Generalized Classical Field Theory

A field is a dynamical system carrying a degree of freedom at every point of space, and this chapter builds the machinery such systems require: the action functional and its Lagrangian density, the Euler–Lagrange field equations, Noether's theorems and the currents and charges they produce [Noether:1918], the energy–momentum tensor in its canonical and improved forms [Belinfante:1940], and the Hamiltonian formulation. The construction is the continuum limit of the machinery of Lagrangian Mechanics and Hamiltonian Mechanics: the label \(a\) that ran over finitely many coordinates becomes the continuous label \(\vect{x}\), sums become integrals, and derivatives become functional derivatives in the sense of Calculus of Variations. What survives the limit is everything that matters — a variational principle, a symmetry-to-conservation-law correspondence, and a phase space.

The chapter sits here, at the end of Part VII, because electromagnetism is the classical field theory for which the evidence is overwhelming and which therefore earns the general formalism. Maxwell's equations [Maxwell:1865] follow from a single Lorentz-scalar Lagrangian density; gauge invariance of that density forces charge conservation; the same variational calculation delivers the energy density, the Poynting flux [Poynting:1884] and the Maxwell stress [Maxwell:1873] that are measured as radiation pressure [Lebedew:1901] [Nichols:1903]. Two further systems are stated for later use: the classical Klein–Gordon field [Gordon:1926], the simplest relativistic field of all and the template for every scalar in Electroweak Unification and the Higgs Boson, and the massive vector field of Proca [Proca:1936], whose comparison with Maxwell is what turns laboratory electrostatics into a bound on the photon mass. Where the Legendre transformation of Hamiltonian Mechanics fails — and for every gauge theory it does fail — the treatment is the constraint formalism already developed in Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism, which this chapter uses rather than repeats. Standard treatments are [Landau:1975] [Jackson:1999] [Weinberg:1995].

Fields as dynamical systems

From many degrees of freedom to a continuum

[Reserved: the loaded string of Oscillations and Mechanical Waves taken to the limit of infinitely many beads at fixed mass density, in which the discrete coordinates \(q_{a}(t)\) become a field \(\phi(t, \vect{x})\) and the coupled equations become a wave equation; the field as the physical object rather than a bookkeeping device, which is Maxwell's own reading of the electromagnetic field as a dynamical system with energy stored in space [Maxwell:1865]; locality as the statement that the Lagrangian is an integral of a density depending on the field and its first derivatives at one point; and why a nonrelativistic mechanical model of that density — an aether — is neither needed nor tenable (Experiments: Light, the Aether, and Time).]

The action functional and the Lagrangian density

[Reserved: the action \(S[\phi]=\int\Lag(\phi,\pp_{\mu}\phi)\, \dd^{4}x\) over a spacetime region, with \(\Lag\) a Lorentz scalar density in the sense of Minkowski Space and Its Symmetries; the three standing restrictions — locality, at most first derivatives, and invariance under the Poincaré group — and what each buys; the dimensions of \(\Lag\) in SI (energy per unit volume) and the normalization conventions fixed here for the whole treatise; the equivalence class of Lagrangians differing by a total divergence [Landau:1975].]

Functional derivatives

[Reserved: the functional derivative \(\delta S/\delta\phi(x)\) defined by the first variation, its delta-function algebra \(\delta \phi(x)/\delta\phi(y)=\delta^{4}(x-y)\), and the conditions under which a functional is differentiable, taken from Calculus of Variations; the field-theoretic version of the Euler operator; the notation used throughout Parts VII and XI is fixed here once [Weinberg:1995].]

The Euler–Lagrange field equations

Variation, boundary terms and natural conditions

[Reserved: the derivation of \(\pp_{\mu}\left(\pp\Lag/\pp(\pp_{\mu} \phi)\right)-\pp\Lag/\pp\phi=0\) by variation at fixed data on the boundary of the region; the surface term and the two ways to kill it — fixed variations or natural boundary conditions; why a total divergence added to \(\Lag\) changes nothing; the extension to several fields and to higher derivatives, with the Ostrogradsky instability as the reason the treatise stops at first order [Landau:1975].]

