Noether's First Theorem

Noether’s first theorem turns a continuous variational symmetry into a locally conserved current. Time-translation symmetry gives energy conservation; spatial-translation symmetry gives momentum conservation. The precise construction starts from a differentiable local action, a finite-dimensional continuous symmetry up to a boundary term, and admissible variations. It first gives an identity for arbitrary fields (off shell); the current is conserved when the fields solve their equations of motion (on shell). A conserved integrated charge also needs boundary and integrability conditions.

Required background. Classical Fields and Their Actions provides field actions, Euler–Lagrange equations, and gauge redundancy; Events, Intervals, and Minkowski Geometry provides Minkowski causal geometry and four-vectors. Helpful background. Calculus and Linear Algebra for Relativity provides multivariable differentiation and linear maps; Presymplectic Potentials and Currents develops boundary variations and current ambiguities.

Continuous variational symmetries

Let

$$S[\phi]=\int_\Omega d^dx\,\mathcal L(\phi^a,\partial_\mu\phi^a,x)$$

be a local, differentiable first-order action. “Differentiable” means that admissible variations—compactly supported in $\Omega$, or satisfying specified boundary data—leave no uncontrolled surface term. Higher-derivative actions have an analogous theorem with a larger boundary form.

Assume a finite-dimensional Lie group acts infinitesimally through independent constant parameters $\epsilon^A$,

$$\delta_\epsilon\phi^a=\epsilon^A\Delta_A\phi^a, \qquad \delta_\epsilon\mathcal L=\epsilon^A\partial_\mu K_A^\mu.$$

The second equation is a quasi-symmetry: the density may change by a divergence, provided its boundary integral is compatible with the action’s boundary conditions. Spacetime transformations fit after replacing $\Delta_A\phi^a$ by their active characteristic. Under these hypotheses there is a current $j_A^\mu$ satisfying an identity off shell and conservation on shell. This is the modern finite-parameter form of Noether’s §1, Theorem I; Pons, §2.2.1 emphasizes that the divergence condition holds off shell.

Proof architecture and the off-shell identity

Define the Euler–Lagrange expressions and momenta

$$E_a=\frac{\partial\mathcal L}{\partial\phi^a} -\partial_\mu\frac{\partial\mathcal L}{\partial(\partial_\mu\phi^a)}, \qquad \pi_a^\mu=\frac{\partial\mathcal L}{\partial(\partial_\mu\phi^a)}.$$

For an arbitrary variation, the chain rule and one local product rule give

$$\delta\mathcal L =E_a\,\delta\phi^a +\partial_\mu\!\left(\pi_a^\mu\delta\phi^a\right).$$

This is a pointwise identity: no spacetime integral or boundary condition has yet been used. Integrating it over $\Omega$ is the step that turns the last term into a directed boundary integral and makes admissibility of the variation consequential.

Insert the quasi-symmetry, compare coefficients of each constant $\epsilon^A$, and define

$$j_A^\mu=\pi_a^\mu\Delta_A\phi^a-K_A^\mu.$$

The result is the off-shell Noether identity

$$\boxed{\partial_\mu j_A^\mu=-E_a\Delta_A\phi^a.}$$

Nothing has yet used the equations of motion. Only after imposing $E_a=0$ does one obtain $\partial_\mu j_A^\mu=0$. Pons, §2.2.1, printed pp. 5–6 states this off-shell Noether condition explicitly. Tong, §1.3.1, printed pp. 13–14, Eqs. (1.34)–(1.38) gives the corresponding compact on-shell current derivation.

