Non-fiction: The Quantum Theory of the Electron

Overview
Paul Dirac formulates a relativistic quantum wave equation for the electron that reconciles the principles of quantum mechanics with special relativity. The equation is linear in both time and space derivatives and requires a multi-component wavefunction, leading to the introduction of four-component objects now called spinors. This formulation brings the electron's intrinsic angular momentum (spin) and its magnetic properties into the same mathematical framework as relativity and the quantum wave description.
The argument begins from the requirement that a single-particle wave equation reproduce the relativistic energy-momentum relation while remaining first order in time so probability density has a sensible interpretation. That constraint forces a new algebraic structure and the appearance of matrices that mix components of the wavefunction, with physical consequences that extend far beyond the original aim of describing hydrogen fine structure.

Key ideas and formulation
Dirac constructs matrices, later denoted γ^μ, that satisfy specific anticommutation relations ensuring that squaring the first-order operator returns the Klein-Gordon operator and thus the relativistic dispersion relation E^2 = p^2c^2 + m^2c^4. The resulting equation acts on a four-component spinor field; two components correspond to electron degrees of freedom and two to mathematically required extra components. These extra components are not mere redundancies but encode spin-1/2 behavior and the correct coupling to electromagnetic fields through the minimal substitution p → p − eA.
Lorentz covariance is built into the algebraic structure: the spinor transforms under Lorentz transformations in a way consistent with the linear operator, and the probability current constructed from the spinor is conserved and transforms as a four-vector. The formalism thus gives a consistent relativistic probability interpretation and yields correct relativistic corrections to atomic spectra, including the fine structure of hydrogen and the electron's magnetic moment to leading order.

Physical consequences
A striking consequence is the appearance of solutions with negative energy. Rather than discard these as unphysical, Dirac proposes the "hole" interpretation: imagine all negative-energy states filled in a vacuum sea; a vacancy (hole) in that sea behaves like a positively charged particle with the same mass as the electron. This prediction amounts to the existence of antimatter, and it motivated the search that eventually led to the discovery of the positron. The negative-energy solutions also lead to the phenomenon called zitterbewegung, a rapid trembling motion produced by interference between positive and negative energy components.
The equation naturally explains electron spin and predicts a gyromagnetic ratio g = 2 at tree level, accounting for observed magnetic properties without ad hoc assumptions. It also provides a framework for calculating relativistic corrections to atomic energy levels and scattering processes, making contact with experiment and suggesting new directions for quantum theory that treat creation and annihilation of particles.

Legacy and impact
The Dirac equation establishes the conceptual and mathematical foundation for relativistic quantum mechanics and points directly toward quantum field theory. The algebra of the γ-matrices becomes identified with a Clifford algebra underlying spinor representations, and the reinterpretation of negative-energy states presages field-theoretic notions of antiparticles and vacuum structure. Many later developments, second quantization, the modern formulation of fermionic fields, and the unification of spin and relativistic symmetry, trace back to Dirac's formulation.
Experimental confirmation of the predicted antiparticle and the success in accounting for fine structure and magnetic behavior cement the equation's central role in physics. Beyond immediate phenomenology, the Dirac equation reshapes theoretical expectations about symmetry, quantization, and the possibility that mathematical consistency can demand new forms of matter.