The real scalar field: the Klein–Gordon equation

[Reserved: the Lagrangian density \(\tfrac{1}{2}\pp_{\mu}\phi\, \pp^{\mu}\phi-\tfrac{1}{2}\mu^{2}\phi^{2}\) and its field equation \((\Box+\mu^{2})\phi=0\) with \(\mu=mc/\hbar\), written down in 1926 by Gordon [Gordon:1926] and by Klein [Klein:1926] (the latter in a five-dimensional setting whose extra dimension has no evidence and is not used anywhere in this treatise); plane-wave solutions and the dispersion relation \(\omega^{2}=c^{2}k^{2}+\mu^{2}c^{2}\); the Yukawa static solution \(\ee^{-\mu r}/r\) and its interpretation as a force of finite range [Yukawa:1935]; the classical field is developed here, its quantization in The Klein–Gordon Equation.]

The complex scalar field and internal symmetry

[Reserved: the complex field with \(\Lag=\pp_{\mu}\phi^{*}\, \pp^{\mu}\phi-\mu^{2}\abs{\phi}^{2}\); invariance under the global phase rotation \(\phi\mapsto\ee^{\ii\alpha}\phi\), an internal \(\U(1)\) symmetry in the sense of Lie Groups, Lie Algebras, and Fibre Bundles; the associated conserved current and the charge it integrates to, obtained from the theorem of Section 64.3.1; why the current is not a probability density and what that cost the interpretation of the equation before quantization [Weinberg:1995].]

Symmetries and conservation laws

Noether's first theorem

[Reserved: the theorem of [Noether:1918] — to every continuous group of transformations leaving the action invariant, with \(r\) independent parameters, correspond \(r\) currents \(j^{\mu}_{A}\) satisfying \(\pp_{\mu}j^{\mu}_{A}=0\) on shell; the charge \(Q_{A}=\int j^{0}_{A}\,\dd^{3}x\) is constant provided the flux through the boundary vanishes; invariance up to a total divergence is enough, which is what covers translations and Galilean boosts; the ambiguity \(j^{\mu}\mapsto j^{\mu}+\pp_{\nu}k^{[\nu\mu]}\) that leaves the divergence and the charge untouched, and which the improvements of Section 64.4.2 exploit.]

Poincaré invariance and the ten charges

[Reserved: the four translations giving the energy–momentum tensor and the six Lorentz transformations giving the angular-momentum current \(M^{\mu\nu\rho}\), whose conservation splits into orbital and spin parts only after the tensor is symmetrized; the ten conserved charges as the classical counterpart of the Poincaré generators of Minkowski Space and Its Symmetries and, after quantization, of the Wigner classification [Wigner:1939] used in Particles as Poincaré Representations; conformal invariance of the free massless field and the extra currents Bessel-Hagen extracted for electrodynamics [BesselHagen:1921].]

Internal symmetry and electric charge

[Reserved: the Noether charge of the internal \(\U(1)\) identified with electric charge in coulombs; global versus local invariance; conservation of charge as an experimental claim rather than a convention, with the sharpest laboratory bound coming from searches for charge-violating electron decay — a lifetime exceeding \(6.6\times 10^{28}\,\mathrm{yr}\) for \(e\to\nu\gamma\) from Borexino [Agostini:2015]; the compilation of related bounds in [Navas:2024].]

Noether's second theorem and local symmetries

[Reserved: the second theorem of [Noether:1918] — a symmetry depending on arbitrary functions of spacetime does not give new conservation laws but identities among the field equations, so that the equations are not independent and the evolution is underdetermined; the identity \(\pp_{\mu}\pp_{\nu}F^{\mu\nu}\equiv0\) for electrodynamics and the contracted Bianchi identity for gravitation (The Einstein Field Equations) as the two instances that matter here; the corollary that a locally invariant theory has a singular Legendre transformation, which is where Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism takes over.]

The energy–momentum tensor

The canonical tensor and its defects

[Reserved: \(T^{\mu\nu}_{\text{can}}=\left(\pp\Lag/\pp(\pp_{\mu} \phi)\right)\pp^{\nu}\phi-\eta^{\mu\nu}\Lag\) as the Noether current of translations [Noether:1918]; the interpretation of its components as energy density, momentum density, energy flux and stress, in SI units; the two defects — it is in general neither symmetric, so it cannot serve as the source of The Einstein Field Equations, nor gauge invariant, which for electrodynamics makes even its energy density depend on the gauge chosen [Landau:1975].]

The Belinfante–Rosenfeld improvement

[Reserved: the improvement term built from the spin current, \(T^{\mu\nu}=T^{\mu\nu}_{\text{can}}+\pp_{\rho}k^{[\rho\mu]\nu}\), constructed independently by Belinfante [Belinfante:1940] and Rosenfeld [Rosenfeld:1940]; that it changes neither the conservation law nor the total charges, being of the superpotential form allowed by Section 64.3.1; that the result is symmetric and, for electrodynamics, gauge invariant; and the resulting clean separation of angular momentum into orbital and spin parts, the spin part being what Beth measured for light [Beth:1936].]

