Classical Symmetries, Currents, and Stress Tensors

A rigid variational symmetry produces a current because allowing its constant parameter to become a smooth test function exposes a term proportional to the parameter’s derivative. In the convention used here, $\delta S=-\int j^\mu\partial_\mu\epsilon$, comparison with the first variation gives an off-shell identity whose right-hand side is an Euler–Lagrange expression; the current is conserved only after the field equations are imposed. Rigid translations give the canonical stress tensor by the same mechanism, while improvements change its local representative by an identically conserved term.

This page works that chain for smooth classical fields on fixed Minkowski spacetime: the internal $U(1)$ currents of a free complex scalar and a free Dirac field, followed by the canonical and a flat-space improved stress tensor of a free real scalar. Boundary flux, current normalization, and on-shell qualifications remain explicit. General symmetry actions, quantum current operators, Ward identities, charge algebras, gauge identities, metric stress tensors, and curved-spacetime renormalization belong to later treatments.

Required background. The Action Principle and Field Equations supplies the first-variation formula and the off-shell Euler–Lagrange expressions used to distinguish a Noether identity from on-shell conservation.

Helpful background. Boundaries, Variations, and Well-Posed Actions explains why boundary flux and admissible variations matter when a local current is turned into an integrated charge. Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards supplies the test-function and weak-derivative language used when a constant symmetry parameter is localized.

Localized variations and the Noether current

Begin with bosonic fields $\Phi^A$ and a first-derivative density $\mathcal L(\Phi,\partial\Phi)$. Reuse the derivative momenta and Euler–Lagrange expressions

$$\begin{aligned} \Pi_A^\mu &\equiv \frac{\partial\mathcal L} {\partial(\partial_\mu\Phi^A)}, \\ \mathcal E_A &\equiv \frac{\partial\mathcal L}{\partial\Phi^A} -\partial_\mu\Pi_A^\mu . \end{aligned}$$

Suppose an infinitesimal transformation with constant parameter $\epsilon$ has

$$\delta_\epsilon\Phi^A = \epsilon\,\Delta^A, \qquad \delta_\epsilon\mathcal L = \epsilon\,\partial_\mu K^\mu .$$

Thus the density may change by a divergence even though the action is invariant under the declared boundary or falloff conditions. Now replace $\epsilon$ by a smooth compactly supported function. Because a derivative can also act on the parameter,

$$\delta_{\epsilon(x)}\mathcal L = \partial_\mu(\epsilon K^\mu) -j^\mu\partial_\mu\epsilon,$$

where this page fixes the current convention

$$j^\mu \equiv K^\mu-\Pi_A^\mu\Delta^A .$$

The total divergence integrates to zero for compact support. The localized variation can therefore be written in two ways:

$$\begin{aligned} \delta_{\epsilon(x)}S &= -\int\mathrm d^d x\, j^\mu\partial_\mu\epsilon \\ &= \int\mathrm d^d x\, \epsilon\,\partial_\mu j^\mu , \end{aligned}$$

and, from the first-variation formula,

$$\delta_{\epsilon(x)}S = \int\mathrm d^d x\, \epsilon\,\mathcal E_A\Delta^A .$$

Since $\epsilon(x)$ is arbitrary,

$$\boxed{ \partial_\mu j^\mu = \mathcal E_A\Delta^A }.$$

This is an off-shell identity. The equations $\mathcal E_A=0$ imply the on-shell conservation law $\partial_\mu j^\mu=0$. Making $\epsilon$ position dependent was only a diagnostic variation; it did not turn the rigid symmetry into a gauge redundancy. The derivation also assumes that the transformation contains no derivatives of the localized parameter. More general variational symmetries require an enlarged formula.

This is the first-theorem side of Noether’s 1918 result: finite-parameter variational invariance produces divergence relations among the Euler–Lagrange expressions. Noether’s second theorem instead concerns symmetries depending on arbitrary functions and the identities among field equations that follow from them; that gauge-theoretic direction is outside this page. Noether 1918, § 1, pp. 1–3, Tavel translation, PDF states the two results separately, and Weinberg 1995, Vol. I, § 7.3, pp. 306–309 develops the localized-parameter construction for fields.

