Higher-Spin Wave Equations

Wigner's classification [Wigner:1939] assigns a mass and a spin to every unitary irreducible representation of the Poincaré group, and Particles as Poincaré Representations shows that the list is unbounded in spin. A relativistic wave equation is the covariant field-theoretic carrier of one entry in that list, and the covariant carrier is always larger than the entry: a symmetric tensor or tensor–spinor has far more components than the representation has states, so every higher-spin equation is a field equation plus a set of constraints that remove the surplus. This chapter follows that pattern up the spin ladder — Proca's spin-1 equation [Proca:1936], the Rarita–Schwinger spin-\(\tfrac{3}{2}\) equation [Rarita:1941], the Fierz–Pauli spin-2 equation [Fierz:1939], and the general constructions of Singh and Hagen [Singh:1974] and Fronsdal [Fronsdal:1978] [Fang:1978] — and it follows the pathologies with equal care, because the constraints are what break first when the field is coupled to anything [Velo:1969a] [Velo:1969b] [Johnson:1961]. It therefore sits after The Klein–Gordon Equation and The Dirac Equation, which are the spin-0 and spin-\(\tfrac{1}{2}\) members of the same family, and after The Weyl Equation and Neutrinos, which decomposed the spin-\(\tfrac{1}{2}\) case.

The honest experimental summary is short and is the reason the chapter ends where it does. Elementary massive spin-1 particles exist and are measured to high precision — the \(W\) and \(Z\) bosons discovered at the CERN \(\mathrm{p}\bar{\mathrm{p}}\) collider [Arnison:1983a] [Banner:1983] [Arnison:1983b] [Bagnaia:1983], with the Proca mass supplied by the mechanism of Electroweak Unification and the Higgs Boson. A massless spin-2 field is required by the observed gravitational radiation of Part V — General Relativity and Cosmology, treated in Gravitational-Wave Theory and measured in Experiment: Gravitational Waves, and its interacting completion is general relativity itself. Spin \(\tfrac{3}{2}\) is realized only as a composite hadronic resonance, the \(\Delta(1232)\) [Anderson:1952] [Navas:2024]. No elementary particle of spin greater than 2 has ever been observed, and Section 97.6 collects the theorems — Weinberg's soft theorems [Weinberg:1964b], Weinberg–Witten [Weinberg:1980] and Coleman–Mandula [Coleman:1967] — which say why that absence is expected rather than accidental. Speculative higher-spin programmes are named in Section 97.7.2 only to record that they have no evidential support, per the scope rule stated in the front matter.

The classification problem

From representations to field equations

[Reserved: Wigner's little-group construction and the mass–spin labelling of unitary irreducible representations [Wigner:1939]; the Bargmann–Wigner equations, which build the spin-\(s\) representation as a totally symmetric product of \(2s\) Dirac indices, each satisfying the Dirac equation [Bargmann:1948]; why a massless representation has two helicity states rather than \(2s+1\); the counting that every later section must reproduce, with the group theory in Lie Groups, Lie Algebras, and Fibre Bundles.]

What a wave equation must deliver

[Reserved: the four requirements — the correct on-shell degree of freedom count, a mass-shell condition \((\Box+m^{2})\phi=0\) on every component, positive-definite energy, and hyperbolic propagation inside the light cone; auxiliary conditions \(\pp^{\mu}\phi_{\mu\nu\ldots}=0\) and tracelessness as the mechanism that removes lower-spin content; the pattern, common to [Proca:1936] [Rarita:1941] [Fierz:1939], that the constraints follow from the field equations only in the free theory — the observation Section 97.3.2 then turns into a no-go.]

Spin 1: the Proca field

The Proca equation

[Reserved: the Lagrangian \(-\tfrac{1}{4}F_{\mu\nu}F^{\mu\nu}+\tfrac{1}{2}m^{2}A_{\mu}A^{\mu}\) and the field equation \(\pp_{\mu}F^{\mu\nu}+m^{2}A^{\nu}=0\) written down by Proca [Proca:1936]; the divergence of the field equation giving the Lorenz condition \(\pp_{\mu}A^{\mu}=0\) as a consequence rather than a gauge choice, so that a Proca field carries three states; the propagator and its \(k^{\mu}k^{\nu}/m^{2}\) term; the \(m\to0\) discontinuity in the state count against Maxwell theory (The Maxwell Equations), the spin-1 ancestor of Section 97.4.2.]

The Duffin–Kemmer form

[Reserved: the first-order matrix formulation with \(\beta\)-matrices satisfying \(\beta^{\mu}\beta^{\nu}\beta^{\lambda}+\beta^{\lambda}\beta^{\nu}\beta^{\mu} =\eta^{\mu\nu}\beta^{\lambda}+\eta^{\lambda\nu}\beta^{\mu}\), due to Duffin [Duffin:1938] and Kemmer [Kemmer:1939]; its five-dimensional and ten-dimensional representations describing spin 0 and spin 1 in a single Dirac-like formalism; what is gained (Dirac-style manipulations, a clean current) and what is not (the algebra has no inverse element, so the analogy is incomplete).]