The Hilbert tensor and the gravitational source

[Reserved: the definition of \(T^{\mu\nu}\) as the response of the matter action to a variation of the metric [Hilbert:1915], which is symmetric and gauge invariant by construction; the theorem that it coincides with the Belinfante tensor for every field of this chapter; its conservation as a consequence of diffeomorphism invariance by Noether's second theorem; and the statement that this is the object appearing on the right-hand side of the Einstein equations of The Einstein Field Equations, which is why the choice of improvement is not a matter of taste.]

What is actually measured

[Reserved: the measured consequences of the electromagnetic stress–energy tensor — radiation pressure on a mirror, detected by Lebedev [Lebedew:1901] and by Nichols and Hull [Nichols:1903]; the Maxwell stress on conductors and dielectrics [Maxwell:1873]; the angular momentum of circularly polarized light [Beth:1936]; and the momentum of light in a medium, where the rival Minkowski [Minkowski:1908] and Abraham [Abraham:1909] forms differ by exactly a division of the total between field and matter, a question belonging to Electrodynamics in Matter.]

The electromagnetic field as a Lagrangian system

The free-field Lagrangian

[Reserved: \(\Lag=-\tfrac{1}{4\mu_{0}}F_{\mu\nu}F^{\mu\nu} -A_{\mu}J^{\mu}\) as the unique Poincaré- and gauge-invariant density quadratic in first derivatives of \(A_{\mu}\); the Euler–Lagrange equations returning the inhomogeneous Maxwell equations of The Maxwell Equations while the homogeneous pair is the identity \(\pp_{[\rho}F_{\mu\nu]}=0\) satisfied by any \(F=\dd A\) — so half of Maxwell's system is a consequence of using a potential at all; the second invariant \(F_{\mu\nu}\tilde{F}^{\mu\nu}\), a total divergence classically and parity-odd, connecting to Discrete Symmetries and CPT [Landau:1975] [Jackson:1999].]

Minimal coupling and the gauge principle

[Reserved: the replacement \(\pp_{\mu}\to\pp_{\mu}+\ii qA_{\mu} /\hbar\) that promotes a global phase symmetry to a local one, introduced for the relativistic charged particle by Fock [Fock:1926] and raised to a principle by Weyl [Weyl:1929]; the interaction term \(-A_{\mu}J^{\mu}\) and the Lorentz force recovered from it, matching Relativistic Dynamics and the electron theory of [Lorentz:1909]; the honest statement that the gauge principle predicts the form of the coupling but not the value of \(q\), whose quantization is an experimental fact (Experiments: Electromagnetism).]

Gauge invariance and charge conservation

[Reserved: the demonstration that invariance of the action under \(A_{\mu}\mapsto A_{\mu}+\pp_{\mu}\Lambda\) requires \(\pp_{\mu}J^{\mu}=0\) identically, not merely on shell — charge conservation is a consistency condition on the coupling, by Noether's second theorem [Noether:1918] [BesselHagen:1921]; gauge fixing as the removal of the resulting redundancy, with the Lorenz condition [Lorenz:1867] and the Coulomb and temporal gauges; and the count of physical polarizations deferred to Section 64.7.3.]

The electromagnetic energy–momentum tensor

[Reserved: the canonical tensor of the Maxwell field, which is neither symmetric nor gauge invariant, and the Belinfante improvement [Belinfante:1940] that yields \(T^{\mu\nu}=\mu_{0}^{-1}\left( F^{\mu\rho}F^{\nu}{}_{\rho}-\tfrac{1}{4}\eta^{\mu\nu}F^{\rho\sigma} F_{\rho\sigma}\right)\); identification of its components with the energy density \(\tfrac{1}{2}(\epsilon_{0}E^{2}+B^{2}/\mu_{0})\), the Poynting vector [Poynting:1884] and the Maxwell stress [Maxwell:1873]; tracelessness as the classical shadow of conformal invariance [BesselHagen:1921]; and the use of the same tensor for the radiated power of Radiation and Scattering of Electromagnetic Waves.]

