2.2.1 Dirac and the failure of single-particle relativistic QM

Dirac’s 1928 relativistic wave equation resolved the limitations of the non-relativistic Schrödinger equation and the spinless Klein–Gordon equation by introducing a first-order, Lorentz-covariant framework using four-component complex spinors. By factorizing the energy–momentum relation through 4×4 matrices satisfying a Clifford algebra, the equation naturally accounts for spin-1/2 and accurately reproduces the hydrogen fine structure. However, it also yielded unavoidable negative-energy solutions, which Dirac initially addressed via hole theory—postulating a vacuum sea where missing negative-energy electrons manifest as positrons. Ultimately, single-particle relativistic quantum mechanics fails because localizing a particle below its Compton wavelength causes energy fluctuations that trigger virtual pair creation. This multi-particle regime renders the single-particle wavefunction framework provisional, necessitating the transition to quantum field theory to accommodate variable particle numbers.

The relativistic wave equation

By 1928, the non-relativistic quantum mechanics was empirically successful but structurally incomplete. The Schrödinger equation is first-order in time and second-order in space; it is not Lorentz-covariant. The Klein–Gordon equation is Lorentz-covariant but second-order in time, and it failed to reproduce the hydrogen fine structure because it does not describe spin. What was needed was a first-order, Lorentz-covariant wave equation that incorporated spin naturally and recovered the correct hydrogen spectrum.

Dirac’s 1928 papers—The Quantum Theory of the Electron—supplied exactly this. The equation is:

iħ γμ ∂μψ - mc ψ = 0

where ψ is a four-component complex spinor, ∂μ = ∂/∂xμ, and the γμ are four 4×4 matrices satisfying the Clifford algebra:

{γμ, γν} = 2ημν 1_4

with ημν = diag(-1, +1, +1, +1) the Minkowski metric and 1_4 the 4×4 identity.

The anticommutator structure is here the defining property of the relativistic theory. Derivation:

Dirac’s strategy was to factorise the relativistic energy–momentum relation. The Klein–Gordon equation follows from E² = (pc)² + (mc²)² by the operator substitution E → iħ∂t, p → -iħ∇. Dirac sought a linear equation in E and p whose squaring recovered the Klein–Gordon form. Writing:

E = c α·p + β mc²

with α = (α¹, α², α³) and β Hermitian matrices, the requirement that squaring gives E² =(pc)² + (mc²)² forces:

{αi, αj} = 2δij 1, {αi, β} = 0, β² = 1

The smallest matrices satisfying these relations are 4×4. In the Dirac–Pauli representation, β and αi are constructed using Pauli matrices σi.

Spin-1/2 and Hydrogen Fine Structure

The spinor structure is not added by hand; it emerges from the mathematics. The four components of ψ describe two spin states for each of two energy branches. The spin angular momentum operator commutes with the free Hamiltonian and satisfies the SU(2) algebra.

The Dirac equation in a Coulomb potential gives the exact energy spectrum:

En,j = mc² [ 1 + α² / (n - δj)² ]^-1/2

Expanding to order α⁴, the equation contains the relativistic kinetic correction, the spin–orbit coupling, and the Darwin term, providing the complete fine structure missing in the Klein-Gordon approach.

The negative-energy problem

The Dirac equation admits plane-wave solutions with the dispersion relation:

ħω = ±√((ħck)² + (mc²)²)

The negative-frequency branch is mathematically unavoidable. For a free particle, this would imply a catastrophe where particles cascade to infinite negative energy. Dirac’s 1929–1931 resolution was the hole theory, postulating a vacuum sea of negative-energy electrons. A hole in this sea behaves as a positively-charged particle (the positron), confirmed by Anderson in 1932.

The failure of single-particle relativistic QM

The hole theory is inconsistent because a single-particle wave function cannot describe pair creation. The uncertainty principle allows energy fluctuations that can create virtual pairs. Localisation below the Compton wavelength forces the system into a multi-particle regime:

λC = h / (mc) ≈ 2.426 × 10^-12 m

Conclusion: Single-particle relativistic QM is provisional. The correct formalism must allow variable numbers of particles and antiparticles via quantum field theory.

…