The Klein–Gordon Equation
The first relativistic wave equation of quantum mechanics was written down before the nonrelativistic one, and discarded. Schrödinger set up the wave equation of a Coulomb-bound electron in relativistic form, found that it gave the wrong fine structure of hydrogen — the term Sommerfeld's old-quantum formula [Sommerfeld:1916] had already fitted — and published instead the nonrelativistic limit that carries his name [Schroedinger:1926a] [Kragh:1982], returning to the relativistic form only in his fourth communication [Schroedinger:1926d]. Within months the same equation was published by Klein [Klein:1926] and, in its minimally coupled form and applied to the Compton effect, by Gordon [Gordon:1926]; it has carried their two names ever since. This chapter opens Part X — Relativistic Quantum Mechanics because the equation is the simplest possible marriage of Part IX — Quantum Mechanics with Part IV — Special Relativity: it is nothing but the mass-shell relation \(E^{2}=\abs{\vect{p}}^{2}c^{2}+m^{2}c^{4}\) of Relativistic Dynamics with the operator substitutions of The Postulates of Quantum Mechanics, and equivalently the eigenvalue equation of the first Poincaré Casimir \(\hat{P}^{2}=m^{2}c^{2}\) of Particles as Poincaré Representations at spin zero.
It is also where that marriage fails. Being second order in time, the equation admits both signs of the energy, and the density built from its conserved current is not positive: neither object can be read as a probability, so the equation is not a one-particle quantum mechanics. Dirac's response was a different equation (The Dirac Equation); the correct response, given by Pauli and Weisskopf in 1934 [Pauli:1934], was to keep this one and change what it describes — not a wavefunction but a field, whose quanta are particles and antiparticles and whose conserved density is charge rather than probability. Read that way the equation is exact and is used everywhere: it is the free part of every scalar field in Part XI — Quantum Field Theory and the Standard Model, its static source solution is the Yukawa potential of nuclear physics [Yukawa:1935], and its observed quanta are the pion [Lattes:1947] and the Higgs boson [Aad:2012] [Chatrchyan:2012]. Standard treatments are [Bjorken:1964] [Weinberg:1995] [Sakurai:2017].
The Klein–Gordon Equation: all derivations of this chapter are pending.
The equation
Schrödinger's discarded first attempt
[Reserved: the historical order of events, which matters because it explains why a correct equation was abandoned — Schrödinger's relativistic Coulomb calculation of late 1925, its disagreement with the measured fine structure of hydrogen, and the nonrelativistic retreat published as the first communication [Schroedinger:1926a], documented in detail by Kragh [Kragh:1982]; the relativistic equation with electromagnetic coupling restored in the fourth communication [Schroedinger:1926d]. The disagreement was real and its cause was spin, not relativity, as The Dirac Equation shows.]
Klein, Gordon, and the equation itself
[Reserved: the equation \(\left(\Box+m^{2}c^{2}/\hbar^{2}\right)\varphi=0\) with \(\Box=\pp_{\mu}\pp^{\mu}\), as published by Klein [Klein:1926] — whose five-dimensional framing is a unification programme without evidence and is treated as such in Part XIII — Frontiers and Open Problems — and by Gordon [Gordon:1926]; the canonical substitutions \(E\mapsto\ii\hbar\pp_{t}\), \(\vect{p}\mapsto-\ii\hbar\vect{\nabla}\) applied to the squared mass-shell relation, and why squaring is the step that costs the probability interpretation; the Compton wavelength \(\hbar/mc\) as the length scale the mass term introduces.]
The mass shell and the Poincaré Casimir
[Reserved: the equation restated group-theoretically. Every irreducible unitary representation of the Poincaré group carries \(\hat{P}^{2}=m^{2}c^{2}\) [Wigner:1939], so every relativistic field of any spin satisfies the Klein–Gordon equation component by component; what distinguishes spin 0 is that this is the only condition. The classification is carried out in Particles as Poincaré Representations and instantiated for spin \(\tfrac{1}{2}\) in The Dirac Equation and for higher spin in Higher-Spin Wave Equations.]
Every free particle observed in Nature, of whatever spin, carries an energy and a momentum related by
with a single constant \(m\) — its mass — that is the same for every particle of the species and the same in every inertial frame. The relation holds across the whole range of momenta accessible to experiment, and the masses it defines are tabulated for every known species in [Navas:2024]; its most direct laboratory face, the saturation of a charged particle's speed at \(c\) however much kinetic energy is supplied to it, was measured by Bertozzi [Bertozzi:1964].