The non-abelian generalization

[Reserved: Yang and Mills' extension of the gauge principle to a non-abelian group [Yang:1954], with \(F_{\mu\nu}=\pp_{\mu}A_{\nu} -\pp_{\nu}A_{\mu}+g\comm{A_{\mu}}{A_{\nu}}\) and a Lagrangian of the same form; the decisive structural difference — the field strength is not gauge invariant but covariant, and the field carries its own charge, so the free theory is already nonlinear; the classical statement is made here, and the evidence for it is the strong and electroweak sectors of Quantum Chromodynamics and Electroweak Unification and the Higgs Boson.]

Massive vector fields: the Proca theory

The Proca equation

[Reserved: the Lagrangian \(-\tfrac{1}{4\mu_{0}}F_{\mu\nu} F^{\mu\nu}+\tfrac{1}{2}\mu^{2}A_{\mu}A^{\mu}\) of [Proca:1936] and its field equation \(\pp_{\mu}F^{\mu\nu}+\mu^{2}A^{\nu}=0\); the mass term breaks gauge invariance, so \(A_{\mu}\) is no longer redundant and the Lorenz condition \(\pp_{\mu}A^{\mu}=0\) becomes a consequence rather than a choice; three propagating polarizations instead of two; the static potential becomes Yukawa's \(\ee^{-\mu r}/r\) [Yukawa:1935] with range \(\hbar/mc\), which is the observable signature; and the discontinuity of the degree-of-freedom count in the massless limit, already analysed in Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism.]

The photon mass as a measured quantity

[Reserved: how a laboratory test of Coulomb's law converts into an upper bound on \(\mu\), since a Proca field gives a deviation growing with apparatus size — the modern laboratory limit [Williams:1971] and the far sharper bounds from planetary and galactic magnetic fields reviewed in [Goldhaber:2010], with the value adopted by this treatise taken from [Navas:2024], currently of order \(10^{-18}\,\mathrm{eV}/c^{2}\); and the methodological point that “the photon is massless” is a bound, never a measurement, so the correct statement is a number with a confidence level.]

Hamiltonian field theory

Momentum density and the Hamiltonian

[Reserved: the momentum density \(\pi=\pp\Lag/\pp\dot{\phi}\), the Hamiltonian density \(\Ham=\pi\dot{\phi}-\Lag\) and the field equations in first-order form; the loss of manifest covariance under the required 3+1 split of Minkowski Space and Its Symmetries, and the fact that the resulting energy nonetheless agrees with \(T^{00}\) of Section 64.4.2; the scalar and Maxwell fields worked as the two standing examples [Weinberg:1995].]

Poisson brackets for fields

[Reserved: the equal-time bracket \(\pb{\phi(\vect{x})}{\pi( \vect{y})}=\delta^{3}(\vect{x}-\vect{y})\) built from functional derivatives, generalizing Hamiltonian Mechanics; the symplectic structure on the space of solutions; the generators of the Poincaré group realized as brackets, closing on the algebra of Minkowski Space and Its Symmetries; and this bracket as the object that the correspondence rule of Canonical Quantization of Fields turns into a commutator.]

Singular Lagrangians and constraints

[Reserved: the Maxwell Lagrangian is singular — \(\pp\Lag/ \pp\dot{A}_{0}\equiv0\) — so its Hamiltonian formulation is the constrained one of Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism, whose results are used here and not rederived: the primary constraint \(\pi^{0}\approx0\), the secondary Gauss constraint, both first class, and the count of two physical polarizations per point of space, against three for the Proca field of Section 64.6.1; the observed absence of a longitudinal mode of light (Experiment: Wave Optics) is what this count predicts.]

Covariant Hamiltonian formulations

[Reserved: the De Donder–Weyl formalism [DeDonder:1935] [Weyl:1935], in which each field carries one polymomentum per spacetime direction and the Hamilton equations are first-order and manifestly covariant; the covariant phase space as the space of solutions with its symplectic form; what these formulations buy (covariance, a clean statement of the Noether charges as forms) and what they do not (a quantization rule of comparable standing to Canonical Quantization of Fields).]

From classical fields to quanta

[Reserved: the closing section, stating precisely what this chapter hands to Part XI and what it cannot decide. The mode expansion of a free field turns it into a continuum of harmonic oscillators (Oscillations and Mechanical Waves), which is the entire content of free field quantization; the canonical route is Canonical Quantization of Fields and the path-integral route Path-Integral Quantization; the classification of the resulting one-particle states by mass and spin is Wigner's [Wigner:1939], treated in Particles as Poincaré Representations. What classical field theory already fixes — the number of polarizations, the form of the couplings, the conserved charges — and what it cannot — the value of \(\hbar\), the spin–statistics connection, and the very existence of quanta — should be stated plainly here [Weinberg:1995].]