The complex-scalar phase current

For a free complex scalar in $d$ dimensions, take

$$\mathcal L_\Phi = \partial_\mu\Phi^*\partial^\mu\Phi -m^2\Phi^*\Phi .$$

Treat $\Phi$ and $\Phi^*$ as independent variables during the variation and impose complex conjugacy afterward. Fix the phase convention

$$\Phi\longmapsto e^{i\alpha}\Phi, \qquad \Phi^*\longmapsto e^{-i\alpha}\Phi^* .$$

For a local $\alpha(x)$,

$$\begin{aligned} \delta_\alpha\Phi &=i\alpha\Phi, & \delta_\alpha\Phi^* &=-i\alpha\Phi^*, \\ \delta_\alpha\mathcal L_\Phi &= -j_\Phi^\mu\partial_\mu\alpha , \end{aligned}$$

with

$$j_\Phi^\mu = i\left( \Phi^*\partial^\mu\Phi -(\partial^\mu\Phi^*)\Phi \right).$$

The divergence can be checked without first imposing either field equation:

$$\partial_\mu j_\Phi^\mu = i\left[ \Phi^*(\Box+m^2)\Phi -\Phi(\Box+m^2)\Phi^* \right].$$

Both Klein–Gordon equations are therefore needed for conservation. The current is real, and the engineering dimensions $[\Phi]=(d-2)/2$ and $[\partial_\mu]=1$ give $[j_\Phi^\mu]=d-1$, as a current density must. The worked derivation in Schwartz 2014, § 3.3, pp. 32–34 uses the opposite parameter orientation, $\Phi\mapsto e^{-i\alpha}\Phi$, and defines $J^\mu=+\Pi_A^\mu(\partial\Phi^A/\partial\alpha)$, whereas this page’s $\delta\mathcal L=-j^\mu\partial_\mu\alpha$ convention gives $j^\mu=-\Pi_A^\mu\Delta^A$ when $K^\mu=0$. The two sign reversals yield the same displayed current; the convention here matches the later symmetry treatment.

On an equal-time region $\Sigma$, define the candidate bulk charge

$$Q_\Phi(t) = \int_\Sigma\mathrm d^{d-1}x\,j_\Phi^0 .$$

The local continuity equation gives

$$\frac{\mathrm dQ_\Phi}{\mathrm dt} = -\int_{\partial\Sigma} \mathrm dS\,n_i j_\Phi^i .$$

Thus on-shell local conservation makes $Q_\Phi$ time independent only when the flux through the spatial boundary vanishes or is included in a larger bulk-plus-boundary balance law. For an unbounded region, existence of the integral also requires adequate falloff. The charge is dimensionless when it exists, but this classical integral is not yet a renormalized quantum operator.

The Dirac phase current

The spinor check specializes to four-dimensional Minkowski spacetime. Use the Hermitian first-derivative density

$$\mathcal L_D = \frac{i}{2} \left[ \overline\psi\gamma^\mu\partial_\mu\psi -(\partial_\mu\overline\psi)\gamma^\mu\psi \right] -m\overline\psi\psi .$$

For the variation, $\psi$ and $\overline\psi$ are independent Grassmann-odd variables and the phase parameter is even. Under

$$\psi\longmapsto e^{i\alpha}\psi, \qquad \overline\psi\longmapsto \overline\psi e^{-i\alpha},$$

direct localization gives

$$\delta_\alpha\mathcal L_D = -j_D^\mu\partial_\mu\alpha, \qquad j_D^\mu = \overline\psi\gamma^\mu\psi .$$

The bulk field equations are

$$\begin{aligned} \left(i\gamma^\mu\partial_\mu-m\right)\psi &=0, \\ i(\partial_\mu\overline\psi)\gamma^\mu +m\overline\psi &=0. \end{aligned}$$

Using both equations,

$$\begin{aligned} \partial_\mu j_D^\mu &= (\partial_\mu\overline\psi)\gamma^\mu\psi +\overline\psi\gamma^\mu\partial_\mu\psi \\ &= im\overline\psi\psi -im\overline\psi\psi =0 . \end{aligned}$$

The normalization check is $j_D^0=\overline\psi\gamma^0\psi=\psi^\dagger\psi$. Also $[\psi]=3/2$ gives $[j_D^\mu]=3=d-1$. The free Dirac phase current is identified in Schwartz 2014, § 10.4, p. 174.

The compact density $i\overline\psi\gamma^\mu\partial_\mu\psi-m\overline\psi\psi$ differs from $\mathcal L_D$ by $\tfrac{i}{2}\partial_\mu(\overline\psi\gamma^\mu\psi)$. They give the same bulk Dirac equations, and the phase-current representative above agrees, but the total divergence remains relevant when boundary variations are allowed. Quantizing this bilinear, defining its charge, and checking possible anomalies require the later quantum-current treatment.

