What relativistic equation describes spin-1/2 particles like electrons?

The Dirac equation is the relativistic quantum equation that describes spin-1/2 particles such as electrons.

Paul Dirac developed the equation in 1928 to reconcile quantum mechanics with special relativity. Unlike the earlier Schrödinger equation, it treats space and time in a relativistically consistent way and naturally incorporates the electron's intrinsic spin. Its wavefunction has multiple components, allowing spin states and the electron's antiparticle, the positron, to emerge from the theory.

The Pauli equation is related but is a non-relativistic approximation used when particle speeds are much lower than the speed of light while spin and magnetic interactions still matter. The Klein–Gordon equation is relativistic, but it is primarily associated with spin-0 particles. Dirac matrices are mathematical objects appearing inside the Dirac equation, not the equation itself.

The theory also predicts the electron's magnetic moment at leading order, including a gyromagnetic factor of approximately 2. Dirac's work earned him the 1933 Nobel Prize in Physics.