Classical Local Fields and Actions

A classical local field model begins with fields over spacetime, a local Lagrangian density, and a declared class of boundary or initial data. This chapter explains how those data produce three structures used throughout QFT: Euler–Lagrange equations, Hamiltonian initial data, and the first classical currents and stress tensors. Its recurring question is not merely “what is the action?” but “which conclusion follows from it, under which boundary and regularity assumptions?”

The free real scalar supplies the regular main line. Maxwell theory supplies the essential counterexample: a perfectly useful local action can have a degenerate Legendre map, so the ordinary Hamiltonian construction cannot simply be copied from the scalar. A complex scalar supplies the first symmetry application. The chapter stops before general PDE theory, constrained reduction, quantum Ward identities, and renormalized stress tensors in curved spacetime.

From local field data to equations, initial data, and currents

For fields $\phi^A$ on a spacetime region $\Omega$, a first-derivative local action has the form

$$S[\phi] = \int_\Omega \mathrm d^d x\, \mathcal L(\phi^A,\partial_\mu\phi^A;x).$$

Here $\phi^A(x)$ is a field value, while a configuration is an entire admissible assignment $x\mapsto\phi^A(x)$. A solution is a configuration that also satisfies the equations of motion and the chosen boundary or initial conditions. Keeping those three notions separate prevents kinematic data from being confused with dynamics. Local field data and the relativistic action principle are developed in Schwartz 2014, §§ 3.1–3.2, pp. 29–32 and Weinberg 1995, §§ 7.1–7.2, pp. 293–305.

For a first-derivative density, integration by parts gives

$$\begin{aligned} \delta S &= \int_\Omega \mathrm d^d x\, \mathcal E_A(\phi)\,\delta\phi^A \\ &\quad+ \int_{\partial\Omega}\mathrm d\Sigma\, n_\mu \frac{\partial\mathcal L} {\partial(\partial_\mu\phi^A)} \delta\phi^A, \\[2pt] \mathcal E_A &= \frac{\partial\mathcal L}{\partial\phi^A} - \partial_\mu \frac{\partial\mathcal L} {\partial(\partial_\mu\phi^A)}. \end{aligned}$$

The outward normal $n_\mu$ and induced measure $\mathrm d\Sigma$ fix the displayed boundary orientation. The equation $\mathcal E_A=0$ follows from stationary action only after the allowed variations and any boundary functional make the surface contribution vanish or cancel. This is variational differentiability; it does not by itself prove existence, uniqueness, or continuous dependence for the resulting PDE. Boundary-sensitive variational principles and a scalar example are treated in Harlow and Wu 2020, § 2.2, pp. 9–12; § 3.2, p. 21.

After choosing a time coordinate, define

$$\pi_A = \frac{\partial\mathcal L}{\partial\dot\phi^A}, \qquad W_{AB} = \frac{\partial^2\mathcal L} {\partial\dot\phi^A\,\partial\dot\phi^B}.$$

If the velocity Hessian $W_{AB}$ is invertible at a configuration, the inverse-function theorem permits the velocities to be expressed locally in terms of $(\phi^A,\pi_A)$, and the ordinary Legendre transform produces a local Hamiltonian description. If the Hessian is singular but has locally constant rank and a smooth image, relations among the canonical data appear as primary constraints; variable-rank cases can instead have stratified images and need separate analysis. Degeneracy diagnoses the need for constraint analysis; it does not by itself decide which variables are gauge, auxiliary, or physical. The regular field construction and the constrained case are separated in Weinberg 1995, §§ 7.1 and 7.6, pp. 293–297 and 325–328, while the stages of the Dirac–Bergmann analysis are distinguished in Brown 2022, §§ IV–X, pp. 5–11.