Translations and the scalar stress tensor

For the real free scalar,

$$\mathcal L_\phi = \frac12\partial_\rho\phi\,\partial^\rho\phi -\frac12m^2\phi^2 .$$

Choose the active infinitesimal translation

$$\delta_a\phi = -a^\nu\partial_\nu\phi .$$

For constant $a^\nu$, the density changes by $-a^\nu\partial_\nu\mathcal L_\phi$, a total divergence. Localizing $a^\nu$ and applying the preceding construction gives

$$\delta_a S = -\int\mathrm d^d x\, T^\mu{}_\nu\, \partial_\mu a^\nu ,$$

where

$$\begin{aligned} T^\mu{}_\nu &= \frac{\partial\mathcal L_\phi} {\partial(\partial_\mu\phi)} \partial_\nu\phi -\delta^\mu{}_\nu\mathcal L_\phi, \\ T^{\mu\nu} &= \partial^\mu\phi\,\partial^\nu\phi -\eta^{\mu\nu}\mathcal L_\phi . \end{aligned}$$

This is the canonical translation current. Its divergence is the off-shell identity

$$\partial_\mu T^\mu{}_\nu = -\mathcal E_\phi\,\partial_\nu\phi = (\Box+m^2)\phi\,\partial_\nu\phi .$$

Hence $\partial_\mu T^{\mu\nu}=0$ on a Klein–Gordon solution. The translation derivation and canonical formula appear in Schwartz 2014, § 3.3.1, pp. 34–36 and Weinberg 1995, Vol. I, § 7.3, pp. 310–312.

The time-time component supplies a stringent sign check:

$$T^{00} = \frac12\dot\phi^{\,2} +\frac12(\boldsymbol\nabla\phi)^2 +\frac12m^2\phi^2 .$$

It is exactly the Hamiltonian density derived on Hamiltonian Initial Data and Phase Space. The equality would fail if the metric sign, translation convention, or Legendre transform were inconsistent. Since $[\mathcal L_\phi]=d$, every component of $T^{\mu\nu}$ also has engineering dimension $d$.

On an equal-time region, the candidate momentum is

$$P^\nu(t) = \int_\Sigma\mathrm d^{d-1}x\,T^{0\nu},$$

and on shell

$$\frac{\mathrm dP^\nu}{\mathrm dt} = -\int_{\partial\Sigma} \mathrm dS\,n_iT^{i\nu}.$$

Translation invariance therefore gives a local stress-tensor conservation law. Conservation of the integrated energy–momentum again needs a vanishing or otherwise accounted-for boundary flux.

A flat-space scalar improvement

A current or stress tensor is not unique. For any $B^{\rho\mu\nu}=-B^{\mu\rho\nu}$,

$$T^{\prime\mu\nu} = T^{\mu\nu} +\partial_\rho B^{\rho\mu\nu}$$

has the same divergence because

$$\partial_\mu\partial_\rho B^{\rho\mu\nu} =0$$

identically, before using any field equation. For the scalar, a useful one-parameter family is

$$\begin{aligned} T_\xi^{\mu\nu} &= T^{\mu\nu} +\xi \left( \eta^{\mu\nu}\Box -\partial^\mu\partial^\nu \right)\phi^2, \\ B^{\rho\mu\nu} &= \xi \left( \eta^{\mu\nu}\partial^\rho -\eta^{\rho\nu}\partial^\mu \right)\phi^2 . \end{aligned}$$

The displayed $B$ is antisymmetric in $\rho,\mu$, which makes the added term identically conserved. Independently, the operator $\eta^{\mu\nu}\Box-\partial^\mu\partial^\nu$ is symmetric in $\mu,\nu$, so $T_\xi^{\mu\nu}$ remains symmetric for this scalar model. It has dimension $d$, just like the canonical tensor. On an equal-time surface its energy shift is

$$\Delta P^0 = -\xi \int_{\partial\Sigma} \mathrm dS\, n_i\partial_i\phi^2 .$$

The canonical and improved energies agree only when this surface term vanishes. More generally, improvements preserve the local divergence but can move charge between bulk and boundary descriptions. The Belinfante construction is a concrete antisymmetric-superpotential example; Weinberg derives it and proves equality of four-momenta under the surface hypothesis in Weinberg 1995, Vol. I, § 7.4, pp. 315–316.

There is one useful controlled check. For the free massless scalar,

$$\begin{aligned} T^\mu{}_\mu &= \left(1-\frac d2\right) (\partial\phi)^2, \\ \Delta T^\mu{}_\mu &= \xi(d-1)\Box\phi^2 . \end{aligned}$$

On shell, $\Box\phi=0$ gives $\Box\phi^2=2(\partial\phi)^2$. Thus, for $d>1$, choosing

$$\xi = \frac{d-2}{4(d-1)}$$

makes $T_{\xi\,\mu}{}^\mu=0$ for this free massless model; in $d=4$, $\xi=1/6$. The four-dimensional scalar improvement and its mass and trace qualifications are given in Zinn-Justin 2021, Appendix A13, § A13.2, pp. 319–320. This does not make the representative unique, prove a general conformal theorem, or identify it with a metric or renormalized stress tensor.

