The Dirac Equation and Spinors

The Klein-Gordon equation carried a negative-definite density because it is second order in time. Dirac's idea, in 1928, was to insist on an equation first order in time, which admits a positive density $\rho ={\psi }^{†}\psi$, and then to make it relativistic by requiring first order in space as well. That single demand, pursued honestly, forces almost everything about the electron: the wavefunction must have four components, it must transform as a spinor rather than a scalar, spin $\frac{1}{2}$ appears without being put in, the magnetic moment comes out with $g=2$, and the equation predicts antiparticles. The Dirac equation is the most productive guess in the history of the subject. This lesson follows the demand to its consequences.

We keep $\hslash =c=1$ and ${\eta }^{\mu \nu }=\mathrm{diag}(+,-,-,-)$, with the Pauli matrices ${\sigma }^i$ and the $2\times 2$ identity as building blocks.

Factorizing the energy relation

Dirac wanted a Hamiltonian linear in the momentum,

with constant coefficients ${\alpha }^1,{\alpha }^2,{\alpha }^3,\beta$. Any solution of a sensible relativistic equation must also satisfy $E^2={\vec{p}}^2+m^2$, so applying $H$ twice must reproduce ${\vec{p}}^2+m^2$. Square it:

Symmetrizing the first term over $i,j$, this equals ${\vec{p}}^2+m^2$ for all momenta only if the coefficients obey

No ordinary numbers satisfy these: two numbers cannot anticommute. The coefficients must be matrices. The smallest matrices that work are $4\times 4$, so $\psi$ must be a four-component column, a Dirac spinor. A common explicit choice is the Dirac representation,

each block $2\times 2$. That the factorization of a scalar equation demands matrices, and matrices of size four, is the origin of every structural feature that follows.

Gamma matrices and the Clifford algebra

The equation looks cleaner in covariant form. Multiply the Dirac equation by $\beta$ and define the gamma matrices

Then $i{\gamma }^0{\partial }_t\psi =(\dots )$ tidies into the covariant Dirac equation

often abbreviated with the Feynman slash notation ${\gamma }^{\mu }{\partial }_{\mu }$. The anticommutation relations above collapse into the single defining relation of the Clifford algebra,

This one line contains all the earlier conditions: $\mu =\nu =0$ gives $({\gamma }^0{)}^2=1$; $\mu =\nu =i$ gives $({\gamma }^i{)}^2=-1$; and $\mu \ne \nu$ gives anticommutation. The Clifford algebra has a unique irreducible representation up to change of basis, so different textbooks' explicit gamma matrices (Dirac basis, chiral/Weyl basis) are physically equivalent; only the algebra is fundamental.¹ It is the algebra, not any particular matrix, that makes the equation work.

The plane-wave solutions

Look for plane-wave solutions of definite four-momentum,

where $u$ and $v$ are four-component constant spinors. Substituting the first form into the Dirac equation gives the momentum-space condition

and the second gives $({\gamma }^{\mu }{p}_{\mu }+m)v(p)=0$. Each is a set of four linear equations; each has two independent solutions. So for every momentum there are four solutions in all:

• $u^{(1)},u^{(2)}$ — two positive-energy spinors, the two spin states of the particle;
• $v^{(1)},v^{(2)}$ — two negative-energy spinors, which the next lesson reinterprets as the two spin states of the antiparticle.

In the Dirac representation, writing the four-spinor as two two-component blocks $\psi =\left(\genfrac{}{}{0ex}{}{\varphi }{\chi }\right)$, the positive-energy solutions have a large upper block $\varphi$ and a small lower block $\chi \sim (\vec{\sigma }\cdot \vec{p}/2m)\varphi$ in the nonrelativistic limit. The two-valuedness of $\varphi$ is spin: the electron's spin is not added to the theory, it is the residual freedom in the plane-wave spinor once the Dirac equation is imposed.

Spin and the magnetic moment $g=2$

That spin comes free is confirmed by coupling the equation to electromagnetism. The minimal substitution $p_{\mu }\to p_{\mu }-q{A}_{\mu }$ inserts the electromagnetic potential, and taking the nonrelativistic limit of the resulting equation for the upper block $\varphi$ yields the Pauli equation

The final term is a magnetic moment interacting with the field, and it was not put in by hand — it emerges from the cross terms in $(\vec{\sigma }\cdot (\vec{p}-q\vec{A}){)}^2$ via the identity $(\vec{\sigma }\cdot \vec{a})(\vec{\sigma }\cdot \vec{b})=\vec{a}\cdot \vec{b}+i\vec{\sigma }\cdot (\vec{a}\times \vec{b})$. Reading off the moment,

The Dirac equation predicts the electron's gyromagnetic ratio $g=2$, twice the classical value for orbital motion, with no free parameter. Experiment confirms $g\approx 2$ to leading order; the tiny deviation $g-2$ is a quantum correction computed in the QED module and is one of the most precise tests in all of physics. A spinless Klein-Gordon particle has no such term. Getting $g=2$ right was the decisive early triumph of the equation.

Chirality, helicity, and the fifth gamma matrix

Beyond the four ${\gamma }^{\mu }$ there is a fifth matrix built from their product,

Because ${\gamma }^5$ squares to the identity, its eigenvalues are $\pm 1$, and the operators

are projectors ($P_L^2=P_L$, $P_R^2=P_R$, $P_L+P_R=1$, $P_L{P}_R=0$) that split any Dirac spinor into left-handed and right-handed chiral parts,

Chirality is the eigenvalue of ${\gamma }^5$; it is Lorentz invariant. Helicity, the projection of spin along the momentum $h=\vec{S}\cdot \hat{p}$, is a distinct but related quantity: for a massless particle chirality and helicity coincide, while for a massive particle they differ by terms of order $m/E$ because one can always boost past a massive particle and reverse its helicity, but not its chirality. This distinction is not idle bookkeeping — the charged weak interaction couples only to ${\psi }_L$, the left-chiral projection, which is why parity is violated and why the weak force treats the two handednesses so differently. The electroweak module builds its doublets out of the projectors introduced here for free spinors.

Summary

Demanding a first-order, positive-density relativistic equation forces the coefficients ${\gamma }^{\mu }$ to obey the Clifford algebra $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2{\eta }^{\mu \nu }$, which pushes the wavefunction up to four components — a Dirac spinor. The covariant equation $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$ has four plane-wave solutions per momentum: two spin states of a particle and two of an antiparticle. Spin $\frac{1}{2}$ is not inserted; it is the residual freedom in the spinor, and coupling to electromagnetism reproduces the electron's magnetic moment with $g=2$ automatically. The fifth gamma matrix ${\gamma }^5$ supplies the chiral projectors $\frac{1}{2}(1\mp {\gamma }^5)$ that the weak interaction later needs. The one loose end — the negative-energy solutions $v(p)$ — is the subject of the next lesson.