Written with $\alpha$ and $\beta$, the Dirac equation singles out time. Multiplying it by $\beta$ and naming $\gamma^{0}=\beta$, $\gamma^{i}=\beta\alpha_{i}$ puts time and space on one footing: $(i\hbar\gamma^{\mu}\partial_{\mu}-mc)\psi=0$. The four gamma matrices obey one compact rule, $\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}$, which contains all the anticommutation relations at once; it defines what mathematicians call a Clifford algebra. With the Dirac adjoint $\bar{\psi}=\psi^{\dagger}\gamma^{0}$, the equation has a conserved current $j^{\mu}=c\bar{\psi}\gamma^{\mu}\psi$ whose time component is $\psi^{\dagger}\psi$ — never negative. That was what Dirac wanted: a density that can be a probability, at least while no pairs are made. The gammas also build every other quantity that behaves simply under Lorentz transformations, and a fifth matrix, $\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$, tells left-handed from right-handed particles, which the weak force treats differently.

Rewriting an account in a common currency lets its entries be added and compared; the gamma matrices rewrite the Dirac equation in a currency shared by time and space.

With $\gamma^{0}=\beta$ and $\gamma^{i}=\beta\alpha_{i}$, the Dirac equation and its conserved current are

• $(\gamma^{0})^{2}=\hat{1}$ and $(\gamma^{i})^{2}=-\hat{1}$; is Hermitian and the anti-Hermitian: .

Take the Dirac equation and its adjoint, which with $\gamma^{0}\gamma^{\mu\dagger}\gamma^{0}=\gamma^{\mu}$ reads as an equation for acted on from the right. Multiply the first by on the left, the second by on the right, and subtract: the mass terms cancel and a total divergence is left. The time component uses .

The Dirac equation.

Take the Hermitian adjoint, multiply by $\gamma^{0}$ on the right, and use $\gamma^{\mu\dagger}\gamma^{0}=\gamma^{0}\gamma^{\mu}$.

$\bar{\psi}\times$(first) minus (second)$\times\psi$.

The bracket is a product-rule derivative.

$(\gamma^{0})^{2}=\hat{1}$.

The four gammas anticommute and square to the metric: $(\gamma^{0})^{2}=1$, $(\gamma^{i})^{2}=-1$.

Multiplying the Dirac equation by $\beta$ puts all four derivatives on one footing, each with its own gamma matrix.

The time component of the Dirac current is $\psi^{\dagger}\psi$, a sum of squared moduli: never negative.

$\bar{\psi}=\psi^{\dagger}\gamma^{0}$ is the combination that makes $\bar{\psi}\psi$ a Lorentz scalar and a four-vector.

Great work! You now know how to:

• write the Dirac equation with gamma matrices
• derive the conserved current and its positive density
• state the anticommutation rule and the role of $\gamma^{5}$