Mechanics check: time translation

For a regular mechanical Lagrangian $L(q^i,\dot q^i)$ with no explicit time dependence, an infinitesimal time translation along the trajectory has

$$\delta q^i=\epsilon\dot q^i, \qquad \delta L=\epsilon\frac{dL}{dt}.$$

Thus $K=L$ and the Noether quantity is

$$j=p_i\dot q^i-L\equiv H, \qquad p_i=\frac{\partial L}{\partial\dot q^i}.$$

The off-shell identity becomes

$$\frac{dH}{dt} =-\left(\frac{\partial L}{\partial q^i}-\frac{d p_i}{dt}\right)\dot q^i.$$

It vanishes on Euler–Lagrange trajectories. For $L=m|\dot{\mathbf q}|^2/2-V(\mathbf q)$, this gives $H=m|\dot{\mathbf q}|^2/2+V$, recovering the familiar mechanical energy with the correct sign. Tong, Classical Dynamics, §2.4, printed pp. 23–25 develops this finite-dimensional symmetry–charge correspondence. If $L$ depends explicitly on time, this simple time-translation argument no longer applies and on solutions $dH/dt=-\partial L/\partial t$; writing down the same algebraic expression for $H$ does not make it conserved. A different quasi-symmetry may still exist, for example after subtracting a total time derivative from the action.

From a current to a charge

In Cartesian coordinates define $Q_A(t)=\int_V d^{d-1}x\,j_A^0$. On shell,

$$\frac{dQ_A}{dt} =-\int_{\partial V}j_A^i n_i\,dA.$$

Thus $Q_A$ is constant only if the integral converges and the outward boundary flux vanishes, cancels by periodicity, or is included as part of a larger closed system. A current may also be shifted by $\partial_\nu U_A^{[\nu\mu]}$ without changing its divergence; the charge changes by a surface integral and is equal only under suitable falloff. Local conservation, global charge existence, and charge constancy are therefore three separate statements.

Gauge transformations depending on arbitrary functions of position are not covered by the finite-dimensional hypothesis. They generate Noether identities among field equations and boundary-sensitive charges rather than one ordinary charge per arbitrary function.

Exercise: global phase symmetry

For

$$\mathcal L=-\partial_\mu\psi^*\partial^\mu\psi-m^2\psi^*\psi,$$

use $\delta\psi=-i\epsilon\psi$ and $\delta\psi^*=i\epsilon\psi^*$ to find the current and check a positive-frequency homogeneous solution.

Solution

The density is exactly invariant, so $K^\mu=0$. Since

$$\frac{\partial\mathcal L}{\partial(\partial_\mu\psi)} =-\partial^\mu\psi^*, \qquad \frac{\partial\mathcal L}{\partial(\partial_\mu\psi^*)} =-\partial^\mu\psi,$$

the theorem gives

$$j^\mu=i\left(\psi\partial^\mu\psi^* -\psi^*\partial^\mu\psi\right).$$

Direct differentiation yields

$$\partial_\mu j^\mu =i\left[\psi(\Box-m^2)\psi^* -\psi^*(\Box-m^2)\psi\right],$$

which displays the off-shell identity and vanishes when both field equations hold. For $\psi=Ae^{-imt}$, $\partial^0=-\partial_t$ and

$$j^0=2m|A|^2\gt0.$$

The sign checks both the $(-,+,+,+)$ convention and the chosen orientation of the $U(1)$ generator. Reversing the generator reverses the current and charge without changing conservation.

Failure modes and next step

A transformation of the equations alone need not be a variational quasi-symmetry. A nondifferentiable action, omitted corner term, anomalous quantum symmetry, explicit external source, or nonzero boundary flux can invalidate the desired conclusion at a different logical step. The theorem is classical; quantum anomalies require a separate measure and regularization analysis.

Canonical Energy-Momentum and Improvement now applies the theorem to translations and explains why the resulting tensor is not unique.

References

• Emmy Noether, “Invariant Variation Problems,” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, 235–257 (1918), English translation by M. A. Tavel, §1, Theorem I, and §2.
• Josep M. Pons, “Noether Symmetries, Energy-Momentum Tensors and Conformal Invariance in Classical Field Theory,” Journal of Mathematical Physics 52, 012904 (2011), doi:10.1063/1.3532941, Open PDF, §2.2.1, printed pp. 5–6.
• David Tong, Classical Dynamics, University of Cambridge lecture notes (2004/2005), PDF, §2.4, printed pp. 23–25.
• David Tong, Quantum Field Theory, University of Cambridge Part III lecture notes (2006/2007), official PDF, §1.3.1, printed pp. 13–14, Eqs. (1.29)–(1.38).