Why Klein-Gordon is not a one-particle wave equation

Units are SI throughout, with x^μ = (ct, x) and metric diag(1, −1, −1, −1). Reading ψ as the amplitude of one particle commits you to three things. First, a conserved density positive everywhere, so ∫ρ d³x can be a probability. Second, a spectrum bounded below, so the particle has a ground state and a weak, slowly varying perturbation moves it among nearby levels rather than dropping it without limit. Third, states localised near a point and built from the equation's own physical solutions, so position is a question the theory can ask. Schrödinger's equation passes all three. The Klein-Gordon equation fails each, and the failures are independent: Dirac's equation in Unit 4 repairs the first but keeps the other two, and discarding negative frequencies helps the second only until a field mixes them back in. Each failure carries a scale, 2mc²/ħ in frequency or ħ/mc in length, and the working skill is to compute that scale and set it beside the problem in hand.

Quote the density from topic 3.02, with the minimal coupling of 3.04: ρ = (iħ/2mc²)(ψ*∂ₜψ − ψ∂ₜψ*) − (qΦ/mc²)|ψ|². Since (ħ/mc²)∂ₜ is a pure number, ρ has the units of |ψ|². For a stationary state ψ = φ(x) e^(−iEt/ħ) the bracket gives −2iE/ħ, and ρ = [(E − qΦ)/mc²]|φ|². That is negative on every free negative-energy wave and, less obviously, on positive-energy waves wherever qΦ exceeds E. Two facts close the case. The equation is second order in time, so ψ and ∂ₜψ at t = 0 are independent data and ρ(x, 0) can be given any sign at any point. And a real solution, ψ* = ψ, has ρ ≡ 0: no probability density can vanish identically for a particle that exists. The limit check: with ψ = e^{(−imc²t/ħ)}φ and φ slow, ρ = |φ|²[1 + O(v²/c²)]. At the Bohr radius of pionic hydrogen (E − qΦ)/mc² = 1 + 2.3 × 10⁻⁵, so the defect is invisible exactly where topic 3.07's levels are tested.

The free energies E = ±√(p²c² + m²c⁴) form a positive band from +mc² upwards, a negative band from −mc² downwards, and nothing between; the negative band has no bottom. Put a particle in a field V cos ωt: the golden rule, quoted from Quantum Mechanics II, connects E to E ± ħω, so one quantum lands in the negative band once ħω ≥ E + mc², which is 2mc² for a particle at rest. For a charged pion that is 279.14 MeV, ω = 4.24 × 10²³ s⁻¹, a gamma ray of wavelength πħ/m_πc = 4.44 fm. Once there nothing stops the next step: emitted photons can carry the particle to −√(p²c² + m²c⁴) for any p, with more final states at every larger p, so a one-particle Klein-Gordon world coupled to radiation has no stable ground state. Dirac blocks the cascade for electrons by filling the negative band and invoking exclusion; pions are bosons, so a filled sea blocks nothing. The threshold is measured: π⁺π⁻ pairs appear in e⁺e⁻ collisions only above 279 MeV, and photons convert to e⁺e⁻ only above 1.022 MeV.

A static field conserves energy, so it mixes the bands only where they overlap at one E. In a potential energy V(x) the local bands are E ≥ V + mc² and E ≤ V − mc², and a particle of energy E finds negative-energy states wherever V(x) ≥ E + mc², a rise of at least 2mc². Across the gap the equation reads −ħ²c²ψ″ = [(E − V)² − m²c⁴]ψ, so κ = √(m²c⁴ − (E − V)²)/ħc. On a ramp of constant force F put u = E − V: 2∫κ dx = (2/ħcF) ∫√(m²c⁴ − u²) du from −mc² to mc², which is π m²c³/(ħF), Sauter's 1931 exponent, derived here at WKB accuracy. Dimensions: kg² m³ s⁻³ over (J s)(N) is a pure number. Limits: as F → 0 the leak dies exponentially, and a rise of 2mc² within ħ/mc gives T of order one. A pion needs F ≈ 98.7 MeV fm⁻¹ for T = e^(−π) = 0.043; the surface of lead, a uniform sphere of radius 7.10 fm for its measured rms charge radius of 5.50 fm, offers 2.34 MeV fm⁻¹ and a total rise of only 25 MeV, against the 279 MeV required.

Discard the negative band and keep ψ = ∫d³p φ(p) e^{(i(p⋅x − Eₚt)/ħ)} with Eₚ = c√(p² + m²c²). Then iħ∂ₜψ = Êψ with Ê = c√(p̂² + m²c²), and ρ = (1/2mc²)[ψ*Êψ + (Êψ)*ψ]. Ê is not a differential operator. Make ψ vanish outside a sphere of radius a: by the Paley-Wiener theorem φ(p) is then analytic everywhere, but Eₚφ(p) inherits the branch points of √(p² + m²c²) at p = ±imc, and a Fourier integral whose nearest singularity sits mc off the real axis falls as e^(−mcr/ħ), no faster. So Êψ, which is iħ∂ₜψ, leaks out of the sphere over a distance ħ/mc. At that instant ρ still vanishes outside, since ψ* does, but an instant later ψ, and with it ρ, is non-zero there. The kernel of Ê falls as r^(−5/2) e^(−mcr/ħ), quoted, as is the width ħ/mc of the Newton-Wigner states, the closest positive-frequency approach to a point. Checks: mcr/ħ is a pure number; as c → ∞ the range ħ/mc shrinks to zero and Schrödinger's point-localised states return; and Δx below ħ/mc needs Δp above mc, energies enough to make pairs. For a pion ħ/mc = 1.41 fm, larger than the measured proton charge radius of 0.841 fm.

Pauli and Weisskopf kept the equation and changed its reading: qj^μ is an electric current, qρ a charge density, and ψ a field whose quanta are particles of charge q and antiparticles of charge −q, all of positive energy. A negative ρ then means more antiparticles than particles; a Klein step that reflects more current than it receives is making pairs; and a real field describes a neutral particle that is its own antiparticle, the π⁰, whose charge density must vanish identically. That reading is stated here and carried out for the free field in Unit 9. What survives it, also stated, is the energies: the relativistic Landau levels of topic 3.05 and the Coulomb levels of topic 3.07 become the energies of one-quantum states of the field in the external potential, unchanged while no level reaches the negative band. The evidence is quantitative: the charged-pion mass, 139.570 MeV, is extracted from pionic-atom X-ray energies analysed with those Klein-Gordon levels plus small QED corrections. What does not survive is ρ as the probability of finding one pion at a point.