Stueckelberg's restoration of gauge invariance

[Reserved: the introduction of a compensating scalar \(A_{\mu}\to A_{\mu}+m^{-1}\pp_{\mu}\varphi\) that makes the massive vector theory gauge invariant [Stueckelberg:1938]; the consequences — a smooth massless limit, power counting improved to renormalizable, and the longitudinal mode exhibited as the would-be Goldstone boson; the relation to the Higgs mechanism [Englert:1964] [Higgs:1964] taken up in Electroweak Unification and the Higgs Boson; why this trick works for spin 1 and has no unproblematic analogue at spin 2.]

Massive vectors in Nature

[Reserved: the \(W^{\pm}\) and \(Z^{0}\) bosons as the observed elementary Proca fields — discovery in \(\mathrm{p}\bar{\mathrm{p}}\) collisions by UA1 [Arnison:1983a] [Arnison:1983b] and UA2 [Banner:1983] [Bagnaia:1983], with the missing-energy and dilepton-mass signatures; the present masses \(m_{W}=80.37\,\mathrm{GeV}/c^{2}\) and \(m_{Z}=91.19\,\mathrm{GeV}/c^{2}\) [Navas:2024]; the photon as the massless limit, with the experimental bound on its mass; the \(\rho\) meson as a composite spin-1 state, not a fundamental Proca field.]

Spin 3/2: the Rarita–Schwinger field

The equation and its constraints

[Reserved: the vector–spinor \(\psi_{\mu}\) and the field equation \((\ii\gamma^{\nu}\pp_{\nu}-m)\psi_{\mu}=0\) supplemented by \(\gamma^{\mu}\psi_{\mu}=0\) and \(\pp^{\mu}\psi_{\mu}=0\), formulated by Rarita and Schwinger [Rarita:1941]; the covariant Lagrangian with its \(\gamma^{\mu\nu\rho}\) kinetic operator; the counting from sixteen components to four physical states; the massless limit and its gauge invariance \(\psi_{\mu}\to\psi_{\mu}+\pp_{\mu}\epsilon\), stated here as a property of the free wave equation only.]

The Velo–Zwanziger causality problem

[Reserved: coupling \(\psi_{\mu}\) minimally to an external electromagnetic field spoils the constraint algebra — the characteristic surfaces of the resulting system lie outside the light cone, so wave fronts propagate superluminally [Velo:1969a] [Velo:1969b]; the earlier indefinite-anticommutator inconsistency found by Johnson and Sudarshan [Johnson:1961]; the general lesson that the constraints of a higher-spin equation are fragile under interaction, and the survey of which couplings evade the problem and at what cost.]

The $\Delta(1232)$ resonance

[Reserved: the first pion–nucleon resonance, found as a peak in the \(\pi^{+}\mathrm{p}\) total cross-section by Anderson, Fermi, Long and Nagle [Anderson:1952]; its quantum numbers \(J^{P}=\tfrac{3}{2}^{+}\), \(I=\tfrac{3}{2}\), mass near \(1232\,\mathrm{MeV}/c^{2}\) and width near \(117\,\mathrm{MeV}\) [Navas:2024]; the point of principle that this is a composite three-quark state described by Quantum Chromodynamics, so it is evidence for spin \(\tfrac{3}{2}\) existing in Nature but not for an elementary Rarita–Schwinger field; higher-spin baryon and meson recurrences organized on Regge trajectories [Regge:1959] [Chew:1961].]

Spin 2: the Fierz–Pauli field

The Fierz–Pauli Lagrangian

[Reserved: the symmetric tensor \(h_{\mu\nu}\) and the unique quadratic Lagrangian free of ghosts, with the relative coefficient of \(h^{2}\) and \(h_{\mu\nu}h^{\mu\nu}\) in the mass term fixed to \(-1\) [Fierz:1939]; the constraints \(\pp^{\mu}h_{\mu\nu}=0\) and \(h^{\mu}{}_{\mu}=0\) reducing ten components to five; the massless case with its linearized diffeomorphism invariance \(h_{\mu\nu}\to h_{\mu\nu}+\pp_{(\mu}\xi_{\nu)}\) and two helicity states, which is the linearized Einstein equation of The Einstein Field Equations.]

The vDVZ discontinuity

[Reserved: the massless limit of Fierz–Pauli theory does not reproduce linearized general relativity — the extra helicity-0 mode survives and couples to the trace of the source, changing the light-bending prediction relative to the Newtonian potential by a factor \(\tfrac{3}{4}\), discovered independently by van Dam and Veltman [vanDam:1970] and by Zakharov [Zakharov:1970]; why this is a discontinuity in the propagator and not in the spectrum alone; the measured light deflection of The Equivalence Principle and Classical Tests as the datum it contradicts.]

The Boulware–Deser ghost and the Vainshtein radius

[Reserved: Vainshtein's observation that the linear analysis breaks down inside a radius \(r_{V}\) around a source, so the discontinuity may be an artefact of perturbation theory [Vainshtein:1972]; Boulware and Deser's counter-result that generic nonlinear completions propagate a sixth, ghost-like mode [Boulware:1972]; the ghost-free tuning of de Rham, Gabadadze and Tolley [deRham:2011] and the review of the subject [Hinterbichler:2012], named with the honest statement that no observation requires a graviton mass.]