The symmetry branch begins from the same variation. When a constant transformation parameter defines a symmetry, allowing that parameter to depend on spacetime exposes a current. Its divergence vanishes after imposing the equations of motion, subject to the usual qualifications about boundary flux and improvement terms. This chapter works that mechanism in free scalar and spinor examples; general current operators, Ward identities, charge algebras, and gauge symmetry are developed later. The classical Noether construction and canonical stress tensor are presented in Schwartz 2014, §§ 3.3–3.3.1, pp. 32–35 and Weinberg 1995, § 7.3, pp. 306–313.

Stage	Governing question	Check before continuing	Result or next branch
Field data	What fields, locality assumptions, derivative order, scales, and admissible configurations define the model?	Distinguish a field value, a configuration, and a solution	A candidate local action and dimensional consistency conditions
Variation	What bulk and surface terms occur in $\delta S$?	Retain the total derivative and state the allowed variations	Euler–Lagrange equations only after the boundary term is controlled
Boundary choice	Which data are fixed, and is a boundary functional required?	Test differentiability on the declared class of fields	A well-defined variational problem, not yet a PDE theorem
Hamiltonian branch	Can the momenta be inverted for the velocities?	Compute the rank of $W_{AB}$	Canonical evolution when regular; constraint analysis when singular
Symmetry branch	Does the action change by a boundary term under the transformation?	Separate off-shell identities from on-shell conservation	A scope-limited current or stress-tensor calculation and a later symmetry handoff

Three entry routes

The chapter overview has no prerequisite. Choose the route from the question you want to answer, then close only the dependencies of the leaf you enter.

Reader’s question	Start and continue	Dependency to close	Observable exit
How does a local action produce field equations without hiding surface terms?	Fields, Configurations, Dimensions, and Local Dynamics → The Action Principle and Field Equations → Boundaries, Variations, and Well-Posed Actions	The boundary page uses the action-principle page; the field-data page is useful orientation rather than a requirement	Derive the bulk equation and state exactly why the surface term vanishes or cancels
What are the canonical initial data, and when does the Legendre transform fail?	The Action Principle and Field Equations → Hamiltonian Initial Data and Phase Space	Close the action-principle page before the Hamiltonian page	Construct scalar canonical data and recognize the singular Maxwell Hessian without prematurely declaring its physical degrees of freedom
How does a continuous transformation produce a first current or stress tensor?	The Action Principle and Field Equations → Classical Symmetries, Currents, and Stress Tensors	Close the action-principle page; the boundary page is helpful for surface terms and charges	Derive a free-model current, state whether conservation is on shell, and identify the later theory that controls improvements and charges

For a first pass through the whole chapter, follow the coordinate route through boundary terms, take the Hamiltonian branch, and finish with the symmetry application. The Hamiltonian and symmetry pages are parallel consequences of the action principle; neither requires the other.

Check your preparation

Use a missed prompt to identify a local repair rather than postponing the entire chapter.

Try this	Sufficient answer	If unsure
Distinguish $\phi(x)$, a configuration $\phi$, and a classical solution	They are respectively a value at one point, an admissible spacetime assignment, and an assignment satisfying both dynamics and declared data	Begin with Fields, Configurations, Dimensions, and Local Dynamics; for the linear structure, review Vector Spaces, Duals, and Linear Maps
Integrate $\partial_\mu\phi\,\partial^\mu\delta\phi$ by parts on a region with boundary	You produce both a bulk term and an outward-normal surface term	Review Field Variations and Boundary Terms and, if orientation is the obstacle, Differential Forms, Integration, Orientation, and Stokes Theorem
Check the dimensions of $\frac12(\partial\phi)^2$ in $d$ dimensions	From $[S]=0$ you infer $[\mathcal L]=d$ and $[\phi]=(d-2)/2$ for the canonically normalized scalar	Start with the chapter’s field-data page
Test whether $p_i=\partial L/\partial\dot q^i$ can be inverted	You examine the velocity Hessian rather than assuming inversion	Review Symplectic Forms, Hamiltonian Flows, and Poisson Brackets
Explain the difference between an off-shell identity and an on-shell conservation law	The identity holds for arbitrary configurations; the conservation law may use $\mathcal E_A=0$ and still requires control of boundary flux to define a conserved charge	Use the action-principle page, then the classical-symmetry page; Delta Distributions, Weak Derivatives, Pullbacks, and Pushforwards supplies useful distributional preparation

For a longer bridge, Variational and classical-field repair and Classical fields, actions, and local dynamics provide site-wide routes through the required variation, equation-of-motion, and classical-field ideas.

