The Klein-Gordon and Dirac Equations

Relativity says energy and momentum obey E² = p²c² + m²c⁴. Quantize with E → iħ∂/∂t and p → −iħ∇, where ħ is the reduced Planck constant. Applying both operators twice gives (iħ∂/∂t)²ψ = −ħ²∂²ψ/∂t² and (−iħ∇)²ψ = −ħ²∇²ψ, so −ħ²∂²ψ/∂t² = −ħ²c²∇²ψ + m²c⁴ψ. Dividing by −ħ² and rearranging gives (□ + (mc/ħ)²)ψ = 0, with □ = (1/c²)∂²/∂t² − ∇². For a complex scalar of mass m > 0, the conserved density (iħ/2mc²)(ψ*∂ψ/∂t − ψ∂ψ*/∂t) is proportional to the U(1) charge density, not positive definite for arbitrary solutions. A real neutral scalar has no such charge density.

Klein-Gordon remains a valid relativistic spin-zero field equation, but its unrestricted one-particle probability interpretation fails. The formal one-particle energies E = ±√(p²c² + m²c⁴) have a negative branch with no bottom. The conserved density is not positive definite for arbitrary solutions, so it cannot serve as a one-particle probability density. The resolution is the field reading: quantize the scalar field, and the negative-frequency modes become antiparticle excitations with positive energy, leaving the physical Hamiltonian bounded below.

iħ∂ψ/∂t = Hψ, H = cα·p + βmc², p = −iħ∇, α = (α₁, α₂, α₃) is a triple of matrices; the αᵢ and β are the coefficient matrices acting on multi-component ψ. H² = c²αᵢαⱼpᵢpⱼ + mc³(αᵢβ + βαᵢ)pᵢ + β²m²c⁴. Matching p²c² + m²c⁴ for every momentum forces the Clifford relations αᵢ² = β² = I, {αᵢ, αⱼ} = 2δᵢⱼI for i ≠ j, and {αᵢ, β} = 0, where I is the identity matrix. Nonzero scalars cannot meet these: a scalar with αᵢ² = β² = 1 commutes with itself, so αᵢβ + βαᵢ = 2αᵢβ cannot be zero. The smallest matrices satisfying them are 4x4.

Requiring first order in both time and space for a local massive 3+1D field forces matrix coefficients, while first order in time alone does not (Schrödinger is a scalar counterexample). The smallest 4x4 matrices mean the wavefunction carries four components. Those components are representation-dependent entries, not universal particle or antiparticle labels. At fixed momentum there are two positive-frequency and two negative-frequency independent plane-wave solutions. In the standard Dirac representation, S_z = (ħ/2)diag(1, −1, 1, −1) has eigenvalues ±ħ/2, with total angular momentum J = L + S.

In the standard Dirac representation, S_z = (ħ/2)diag(1, −1, 1, −1) squares to (ħ²/4)I and has eigenvalues ±ħ/2, matching the two Stern-Gerlach bands of neutral silver atoms in the orbital ground state, whose single unpaired electron carries the splitting angular momentum. For a massive charged Dirac field, negative-frequency solutions become positive-energy antiparticles of the same mass and opposite charge after quantization. Spin is intrinsic angular momentum, not a literal twirl.