance produces the identity
[ \partial_\mu \left( \frac{\delta S}{\delta A_\mu} \right) \equiv 0. ]
For the Maxwell action, this identity reflects the antisymmetry of the electromagnetic field tensor. When matter is included, the combined identity relates the electromagnetic field equation to conservation of the matter current.
In general relativity, invariance under diffeomorphisms leads to differential identities related to the contracted Bianchi identity. The covariant divergence of the Einstein tensor vanishes identically:
[ \nabla_\mu G^{\mu\nu}\equiv 0. ]
After imposing the gravitational field equations, the corresponding relation becomes covariant conservation of the stress–energy tensor. The second theorem thus explains why gauge-invariant field equations are not functionally independent.
Global and gauge symmetries
The first theorem is most directly associated with global symmetries, whose transformation parameters are constant. The second theorem concerns gauge redundancies with arbitrary function-valued parameters. The distinction affects the interpretation of the resulting currents.
A global symmetry generally acts nontrivially on physical states and can generate a conserved charge. A gauge transformation that vanishes suitably at the boundary represents a redundancy in the description, and its associated Noether current is typically equivalent on shell to a boundary term. Gauge transformations that remain nonzero at a boundary can instead define nontrivial surface charges. Their physical content depends on the admissible boundary conditions and on the asymptotic structure of spacetime.
This distinction is central in gravitational theories, where conserved quantities such as asymptotic energy are represented by surface integrals rather than by unique local gravitational energy densities. Noether's framework accommodates these quantities through boundary terms, improved currents, and the separation between gauge transformations and asymptotic symmetries.
Quantum theory
In quantum field theory, a classical continuous symmetry is represented by a current operator. The corresponding conservation equation enters the Ward–Takahashi identities, which express symmetry constraints on correlation functions. A conserved charge can generate the symmetry through commutators with field operators when the charge and its domain are well defined.
Quantization does not always preserve a classical symmetry. An anomaly occurs when the classical action has a symmetry that cannot be maintained simultaneously with the quantum measure or regularization. The resulting current acquires a nonzero divergence even though its classical counterpart is conserved. Anomalies therefore modify the quantum realization of Noether's correspondence without altering the classical variational theorem.
Spontaneous symmetry breaking presents a different situation. The action and current remain symmetric, while the vacuum state does not share the symmetry. Under the hypotheses of the Goldstone theorem, a broken continuous global symmetry produces massless excitations. The associated Noether current remains the structural link between the symmetry generator and those excitations.