Convention and sign bridge

This chapter inherits the site conventions. The compact summary below is enough to compare sources that use different metric or boundary conventions.

Datum	Convention here	Invariant check
Spacetime	$\eta_{\mu\nu}=\operatorname{diag}(+1,-1,-1,-1)$	The free scalar has $\mathcal L=\frac12\dot\phi^2-\frac12(\boldsymbol\nabla\phi)^2-\frac12m^2\phi^2$ and positive Hamiltonian density
Units	$\hbar=c=1$, so $[S]=0$ and $[\mathcal L]=d$	Every term in one density has the same mass dimension
Boundary orientation	$\mathrm d\Sigma_\mu=n_\mu\mathrm d\Sigma$ with $n_\mu$ outward from $\Omega$	Reverse the orientation and the entire surface integral changes sign, not the bulk equation
Canonical variables	$\pi_A=\partial\mathcal L/\partial\dot\phi^A$ after a time slicing is chosen	Hamilton’s equations reproduce the Euler–Lagrange equation when the Legendre map is regular
Equality language	“Off shell” means before imposing $\mathcal E_A=0$; “on shell” means after imposing it	A Noether identity and a conserved current are not silently treated as the same statement
Scaling language	Brackets $[\,cdot\,]$ denote engineering mass dimension here	Engineering dimension is not asserted to equal the full quantum scaling dimension of an interacting operator

For the canonically normalized real scalar in $d$ spacetime dimensions,

$$[\mathcal L]=d, \qquad [\partial_\mu]=1, \qquad [\phi]=\frac{d-2}{2}.$$

This bookkeeping result follows from the kinetic term and is independent of the sign convention for the metric. Its use and its quantum limitation are summarized in Schwartz 2014, Appendix A.1, pp. 815–816.

Three examples carried through the chapter

Free real scalar — the regular line. With the site convention,

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

Its variation is

$$\begin{aligned} \delta S_\phi &= -\int_\Omega \mathrm d^d x\, (\Box+m^2)\phi\,\delta\phi \\ &\quad+ \int_{\partial\Omega}\mathrm d\Sigma\, n_\mu\partial^\mu\phi\,\delta\phi. \end{aligned}$$

Dirichlet variations set $\delta\phi|_{\partial\Omega}=0$; other data may require a different boundary functional. On a constant-time slice, $\pi=\dot\phi$ and

$$\mathcal H_\phi = \frac12\pi^2 + \frac12|\boldsymbol\nabla\phi|^2 + \frac12m^2\phi^2.$$

Thus the same signs are checked three ways: the Klein–Gordon equation is $(\Box+m^2)\phi=0$, the surface term is retained, and the Hamiltonian density is nonnegative for $m^2\geq0$. The complete derivation is reserved for the action, boundary, and Hamiltonian leaves.

Maxwell theory — the degeneracy line. For $\mathcal L_A=-\frac14F_{\mu\nu}F^{\mu\nu}$, variation again produces a bulk equation and a surface term. But no time derivative of $A_0$ occurs, so its canonical momentum vanishes and the velocity Hessian is singular. This is the point where the scalar recipe stops: constraint consistency, gauge transformations, and physical polarization counting require the dedicated constrained and gauge treatments. Calling $A_0$ “unphysical” from the Hessian alone skips that analysis.

Complex scalar — the symmetry line. The density $\partial_\mu\Phi^*\partial^\mu\Phi-m^2\Phi^*\Phi$ is invariant under a constant phase rotation. Promoting its parameter to a spacetime-dependent test function isolates the $U(1)$ current. The divergence vanishes on the equations of motion, while conservation of the integrated charge also requires the relevant flux through the boundary to vanish. Improvement freedom and the quantum definition of composite currents belong to the later symmetry treatment.