Derivation. Insert the plane wave \(\varphi(x)=A\,\ee^{-\ii p\cdot x/\hbar}\), with \(p\cdot x=\eta_{\mu\nu}p^{\mu}x^{\nu}=Et-\vect{p}\cdot\vect{x}\), into the Klein–Gordon equation \(\left(\Box+m^{2}c^{2}/\hbar^{2}\right)\varphi=0\) of Section 92.1.2, where \(\Box=\pp_{\mu}\pp^{\mu}\). Each derivative brings down a factor, \(\pp_{\mu}\varphi=-\ii p_{\mu}\varphi/\hbar\), so that \(\Box\varphi=-p_{\mu}p^{\mu}\varphi/\hbar^{2}\) and
A nonzero solution therefore requires \(p_{\mu}p^{\mu}=m^{2}c^{2}\), which written out is \(E^{2}/c^{2}-\abs{\vect{p}}^{2}=m^{2}c^{2}\) and hence Equation (92.1). Both sides are Lorentz scalars, so every frame returns the same \(m\). The universality is a group-theoretic fact rather than a property of spin zero: every irreducible unitary representation of the Poincaré group is labelled by the eigenvalue of the Casimir \(\hat{P}^{2}\) [Wigner:1939], so a field of any spin satisfies this same equation component by component, and what singles out spin zero is only that no further condition is imposed.
∎Lorentz covariance and the scalar transformation law
[Reserved: \(\varphi'(x')=\varphi(x)\) as the defining transformation law of a scalar; invariance of the d'Alembertian under the transformations of Lorentz Transformations; the Lagrangian density \(\Lag=\tfrac{1}{2}\pp_{\mu}\varphi\,\pp^{\mu}\varphi -\tfrac{1}{2}(mc/\hbar)^{2}\varphi^{2}\) whose Euler–Lagrange equation this is, connecting to the field-theoretic variational formalism of Generalized Classical Field Theory; the distinction between a scalar and a pseudoscalar under parity, which is what the observed pion turns out to be (Discrete Symmetries and CPT).]
Negative energies and the indefinite density
The two defects of the equation as a one-particle theory are usually listed separately. They are the same defect.
Both branches of the dispersion relation
[Reserved: plane-wave solutions \(\ee^{-\ii p\cdot x/\hbar}\) with \(p^{0}=\pm\sqrt{\abs{\vect{p}}^{2}+m^{2}c^{2}}\); the negative-energy branch cannot be discarded because the positive-energy solutions alone are not complete — any localized initial datum excites both — and because a Lorentz transformation cannot remove it, the two branches being separate orbits in Particles as Poincaré Representations; the resulting instability of any system coupled to such a spectrum, and Dirac's statement of the difficulty as the motivation for a first-order equation [Dirac:1928].]
The conserved current and why its density changes sign
[Reserved: the current \(j^{\mu}=\tfrac{\ii\hbar}{2m}\left(\varphi^{*}\pp^{\mu}\varphi -\varphi\,\pp^{\mu}\varphi^{*}\right)\) conserved by \(\pp_{\mu}j^{\mu}=0\); the density \(j^{0}\) containing a time derivative because the equation is second order, so that \(\varphi\) and \(\pp_{t}\varphi\) are independent initial data and \(j^{0}\) takes either sign; the contrast with the positive-definite \(\abs{\psi}^{2}\) of The Postulates of Quantum Mechanics and with the Dirac current of The Dirac Equation [Dirac:1928].]
The quantity conserved in the processes that spin-0 particles undergo is electric charge, which takes both signs, and not the number of particles, which does not survive: charged pions are created singly in weak decays, in pairs of opposite charge in strong collisions, and the neutral pion disappears entirely into two photons. Every charged spin-0 species observed comes with a partner of equal mass and opposite charge, while the neutral pion is its own antiparticle [Navas:2024]. No experiment counts a conserved number of spin-0 quanta; every experiment finds the total charge unchanged.
