Classical fields, actions, and local dynamics

A local action turns a proposed field model into equations of motion, but only after the fields, allowed variations, and boundary behavior have been stated. This page develops that logic for a scalar field: first the general bulk–boundary split, then a complete derivation of the nonlinear Klein–Gordon equation, and finally the connection between symmetries and conserved currents. The discussion is classical and assumes smooth fields on a fixed Minkowski region; quantization and constrained gauge systems come later.

Helpful background. Orientation and conventions sets the site-wide metric and Fourier choices. If varying an ordinary mechanical action is not yet comfortable, use the variational and classical-field review and return to the worked derivation below.

Local actions and admissible variations

In particle mechanics, a history is a curve $q^a(t)$ and the action is $S[q]=\int \mathrm dt\,L(q,\dot q,t)$. A field history assigns components $\phi^A(x)$ throughout spacetime. For a first-derivative local theory on a region $M$,

$$S[\phi] = \int_M \mathrm d^d x\, \mathcal L\!\left(\phi^A,\partial_\mu\phi^A;x\right).$$

“Local” means that the density at $x$ depends on fields and finitely many of their derivatives at that same point. It does not mean that every observable in the eventual quantum theory is pointlike.

Vary the fields through $\phi^A\mapsto\phi^A+\varepsilon\eta^A$ and differentiate at $\varepsilon=0$. The test fields $\eta^A$ must have the same tensor and reality properties as $\phi^A$. One integration by parts gives

$$\begin{aligned} \delta S ={}& \int_M \mathrm d^d x\, \left[ \frac{\partial\mathcal L}{\partial\phi^A} -\partial_\mu \frac{\partial\mathcal L}{\partial(\partial_\mu\phi^A)} \right]\eta^A \\ &+ \int_{\partial M}\mathrm d\Sigma_\mu\, \frac{\partial\mathcal L}{\partial(\partial_\mu\phi^A)} \eta^A . \end{aligned}$$

Here $\mathrm d\Sigma_\mu$ is the outward-directed surface element. If the variation has compact support in the interior, the surface term vanishes. Stationarity for every such variation then implies the local Euler–Lagrange equations

$$\frac{\partial\mathcal L}{\partial\phi^A} -\partial_\mu \frac{\partial\mathcal L}{\partial(\partial_\mu\phi^A)} =0.$$

The quantifier “every admissible variation” matters. Compactly supported variations establish only the interior equation. For a finite region, the remaining surface term must also vanish under the chosen boundary data, or be cancelled by the variation of an added boundary functional. This is the precise sense in which the boundary condition is part of the variational problem Harlow and Wu 2020, § 2.2.

Worked derivation: a self-interacting scalar

Use the inherited mostly-minus metric and consider one real scalar field with

$$\mathcal L = \frac12\partial_\mu\phi\,\partial^\mu\phi -V(\phi), \qquad V(\phi)=\frac12m^2\phi^2+\frac{\lambda}{4!}\phi^4.$$

Under $\phi\mapsto\phi+\varepsilon\eta$,

$$\begin{aligned} \delta S &= \int_M\mathrm d^d x\, \left[ \partial_\mu\phi\,\partial^\mu\eta -V'(\phi)\eta \right] \\ &= -\int_M\mathrm d^d x\, \left[\Box\phi+V'(\phi)\right]\eta +\int_{\partial M}\mathrm d\Sigma_\mu\, \partial^\mu\phi\,\eta . \end{aligned}$$

The second line is the decisive step: it separates the equation in the bulk from the data at the boundary. If $\eta$ vanishes on $\partial M$—for example, because the boundary value of $\phi$ is fixed—stationarity for all interior variations gives

$$\boxed{ (\Box+m^2)\phi+\frac{\lambda}{3!}\phi^3=0 }.$$

If the boundary value is free instead, the original action has an additional natural stationarity condition, $n_\mu\partial^\mu\phi=0$ on a non-null boundary, where $\mathrm d\Sigma_\mu=n_\mu\,\mathrm d\Sigma$ and $n_\mu$ is the outward unit normal on each smooth boundary piece. Other choices, such as mixed data, require testing the total surface variation after the appropriate boundary term is included. They cannot be obtained by silently discarding the last integral.

Three quick checks catch most errors in this derivation:

• Free limit. Setting $\lambda=0$ gives the Klein–Gordon equation.
• Mass shell. For $e^{-ip\cdot x}$, the inherited Fourier convention has $\Box\mapsto-p^2$, so the free equation gives $p^2=m^2$.
• Dimensions. In natural units, $[\mathcal L]=d$ and $[\phi]=(d-2)/2$. Every term in the equation therefore has dimension $(d+2)/2$; in $d=4$, $\lambda$ is dimensionless.