The five pages in order

The order below is the chapter order. “Requires” records only a dependency needed by the page’s central argument; useful orientation is stated separately.

Page	Role	Main task and first check	Preparation inside the chapter	Stopping boundary and continuation
1. Fields, Configurations, Dimensions, and Local Dynamics	Concept and definitions	Specify scalar, spinor, Proca, and Maxwell field data; separate kinematics from dynamics; check engineering dimensions and derivative order	Enter directly	Power counting continues with Power Counting and Superficial Degree of Divergence; bundle and gauge equivalence require their mathematical and symmetry treatments
2. The Action Principle and Field Equations	Derivation of field equations	Derive the Klein–Gordon and free Maxwell equations while retaining every integration-by-parts term	Enter directly; page 1 is helpful orientation	Functional quantization comes later in Foundations; higher-derivative stability requires a separate treatment
3. Boundaries, Variations, and Well-Posed Actions	Method for boundary control	Match the scalar or Maxwell surface term to admissible data or a boundary functional	Requires the action-principle page	Analytic well-posedness continues with Weak Solutions, Sobolev Spaces, and Well-Posedness; boundary QFT and gravitational boundary charges lie beyond this chapter
4. Hamiltonian Initial Data and Phase Space	Method for canonical data	Construct scalar canonical data, equal-time brackets, and Hamiltonian evolution; then diagnose Maxwell degeneracy	Requires the action-principle page	General symplectic geometry and constrained reduction continue in Mathematical Methods; gauge generators and BRST/BV continue in Symmetry and Gauge Structure
5. Classical Symmetries, Currents, and Stress Tensors	First symmetry application	Work scalar and spinor symmetry variations with on-shell and improvement qualifications explicit	Requires the action-principle page; the boundary page is helpful	General quantum currents and charge algebras continue in Symmetry and Gauge Structure; Hilbert and renormalized stress tensors continue in Curved Spacetime QFT

Exact continuations

Question leaving the chapter	Continue with
Quantize the regular scalar canonical data	Canonical Quantization and the Free Scalar
Develop variational calculus and boundary geometry abstractly	Field Variations and Boundary Terms and Differential Forms, Integration, Orientation, and Stokes Theorem
Prove existence, uniqueness, or stability for field equations	Weak Solutions, Sobolev Spaces, and Well-Posedness
Develop field phase space and Poisson geometry	Symplectic Forms, Hamiltonian Flows, and Poisson Brackets
Classify and reduce constrained systems	Constraints, Dirac Brackets, and Symplectic Reduction
Determine gauge orbits, Gauss constraints, and physical variables	Gauge Orbits, Gauss Constraints, and Stabilizers
Develop continuous symmetries and classical charges systematically	Continuous Symmetries, Generators, and Charges
Define quantum currents and their improvements	Quantum Currents, Improvements, and Conservation
Relate spacetime currents, stress tensors, and charge algebras	Spacetime Currents, Stress Tensors, and Charge Algebras
Treat Hilbert and renormalized stress tensors in curved spacetime	Renormalized Stress Tensor: Axioms and Curvature Ambiguities
Turn engineering dimensions into an EFT expansion	Power Counting and Predictive Order

Four distinctions that survive the chapter

A local density is not a complete theory. It must be paired with a spacetime setting, a field configuration space, admissible data, and an interpretation of any redundancy. In quantum theory it will need still more: a state or measure, observables, a regulator or limiting prescription where applicable, and renormalization data.

A differentiable action is not a well-posed PDE theorem. Controlling $\delta S$ identifies the equations and compatible variational data. Analytic well-posedness asks whether solutions exist, are unique, and depend continuously on their data in specified function spaces.