What conservation does—and does not—mean

The examples establish three distinct statements:

Statement	Required input	What can still fail
Off-shell identity, such as $\partial_\mu j^\mu=\mathcal E_A\Delta^A$	A differentiable variational symmetry and the localized-parameter calculation	The right-hand side need not vanish away from a solution
On-shell local conservation, $\partial_\mu j^\mu=0$	The relevant field equations	The spatial integral may diverge or exchange flux with a boundary
Conservation of an integrated charge	Local conservation plus existence of the integral and vanishing or accounted-for flux	Improvements or boundary degrees of freedom can change the bulk charge

The complex-scalar and Dirac calculations are first classical current examples; the scalar translation calculation is a first stress-tensor example. These calculations do not supply the definition of a renormalized composite current operator, a charge algebra, or a gravitational stress tensor.

Common pitfalls

Localizing a parameter is not gauging the symmetry. The function $\epsilon(x)$ is used as a test variation to isolate a current. A genuine gauge redundancy brings identities of Noether’s second-theorem type and requires a separate analysis.

On-shell conservation is not an off-shell identity. The identity relates $\partial_\mu j^\mu$ to Euler–Lagrange expressions. It becomes zero only after the required equations are imposed.

A conserved current does not automatically define a conserved charge. The spatial integral must exist, and flux through finite or asymptotic boundaries must vanish or be included in the balance law.

Changing the phase convention changes intermediate signs. The choices $\Phi\mapsto e^{i\alpha}\Phi$ and $\Phi\mapsto e^{-i\alpha}\Phi$ label the same $U(1)$ action with opposite parameter orientation. The transformation, localized variation, current, and later generator convention must be changed coherently.

Canonical does not mean unique or universally preferred. A canonical stress tensor is the representative returned by the first-derivative translation calculation. In a general theory it need not be symmetric, gauge invariant, equal to a metric stress tensor, or already defined as a quantum composite operator.

Check your understanding

1. Off-shell scalar current. Starting from $j_\Phi^\mu$, derive its divergence without using the equations of motion. Why must both complex scalar equations be imposed?

Check

The product-rule cross terms cancel, leaving

$$\partial_\mu j_\Phi^\mu = i\left( \Phi^*\Box\Phi-\Phi\Box\Phi^* \right).$$

Adding and subtracting the equal mass terms gives the displayed off-shell identity. One equation removes the first term and the conjugate equation removes the second.

2. Energy-density check. Derive $T^{00}$ from the canonical scalar tensor using the mostly-minus metric.

Check

Since $\mathcal L_\phi= \tfrac12\dot\phi^{\,2} -\tfrac12(\boldsymbol\nabla\phi)^2 -\tfrac12m^2\phi^2$ and $\partial^0\phi=\dot\phi$,

$$T^{00} = \dot\phi^{\,2}-\mathcal L_\phi = \frac12\dot\phi^{\,2} +\frac12(\boldsymbol\nabla\phi)^2 +\frac12m^2\phi^2 .$$

3. Improvement check. Show directly that the scalar improvement has zero divergence and determine when it changes the total energy.

Check

Commuting derivatives gives

$$\partial_\mu \left( \eta^{\mu\nu}\Box-\partial^\mu\partial^\nu \right)\phi^2 = \partial^\nu\Box\phi^2-\Box\partial^\nu\phi^2 =0 .$$

For $\mu=\nu=0$, the improvement is $-\xi\boldsymbol\nabla^2\phi^2$. Its spatial integral is the surface term $-\xi\int_{\partial\Sigma}\mathrm dS\,n_i\partial_i\phi^2$, so the total energy is unchanged only when that term vanishes.

Where the general theory continues

• Continuous Symmetries, Generators, and Charges turns the classical localized variation into the general current-to-charge-to-generator chain.
• Quantum Currents, Improvements, and Conservation treats currents as renormalized operator-valued distributions and separates conservation, contact terms, and improvements.
• Spacetime Currents, Stress Tensors, and Charge Algebras develops Lorentz currents, Belinfante improvement, hypersurface generators, and their qualified algebra.
• Renormalized Stress Tensor: Axioms and Curvature Ambiguities develops metric variation, gravitational coupling, and the renormalized stress tensor in curved spacetime.

The reusable classical lesson is precise: a rigid variational symmetry gives an off-shell divergence identity; the field equations turn it into local conservation; and boundary and improvement conditions decide whether the associated integrated quantity is unchanged.

References

• Noether, Emmy. “Invariant Variation Problems.” Translated by M. A. Tavel. Transport Theory and Statistical Physics 1, no. 3 (1971): 186–207. Translation of “Invariante Variationsprobleme,” Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (1918): 235–257. DOI. Open PDF, arXiv:physics/0503066v3.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.
• Weinberg, Steven. The Quantum Theory of Fields, Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.
• Zinn-Justin, Jean. Quantum Field Theory and Critical Phenomena. 5th ed. Oxford: Oxford University Press, 2021. DOI.