Derivation. That the equation cannot supply a conserved probability is immediate. Evaluate the current of Section 92.2.2 on the plane wave \(\varphi=A\,\ee^{-\ii p\cdot x/\hbar}\) of Phenomenon 92.1. With \(\pp^{0}=c^{-1}\pp_{t}\) and \(\pp_{t}\varphi=-\ii E\varphi/\hbar\),
The sign of \(j^{0}\) is the sign of \(E\), so the two branches of Equation (92.1) carry densities of opposite sign, and no choice of normalization can make \(j^{0}\) positive on both — while the positive-frequency solutions alone are not a complete set, so the negative branch cannot simply be discarded. A quantity that changes sign cannot be a probability density. It can be a charge density, and multiplying \(j^{\mu}\) by the charge \(q\) makes it exactly that; the conserved integral is then a charge, which pair creation leaves unchanged because the pair is neutral. This is the reinterpretation proved in Section 92.6, and it is what the observations above record.
∎The two problems are one problem
[Reserved: that \(j^{0}\) is negative exactly on the negative-frequency solutions, so a theory that could exclude one defect could exclude the other; Feshbach and Villars' explicit exhibition of this correspondence [Feshbach:1958a]; the statement, to be proved in Section 92.6, that the resolution is to keep both branches and reinterpret \(ej^{\mu}\) as an electric current [Pauli:1934], antiparticles being the negative-frequency modes read forwards in time [Stueckelberg:1941] [Feynman:1949a].]
Minimal coupling and the charged scalar
The covariant derivative and gauge invariance
[Reserved: the substitution \(\pp_{\mu}\mapsto D_{\mu}=\pp_{\mu}+\tfrac{\ii q}{\hbar}A_{\mu}\) introduced for this equation by Schrödinger [Schroedinger:1926d] and by Gordon [Gordon:1926]; invariance of the coupled equation under \(\varphi\mapsto\ee^{-\ii q\Lambda/\hbar}\varphi\), \(A_{\mu}\mapsto A_{\mu}+\pp_{\mu}\Lambda\), the potentials being those of The Maxwell Equations; a real field cannot carry charge, so the charged scalar is necessarily complex and its two real components are the particle and the antiparticle; the “seagull” term quadratic in \(A_{\mu}\), absent from the Dirac case, and the fact that gauge invariance forces it.]
The current as charge, not probability
[Reserved: the coupled current \(j^{\mu}=\tfrac{\ii\hbar}{2m}\left(\varphi^{*}D^{\mu}\varphi -\varphi\left(D^{\mu}\varphi\right)^{*}\right)\); its total integral conserved and quantized in units of \(q\); the reinterpretation of a sign-indefinite density as an honest charge density [Pauli:1934]; the corresponding Noether statement in Generalized Classical Field Theory, and the observable consequence that a spin-0 particle and its antiparticle are distinct unless the field is real, as they are for \(\pi^{\pm}\) and are not for \(\pi^{0}\) [Navas:2024].]
The Compton effect after Gordon
[Reserved: Gordon's original application, the Compton effect computed from the minimally coupled equation [Gordon:1926], set against the kinematic treatment of The Photon: Photoelectric and Compton Effects; what the calculation gets right (the kinematics, the shift \(\Delta\lambda=(h/mc)(1-\cos\theta)\)) and what it cannot get right (the cross section, which needs spin \(\tfrac{1}{2}\) and the field quantization of Canonical Quantization of Fields); scalar Compton scattering as a genuine process for charged pions.]
Radiation scattered by a free charged particle of mass \(m\) returns with a wavelength longer than that of the incident beam by
\(\theta\) being the scattering angle. The shift depends on neither the incident wavelength nor the intensity nor the spin of the scatterer, and vanishes in the forward direction. The constant \(h/mc\) is the Compton wavelength, equal to \(2.42631\times 10^{-12}\,\mathrm{m}\) for the electron [Mohr:2025]; the measurement is The Photon: Photoelectric and Compton Effects.
Derivation. The shift is pure four-momentum conservation and needs no dynamics — which is why the spinless equation gets it right while getting the cross section wrong. Let the incident photon have four-momentum \(k\), the scattered photon \(k'\), and the target be at rest with \(p=(mc,\vect{0})\). Conservation gives \(p'=p+k-k'\), and squaring the invariant \(p'\cdot p'=m^{2}c^{2}\) while using \(k\cdot k=k'\cdot k'=0\) leaves
With \(k^{0}=E/c\) and \(k'^{0}=E'/c\) this reads \(mE-mE'=\left(EE'/c^{2}\right)\left(1-\cos\theta\right)\), so that
Substituting \(E=hc/\lambda\) and \(E'=hc/\lambda'\) turns the left side into \(\left(\lambda'-\lambda\right)/hc\) and yields Equation (92.2). The mass enters only through the combination \(h/mc\), so the effect is some 1836 times smaller on a proton than on an electron, and the derivation used nothing about the target beyond its mass — hence the independence from spin.