A singular Hessian is not a completed gauge reduction. It signals primary relations among canonical variables. Consistency conditions, first- and second-class classification, boundary terms, and the action of candidate generators must still be analyzed.

An on-shell conserved current is not automatically a conserved charge. One must specify a hypersurface, falloff or boundary conditions, and possible improvements. At the quantum level the composite current also requires a controlled definition, and anomalies can obstruct the classical conclusion.

What the chapter establishes—and what it does not

At the chapter exit, you should be able to:

• state the data of a classical local field model and distinguish kinematics from dynamics;
• derive Euler–Lagrange equations with dimensions, signs, boundary orientation, and admissible variations explicit;
• distinguish variational differentiability from analytic PDE well-posedness;
• construct canonical momenta, a Hamiltonian density, and equal-time data when the Legendre map is regular;
• use Hessian degeneracy as a signal for constraint analysis without treating it as the analysis itself; and
• derive a qualified free-model current or stress tensor while stating its on-shell, boundary, and improvement conditions.

The chapter does not prove general PDE theorems, develop infinite-dimensional symplectic geometry, carry out Dirac reduction, define interacting quantum currents, establish Ward identities, or renormalize stress tensors in curved spacetime. Those are explicit continuations. The scalar thread next becomes a quantum theory in Canonical Quantization and the Free Scalar.

Review the chapter

Prompt	A satisfactory answer includes	Repair route
A derivation writes $\delta S=-\int(\Box+m^2)\phi\,\delta\phi$ and stops. What is missing?	The outward-normal surface integral is restored, and the answer states which boundary variations or boundary functional remove it	Review The Action Principle and Field Equations and Boundaries, Variations, and Well-Posed Actions
Compare the velocity Hessians of the real scalar and Maxwell field	The scalar Hessian is invertible for its velocity, whereas the absence of $\dot A_0$ makes the Maxwell Hessian singular; further constraint analysis is required before identifying physical variables	Review Hamiltonian Initial Data and Phase Space and the constrained-reduction continuation above
In $d=4$, a scalar kinetic term gives $[\phi]=1$. Does that prove an interacting operator has scaling dimension one?	No. The result is its engineering dimension in the specified normalization; anomalous dimensions can change quantum scaling	Review Fields, Configurations, Dimensions, and Local Dynamics and the EFT continuations above
A current obeys $\partial_\mu j^\mu=0$ on shell. When is $Q=\int_\Sigma j^\mu\mathrm d\Sigma_\mu$ conserved?	The equations hold and the flux through the remaining boundary vanishes or is otherwise accounted for; improvements and quantum definitions remain qualified	Review Classical Symmetries, Currents, and Stress Tensors and the symmetry continuations above
Choose a route for “derive the Klein–Gordon equation, prepare initial data, then quantize.”	Action principle → Hamiltonian initial data → canonical quantization, with the boundary page inserted when the admissible data or surface term are not already controlled	Follow the second entry route, then continue to the free-scalar quantization chapter
Add $-\lambda\phi^4/4!$ to the real-scalar density. Which chapter-scale checks change, and which do not?	In $d=4$, $\lambda$ is engineering-dimension zero; the bulk equation gains $\lambda\phi^3/3!$ and the Hamiltonian gains $+\lambda\phi^4/4!$, while the non-derivative interaction leaves the displayed surface term and velocity Hessian unchanged. Quantum running and predictive power counting are deferred to Renormalization and EFT	Revisit the field-data, action, and Hamiltonian pages, then use Power Counting and Predictive Order

References

• Brown, J. David. “Singular Lagrangians, Constrained Hamiltonian Systems and Gauge Invariance: An Example of the Dirac–Bergmann Algorithm.” Universe 8, no. 3 (2022): 171. DOI. Open PDF, arXiv:2201.06558v3.
• Harlow, Daniel, and Jie-qiang Wu. “Covariant Phase Space with Boundaries.” Journal of High Energy Physics 2020, no. 10 (2020): 146. DOI. Open PDF, arXiv:1906.08616v4.
• 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.