The Dirac Equation

The Klein–Gordon equation's negative-probability problem traces to its second-order time derivative. Dirac asked: can we build a relativistic equation that is first order in time — like Schrödinger's — and so yields a positive-definite probability density? Demanding that, while still respecting $E^2 = p^2c^2 + m^2c^4$ , forces a remarkable structure into existence: spin, four-component wavefunctions, and antimatter all emerge as consequences.

Dirac's factorization

Dirac wanted a Hamiltonian linear in momentum,

i\hbar\,\partial_t\psi = H\psi, \qquad H = c\,\boldsymbol{\alpha}\cdot\mathbf{p} + \beta\,mc^2 ,

with constant coefficients $\alpha_x,\alpha_y,\alpha_z,\beta$ . For consistency, squaring $H$ must reproduce $E^2 = p^2 c^2 + m^2 c^4$ with no cross terms. Working out $H^2$ shows the coefficients cannot be numbers — they must satisfy the anticommutation relations

\{\alpha_i,\alpha_j\} = 2\delta_{ij},\qquad \{\alpha_i,\beta\} = 0,\qquad \beta^2 = 1 .

No scalars do this; the smallest objects that do are $4\times 4$ matrices. The wavefunction $\psi$ is therefore a four-component column — a Dirac spinor.

Covariant form and the gamma matrices

Multiplying through by $\beta/c$ and defining the gamma matrices $\gamma^0 = \beta$ , $\gamma^i = \beta\alpha^i$ , the equation takes its iconic Lorentz-covariant form (natural units $\hbar = c = 1$ ):

\boxed{\;\big(i\gamma^\mu\partial_\mu - m\big)\psi = 0\;}

The gammas obey the Clifford algebra $\{\gamma^\mu,\gamma^\nu\} = 2g^{\mu\nu}\mathbb{1}$ , where $g^{\mu\nu} = \mathrm{diag}(+,-,-,-)$ is the Minkowski metric. This single algebraic relation encodes all of relativity in the equation. A standard $2\times 2$ -block representation is

\gamma^0 = \begin{pmatrix} \mathbb{1} & 0 \\ 0 & -\mathbb{1}\end{pmatrix},\qquad \gamma^i = \begin{pmatrix} 0 & \sigma_i \\ -\sigma_i & 0 \end{pmatrix},

built from the $2\times 2$ Pauli matrices $\sigma_i$ — the link to spin is already visible.

A positive probability density

Because the equation is first order in $\partial_t$ , the conserved current $j^\mu = \bar\psi\gamma^\mu\psi$ (with $\bar\psi = \psi^\dagger\gamma^0$ ) has time component

\rho = j^0 = \psi^\dagger\psi = \sum_{a=1}^{4}|\psi_a|^2 \;\ge\; 0 .

This is manifestly non-negative — Dirac's original goal achieved. The probability problem of Klein–Gordon is solved.

What the four components mean

A four-component spinor seems like too much for a single electron. The structure resolves into two doublets:

Two spin states — the equation automatically describes a spin- $\tfrac12$ particle, with the Pauli matrices appearing inside $\gamma^i$ . Spin is not added by hand; it is forced by relativistic covariance.
Two energy signs — like Klein–Gordon, the spectrum has both $E>0$ and $E<0$ branches. The negative-energy solutions, reinterpreted, become antiparticles (the positron).

The non-relativistic limit

Expanding the Dirac equation for $E \approx mc^2$ reproduces the Pauli equation — Schrödinger's equation for a two-component spinor coupled to electromagnetism — with a magnetic moment $\boldsymbol\mu = -g\, \tfrac{e}{2m}\mathbf{S}$ and $g = 2$ emerging automatically. Going one order further yields the fine-structure corrections (spin–orbit coupling, the Darwin term, relativistic kinetic energy) that we previously bolted onto hydrogen by hand. Here they are derived from a single equation, exactly. The next lessons unpack what the negative-energy solutions and the spin structure really mean.