∎Exact solutions in external fields
[Reserved: the Coulomb problem solved exactly, giving the spinless energy levels with their non-integer effective angular momentum and their \(Z\alpha\) expansion; the failure against hydrogen and the success against pionic atoms [Deser:1954]; the constant magnetic field and the scalar Landau levels; the potential step, whose transmission anomaly is the spin-0 version of the Klein paradox treated in The Dirac Equation [Klein:1929b].]
The Feshbach–Villars form and the nonrelativistic limit
The two-component representation
[Reserved: Feshbach and Villars' rewriting of the second-order equation as a coupled pair of first-order equations for \(\varphi\pm\tfrac{\ii\hbar}{mc^{2}}\pp_{t}\varphi\) [Feshbach:1958a], in which the Hamiltonian is a \(2\times2\) matrix built from Pauli matrices, the norm is the indefinite charge form of Section 92.2.2, and the two components are the particle and antiparticle amplitudes; the Hamiltonian is not Hermitian in the ordinary inner product but is pseudo-Hermitian with respect to that form.]
Recovery of the Schrödinger equation
[Reserved: the substitution \(\varphi=\ee^{-\ii mc^{2}t/\hbar}\psi\) and the neglect of \(\pp_{t}^{2}\psi\) against \(mc^{2}\pp_{t}\psi/\hbar\), which turns the Klein–Gordon equation into the Schrödinger equation of The Postulates of Quantum Mechanics with the rest energy removed and the antiparticle component suppressed by \(\abs{\vect{p}}^{2}/m^{2}c^{2}\) [Feshbach:1958a]; the systematic version of this limit for spin \(\tfrac{1}{2}\) is the Foldy–Wouthuysen transformation of The Dirac Equation [Foldy:1950].]
Relativistic corrections
[Reserved: the leading correction \(-\abs{\vect{p}}^{4}/8m^{3}c^{2}\) obtained by expanding the square root, the same term produced by perturbation theory in Approximation Methods; the absence of any spin–orbit or Darwin term here, which is precisely why the spinless equation misses the hydrogen fine structure and why its failure is evidence for spin rather than against relativity (The Hydrogen Atom).]
Propagation, causality and the propagator
Green functions of the Klein–Gordon operator
[Reserved: the equation \(\left(\Box+m^{2}c^{2}/\hbar^{2}\right)G(x-y)=-\delta^{4}(x-y)\) and the pole prescriptions that distinguish its retarded, advanced and Feynman solutions; the contour integral in the complex \(p^{0}\) plane, using the machinery of Complex Analysis; the retarded Green function supported inside the forward light cone, which is the sense in which the classical equation is causal, and its tail inside the cone — absent for \(m=0\) — as a massive-wave phenomenon.]
The Feynman prescription
[Reserved: the propagator \(\tilde{G}_{\text{F}}(p)=\ii\hbar/\left(p^{2}-m^{2}c^{2}+\ii\epsilon \right)\); the \(\ii\epsilon\) read as the instruction to propagate positive frequencies forwards and negative frequencies backwards in time [Stueckelberg:1941] [Feynman:1949a], which is the same reinterpretation as Section 92.2.3 in a different language; the propagator as the single object that carries this chapter's content into Canonical Quantization of Fields and Quantum Electrodynamics and Renormalization, where it becomes an internal line of a Feynman diagram [Feynman:1949b].]
Localization and its limits
[Reserved: the Newton–Wigner position operator and the localized states it defines [Newton:1949]; the amplitude for propagation outside the light cone, which is nonzero but falls as \(\ee^{-mc\,r/\hbar}\) over the Compton wavelength; Hegerfeldt's theorem that a state localized in a bounded region at one instant, in any theory with energy bounded below, immediately develops tails outside the light cone [Hegerfeldt:1974]; why this is not a violation of relativistic causality, which constrains commutators of observables rather than amplitudes — the statement made precise in Axiomatic Quantum Field Theory.]
The Yukawa potential
[Reserved: the static equation with a point source, whose solution \(\ee^{-mc\,r/\hbar}/4\pi r\) is the Yukawa potential [Yukawa:1935]; the range \(\hbar/mc\) read off from the exponent, \(1.4\,\mathrm{fm}\) for the observed pion mass, how Yukawa predicted a particle of a few hundred \(\mathrm{MeV}/c^{2}\) from the size of the nucleus; the Coulomb limit \(m\to0\); the potential's role in the nuclear force models of Nuclear Forces and Nuclear Structure and its status there as an effective, not fundamental, description.]
The static potential produced by a point source of a field of mass \(m\) is not Coulombic but exponentially cut off,
so the interaction it carries has a definite reach \(\lambda\) instead of extending indefinitely. The force between nucleons behaves this way, with a range of the order of a femtometre; that is the reduced Compton wavelength of a particle of mass \(139.57\,\mathrm{MeV}/c^{2}\), which is the observed mass of the charged pion [Navas:2024]. Electromagnetism, by contrast, has no such cutoff, and the photon has no measured mass.
Derivation. For a time-independent field the Klein–Gordon operator reduces to \(-\vect{\nabla}^{2}+\lambda^{-2}\) with \(\lambda=\hbar/mc\), so a static point source of strength \(g\) gives
Put \(u(r)=\ee^{-r/\lambda}/r\). Away from the origin the radial Laplacian is \(\vect{\nabla}^{2}u =r^{-1}\dd^{2}\left(ru\right)/\dd r^{2} =r^{-1}\dd^{2}\ee^{-r/\lambda}/\dd r^{2}=u/\lambda^{2}\), so \(\left(-\vect{\nabla}^{2}+\lambda^{-2}\right)u=0\) for \(r>0\): the mass term is exactly what the exponential decay cancels. Near the origin \(u\to1/r\), whose Laplacian is \(-4\pi\delta^{3}\left(\vect{x}\right)\) while the mass term stays integrable, so the source is reproduced by
which is Equation (92.3). Setting \(m=0\) removes the exponential and returns the Coulomb form, so the range is finite precisely because the mediating field is massive, and reading \(\lambda\) off a measured range predicts a mass. Note what the argument does not do: it gives the range, not the strength, and it treats the nucleon source as a fixed point charge, so the resulting potential is an effective description and not a fundamental one.
∎Pauli–Weisskopf: the equation is a field
Quantization of the scalar field
[Reserved: Pauli and Weisskopf's 1934 construction [Pauli:1934], in which \(\varphi\) is promoted to an operator expanded in creation and annihilation operators for particles and antiparticles, the negative-frequency coefficients being reinterpreted as creation operators; the Hamiltonian becomes positive definite, the charge operator takes both signs, and the number of particles is no longer conserved; the construction is carried out in full in Canonical Quantization of Fields. Their paper was written to show that Dirac's hole theory was not needed to make sense of negative energies — and it did so for a spin-0 field, where no exclusion principle exists to build a filled sea from.]
Bose statistics and microcausality
[Reserved: commutators rather than anticommutators are forced for spin 0, so scalar quanta are bosons; the field commutator vanishing at spacelike separation as the precise statement of relativistic causality [Pauli:1940]; the spin–statistics theorem, proved from Lorentz invariance, positivity of the energy and microcausality, whose axiomatic form belongs to Axiomatic Quantum Field Theory and whose observational consequences run through Identical Particles and Quantum Statistics.]
What survives of the one-particle reading
[Reserved: an explicit accounting of which results of this chapter remain true in the field theory and which do not — the equation, the plane waves, the propagator, the Yukawa potential and the nonrelativistic limit all survive unchanged; the position operator, the probability density, the single-particle Hilbert space and the external Coulomb spectrum survive only as approximations valid for energies well below \(2mc^{2}\), the threshold at which pair creation makes particle number ill-defined [Pauli:1934] [Bjorken:1964].]
The observed spin-0 sector
A relativistic wave equation earns its place by the particles that obey it. Nature supplies few elementary spin-0 particles and many composite ones, and both kinds are listed here with the measurements that fix their quantum numbers.