The massless graviton and its evidence

[Reserved: the theorem that a self-coupled massless spin-2 field is general relativity — Gupta's iterative construction [Gupta:1954] and Deser's one-step derivation from the first-order form [Deser:1970]; the evidence, which is the quadrupole energy loss of the binary pulsar [Hulse:1975] [Taylor:1982] [Weisberg:2016] and the two transverse-traceless polarizations of the detected waves [Abbott:2016] [Abbott:2017]; the graviton mass bound of order \(10^{-23}\,\mathrm{eV}/c^{2}\) from the arrival-time comparison of GW170817 with its electromagnetic counterpart [Abbott:2017]; full treatment in Part V — General Relativity and Cosmology, quantization status in Quantum Gravity: The Honest Status.]

General spin

Massive fields of arbitrary spin

[Reserved: the Singh–Hagen Lagrangians, which achieve a consistent massive spin-\(s\) description only at the price of a tower of auxiliary traceless fields of ranks \(s-2,s-3,\ldots,0\) [Singh:1974]; the pattern that the number of auxiliary fields grows with spin; the Fierz–Pauli conditions \(\pp^{\mu}\phi_{\mu\ldots}=0\), \(\phi^{\mu}{}_{\mu\ldots}=0\) as the general statement; why no such Lagrangian has ever been needed to describe an observed elementary particle.]

Massless fields of arbitrary spin

[Reserved: Fronsdal's gauge-invariant equations for massless integer spin, with doubly-traceless fields and traceless gauge parameters [Fronsdal:1978], and the half-integer case of Fang and Fronsdal [Fang:1978]; the two helicity states \(\pm s\) that survive; the massless limit of the massive theory as a decomposition into helicities, in contrast with the discontinuity of Section 97.4.2; the free theory as the last point at which higher spin is uncontroversial.]

Coupling obstructions

[Reserved: the systematic obstructions to interacting higher-spin fields — loss of the constraint algebra [Velo:1969a] [Velo:1969b], non-positive commutators [Johnson:1961], and the failure of gauge invariance to survive minimal coupling to gravity; the contrast with spins \(0,\tfrac{1}{2},1,\tfrac{3}{2},2\), each of which has a consistent interacting realization; the statement of the problem in the form the no-go theorems of Section 97.6 then answer.]

No-go theorems

Soft theorems and Weinberg's low-energy argument

[Reserved: Weinberg's covariant Feynman rules for any spin [Weinberg:1964a]; the soft-emission analysis showing that Lorentz invariance of the \(S\) matrix forces a massless spin-1 particle to couple to a conserved charge and a massless spin-2 particle to couple universally with a single constant — the equivalence principle derived rather than assumed [Weinberg:1964b]; the extension to infrared photons and gravitons [Weinberg:1965]; the conclusion that a massless particle of spin \(\geq3\) cannot couple to any conserved quantity that survives at low energy.]

Weinberg–Witten

[Reserved: the theorem that a theory with a Lorentz-covariant, conserved current forbids massless particles of helicity \(\abs{h}>\tfrac{1}{2}\) carrying that charge, and that a Lorentz-covariant, conserved stress tensor forbids massless particles of helicity \(\abs{h}>1\) [Weinberg:1980]; the precise reading of the hypotheses, without which the theorem appears to exclude the graviton and the gluon — neither has a gauge-invariant covariant current or stress tensor of the required kind; what the theorem does and does not exclude, stated carefully.]

Coleman–Mandula

[Reserved: the theorem that under an \(S\)-matrix with a mass gap, analyticity and a finite number of particle types below any mass, the largest symmetry Lie algebra is the direct product of the Poincaré algebra with internal symmetries [Coleman:1967]; the corollary that conserved charges carrying more than one spacetime index would force the scattering amplitude to be trivial; the standard graded-algebra escape route, named here only to record that it lies outside the scope of this book.]

Experimental status

The observed spectrum

[Reserved: the complete list of elementary particles and their spins as measured — spin 0 (the Higgs boson, see Experiment: The Higgs Boson Discovery), spin \(\tfrac{1}{2}\) (quarks and leptons), spin 1 (photon, gluons, \(W\), \(Z\)), and spin 2 inferred for the graviton from the polarization content of gravitational waves [Abbott:2016] [Abbott:2017]; the tabulated composite states of higher spin [Navas:2024], all hadronic, all with Regge-organized recurrences [Regge:1959] [Chew:1961]; how a spin assignment is actually measured, from angular distributions and decay correlations.]

Programmes without evidence

[Reserved: an honest closing statement — no elementary particle of spin \(\tfrac{3}{2}\) or of spin greater than 2 has been observed, and searches place limits but no signal [Navas:2024]; the theoretical programmes that postulate them, including graded extensions of the Poincaré algebra and infinite higher-spin towers, are named here solely to record that they have no observational support and are therefore outside the scope of this treatise; the open questions that remain within scope are collected in What We Observe but Do Not Understand and Quantum Gravity: The Honest Status.]