Dirac

The Dirac equation combines first-order relativistic dynamics with spin one half and a positive conserved density. Its four components belong to a spinor representation, and its free Hamiltonian has two energy sectors. This chapter develops the equation, the modes used in calculations, and the interpretation of their currents and interference. It also identifies the extra structure needed when the desired prediction involves quantum fields or radiation.

Enter this chapter

Use Metric and Units, Four-Vectors, and the Energy–Momentum Relation for the common relativistic conventions. The Klein–Gordon Equation supplies the second-order equation recovered when the Dirac operator is squared. Pauli-matrix algebra and the Hilbert-space interpretation of a wavefunction are assumed.

From the equation to calculations

Task	Reading sequence	Result to carry forward
Understand the construction	Motivation → Gamma Matrices → Covariant Dirac Equation	First-order factorization, covariance, and the conserved positive density
Control the free evolution	Hamiltonian Form → Free Dirac Spinors	Operator domain, energy projectors, normalized modes, and spin sums
Calculate local observables	Dirac Current → Bilinear Covariants	Gordon decomposition, Pauli spin current, and Lorentz transformation types
Interpret the two sectors	Dirac Negative-Energy Solutions → Zitterbewegung	Component blocks versus spectral sectors, and coherent intersector motion
Choose the larger theory	Dirac Equation as Bridge	Which predictions require matter quantization, photon dynamics, or radiative corrections

Keep the Gamma-Matrix Conventions open when checking an explicit matrix calculation. The Clifford-algebra derivation is representation independent; the convention table fixes the basis and signs used to implement it.

Three distinctions that organize the chapter

Density and energy. The density $\psi^\dagger\psi$ is positive in both free energy sectors. Positive norm does not imply a Hamiltonian bounded below, and $\bar\psi\psi$ is a scalar bilinear rather than that norm.

Components and states. Four spinor components do not mean four independent particle species. Upper and lower Dirac-basis blocks differ from positive and negative energy projections except at special momenta.

Equation and interpretation. The same mode equation can describe a c-number amplitude or provide coefficients of a quantum field operator. The latter requires a state space, vacuum, and operator algebra in addition to the differential equation.

Connections and checks

The Pauli limit connects this chapter to low-energy spin dynamics in prescribed fields. The limits of one-particle quantum mechanics explain why a successful wave equation is not the whole interacting theory. The Poincaré group places its spin content in a wider representation framework.

A useful end-of-chapter calculation is to choose a nonzero momentum, construct all four normalized energy eigenvectors, verify their projector completeness, and evaluate their currents. It simultaneously tests signs, normalization, the momentum label of a negative-frequency mode, and the distinction between a component block and an energy eigenspace.

References

• J. D. Bjorken and S. D. Drell, Relativistic Quantum Mechanics, McGraw–Hill, 1964.
• P. A. M. Dirac, “The Quantum Theory of the Electron,” Proceedings of the Royal Society A 117, 610–624, 1928, doi:10.1098/rspa.1928.0023.
• B. Thaller, The Dirac Equation, Springer, 1992.