Pions and the light pseudoscalars
[Reserved: Yukawa's prediction of a meson of nuclear-force range [Yukawa:1935]; the charged pion identified in photographic emulsions exposed to cosmic rays by Lattes, Occhialini and Powell [Lattes:1947], with the muon of Cosmic Rays and Astroparticle Physics as its decay product and as the particle it had been confused with; masses \(139.57\,\mathrm{MeV}/c^{2}\) and \(134.98\,\mathrm{MeV}/c^{2}\) for \(\pi^{\pm}\) and \(\pi^{0}\), spin 0 and odd parity [Navas:2024]; the pions as pseudoscalars, not scalars, and as composite states of quarks rather than elementary fields (Quantum Chromodynamics).]
Three particles of spin 0 and odd intrinsic parity, \(J^{P}=0^{-}\), are observed: a charged pair \(\pi^{\pm}\) of mass \(139.57\,\mathrm{MeV}/c^{2}\) and a neutral \(\pi^{0}\) of mass \(134.98\,\mathrm{MeV}/c^{2}\) [Navas:2024]. The two charged states are distinct particles, each the antiparticle of the other and of equal mass; the neutral state is its own antiparticle and decays to two photons, which no charged particle can do.
Derivation. The structural half of the statement follows from the current of Phenomenon 92.2 alone. If the field is real, \(\varphi^{*}=\varphi\), then \(j^{\mu}\propto\varphi^{*}\pp^{\mu}\varphi -\varphi\,\pp^{\mu}\varphi^{*}\) vanishes identically. A real scalar field carries no conserved charge at all, so its quantum has nothing by which it could be distinguished from its antiparticle: it is self-conjugate, and necessarily neutral. A charged spin-0 particle therefore requires a complex field, whose overall phase is the symmetry whose Noether charge is the conserved one; the quanta associated with \(\varphi\) and with \(\varphi^{*}\) are then two distinct states of opposite charge, and of equal mass, because both obey Equation (92.1) with the one mass parameter appearing in the equation. That is exactly the observed pattern: \(\pi^{+}\) and \(\pi^{-}\) share a mass while \(\pi^{0}\) has no charged partner. What this chapter does not derive is the numerical values of the masses or the odd parity; both are measurements, the parity of the charged pion having been fixed by the capture reaction discussed in Discrete Symmetries and CPT, and the values themselves are consequences of the quark substructure of Quantum Chromodynamics rather than of any free-field equation.
∎Pionic and kaonic atoms
[Reserved: the spin-0 realization of the Coulomb problem of Section 92.3.4 — a negative pion or kaon captured into a hydrogen-like orbit around a nucleus, whose X-ray cascade is described by the Klein–Gordon equation with no spin–orbit structure at all, so that the equation is tested where it applies rather than where it fails [Backenstoss:1970]; the strong-interaction shift and broadening of the lowest level, related to the scattering length by the Deser formula [Deser:1954]; the kaonic hydrogen measurement as the modern precision case [Bazzi:2011]; comparison with the electronic spectra of Experiment: Precision Spectroscopy and Atomic Clocks.]
The Higgs boson
[Reserved: the only elementary spin-0 particle observed to date, found at the LHC in 2012 by ATLAS [Aad:2012] and CMS [Chatrchyan:2012] with a mass near \(125\,\mathrm{GeV}\), whose free field obeys exactly the equation of this chapter; its measured spin-parity \(J^{P}=0^{+}\), distinguishing it from the pseudoscalars above; the experiment is treated in Experiment: The Higgs Boson Discovery and the mechanism that requires the field in Electroweak Unification and the Higgs Boson. Nothing in this chapter predicts the existence of such a particle; the equation says only what a spin-0 particle must satisfy if one exists.]
Exactly one elementary particle of spin 0 has been observed. Its mass is close to \(125\,\mathrm{GeV}/c^{2}\); its spin and parity are measured to be \(J^{P}=0^{+}\), distinguishing it from the pseudoscalar mesons of Phenomenon 92.5; and its production and decay rates agree with those of the Standard Model Higgs boson within present uncertainties [Navas:2024]. Its free field obeys Equation (92.1), and hence the equation of this chapter, with no additional structure whatever — it is the only known particle for which that is the complete free-field description.
Determination of the spin and parity of the boson near 125 GeV from the angular distributions of its four-lepton and diphoton decays, and the electroweak argument that requires an elementary scalar field in the first place. The experiment is the Higgs discovery chapter of Part XII and the mechanism is the electroweak chapter of the same part.