The calculation proves stationarity, not minimization. It also does not prove that a solution exists, is unique, or depends continuously on its initial or boundary data. Those are separate analytic questions. A fuller treatment of the first-variation argument appears in Schwartz 2014, §§ 3.1–3.2.

What symmetry adds

The action is more informative than the field equation alone because its differentiable continuous global symmetries—allowing invariance up to a divergence—produce currents. The scalar bulk kinetic density is Lorentz invariant because its indices are fully contracted. On a finite region, the region, boundary functional, and boundary data must transform compatibly for the complete variational problem to share that symmetry. If $V(\phi)$ is even, the action also has the discrete symmetry $\phi\mapsto-\phi$. That symmetry constrains allowed terms, but it has no infinitesimal parameter and therefore does not produce an ordinary Noether current.

For comparison, a complex scalar with

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

is invariant under the continuous global rephasing $\phi\mapsto e^{-i\alpha}\phi$, $\phi^*\mapsto e^{+i\alpha}\phi^*$. Temporarily allowing $\alpha$ to vary with $x$ gives $\delta\mathcal L=-j^\mu\partial_\mu\alpha$ and isolates the current

$$j^\mu = i\left( \phi^*\partial^\mu\phi -\phi\partial^\mu\phi^* \right),$$

whose divergence vanishes when both complex Klein–Gordon equations hold. Thus invariance of the action gives an off-shell identity, while current conservation is an on-shell statement. Conservation of the integrated charge additionally requires the spatial integral to exist and the boundary flux to vanish or be included in the balance law. This is the classical starting point of Noether’s theorem, not yet a quantum Ward identity Noether 1918; Tavel translation 1971.

Checks and limitations

Keep these distinctions visible when moving to quantum theory:

• Two densities that differ by a total divergence give the same interior equation for compactly supported variations, but can define different boundary problems.
• A field equation is not a complete physical prediction; it still needs initial or boundary data and an observable.
• Gauge fields have degenerate variational directions and constraints. Their action cannot be treated as an ordinary nondegenerate scalar system.
• A classical current may need renormalization and can acquire contact terms or anomalies after quantization.

Exercises

1. Add a source. Replace the scalar density by $\mathcal L+J(x)\phi(x)$, treating $J$ as fixed. Derive the bulk equation and state whether the kinetic surface term changes.

Solution

The source contributes

$$\delta S_J=\int_M\mathrm d^d x\,J(x)\eta(x)$$

and contains no derivative of the variation. Combining it with the worked variation gives

$$\delta(S+S_J) = -\int_M\mathrm d^d x\, \left[\Box\phi+V'(\phi)-J\right]\eta +\int_{\partial M}\mathrm d\Sigma_\mu\, \partial^\mu\phi\,\eta .$$

Hence

$$\Box\phi+V'(\phi)=J.$$

The source changes the bulk equation but not the surface term because it has no derivatives of $\phi$.

2. Transfer to a complex field. Treat $\phi$ and $\phi^*$ as independent variables in the complex-scalar density above. Derive both field equations, write the complete surface variation, and verify $\partial_\mu j^\mu=0$ on shell.

Solution

Independent variations give

$$\begin{aligned} \delta S ={}&- \int_M\mathrm d^d x\, \left[ (\Box+m^2)\phi\,\delta\phi^* +(\Box+m^2)\phi^*\,\delta\phi \right] \\ &+ \int_{\partial M}\mathrm d\Sigma_\mu\, \left( \partial^\mu\phi\,\delta\phi^* +\partial^\mu\phi^*\,\delta\phi \right). \end{aligned}$$

For variations that make the surface term vanish, the two bulk equations are

$$(\Box+m^2)\phi=0, \qquad (\Box+m^2)\phi^*=0.$$

Directly differentiating the current gives

$$\partial_\mu j^\mu = i\left(\phi^*\Box\phi-\phi\Box\phi^*\right).$$

Substituting $\Box\phi=-m^2\phi$ and its conjugate makes the two terms cancel. Also $[j^\mu]=d-1$, the correct dimension for a current density.

The natural next step is to ask what changes when these classical fields are used to build a quantum theory. Continue to Quantum fields, states, and observables, or return to Orientation and conventions if the metric, Fourier, or boundary choices in the checks were unclear.

References

• Harlow, Daniel, and Jie-qiang Wu. “Covariant Phase Space with Boundaries.” Journal of High Energy Physics 2020, no. 10 (2020): 146. DOI.
• 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.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI.