Dirac wanted an equation first order in time, like Schrödinger’s, so that a positive density might survive. Relativity then asks for first order in space too: $i\hbar\partial\psi/\partial t=\bigl(c\,\vec{\alpha}\cdot\hat{\mathbf{p}}+\beta mc^{2}\bigr)\psi$. For this to give the right energies, its Hamiltonian squared must be $p^{2}c^{2}+m^{2}c^{4}$, which works only if the four coefficients $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ and $\beta$ anticommute with one another and square to one. Numbers cannot do that; matrices can, and the smallest that do are four by four. So $\psi$ has four components. That was a surprise with two rewards. Two of the components turn out to be the electron’s two spin states — spin is not added by hand, it is demanded by relativity and first order — and the other two belong to negative energies, which became antiparticles.

Asking for a number whose square is $-1$ forced mathematicians to invent $i$; asking for a square root of $p^{2}c^{2}+m^{2}c^{4}$ linear in forced Dirac to use matrices. The extra room they brought held the spin and the antiparticle.

For a free particle of mass $m$, with four Hermitian matrices $\alpha_{1},\alpha_{2},\alpha_{3},\beta$:

• $\{A,B\}=AB+BA$ is the anticommutator; the relations make $\hat{H}_{D}^{2}=p^{2}c^{2}+m^{2}c^{4}$.

Square the Dirac Hamiltonian, keeping the order of the coefficients, since they need not commute. Matching the terms quadratic in momentum, linear in momentum and free of it gives the three relations. Then count: the relations force each coefficient to be traceless with eigenvalues $\pm1$, so the dimension is even; two dimensions have only three anticommuting matrices, and four are needed.

Expand, keeping the order of the matrices.

$p_{i}p_{j}$ is symmetric in $i,j$.

Match $p^{2}c^{2}+m^{2}c^{4}$ term by term.

Multiply $\{\alpha_{i},\beta\}=0$ by $\beta$ and use $\beta^{2}=\hat{1}$ and the cyclic trace; likewise .

Each squares to one, so its eigenvalues are $\pm1$, as many of each. In two dimensions only the three Pauli matrices anticommute; four are needed.

Dirac asked for an equation first order in time, like Schrödinger’s, and so, by relativity, first order in space as well.

The coefficients must anticommute and square to one. The smallest matrices that do are four by four, so $\psi$ has four components.

Numbers commute, so they cannot anticommute unless they vanish: the coefficients have to be matrices.

Squaring the Dirac Hamiltonian, the cross terms linear in $p$ must cancel: $\alpha$ and $\beta$ anticommute.

Great work! You now know how to:

• write the Dirac equation and its conditions on $\alpha$ and $\beta$
• derive the anticommutation relations from the energy
• explain why $\psi$ has four components