The Dirac equation

The Dirac equation is what you get when you refuse to compromise on two demands at once: a wave equation must be first order in the time derivative — so that specifying the wavefunction now fixes its whole future and the probability density stays positive — and it must respect Einstein's relation $E^2=p^2c^2+m^2c^4$ , so that its solutions describe genuinely relativistic particles. The Klein–Gordon equation honours the second demand but not the first, and pays for it with a probability density $\rho\propto E$ that turns negative. Dirac's move, in 1928, was to write an energy operator linear in momentum, $\hat E\psi=(\boldsymbol\alpha\cdot\hat{\mathbf p}+\beta m)\psi$ , and then ask what the coefficients $\boldsymbol\alpha$ and $\beta$ must be for its solutions to also solve Klein–Gordon. The answer is startling. Squaring the operator and matching term by term forces $\alpha_x^2=\alpha_y^2=\alpha_z^2=\beta^2=1$ together with anticommutation relations, $\{\alpha_i,\alpha_j\}=0$ and $\{\alpha_i,\beta\}=0$ . Ordinary numbers commute, so none can satisfy these — the coefficients must be matrices, and the smallest that work are $4\times4$ . The wavefunction is therefore not a scalar but a four-component object, the Dirac spinor. That structure is not bookkeeping: reduce the equation to its non-relativistic limit and out fall the electron's spin, its magnetic moment $g=2$ , and the fine structure of hydrogen — none of it put in by hand. Spin and antimatter, grafted onto the Schrödinger theory by hand, are here consequences of insisting quantum mechanics and relativity hold together.

Key takeaways

Dirac demanded a wave equation first order in time — curing Klein–Gordon's negative probability density — while still enforcing $E^2=p^2c^2+m^2c^4$ .
Requiring the solutions to also satisfy Klein–Gordon forces $\alpha_i^2=\beta^2=1$ and the anticommutation relations $\{\alpha_i,\alpha_j\}=\{\alpha_i,\beta\}=0$ .
No numbers anticommute, so $\boldsymbol\alpha,\beta$ must be $4\times4$ Hermitian, traceless matrices; the wavefunction becomes a four-component Dirac spinor.
In covariant form the equation is $(i\gamma^\mu\partial_\mu-m)\psi=0$ with the Clifford algebra $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}$ — Griffiths reaches it by factoring $p^\mu p_\mu-m^2$ .
The probability density is now $\rho=\psi^\dagger\psi\ge0$ , positive definite — exactly the flaw Klein–Gordon could not fix.
The non-relativistic limit delivers spin-½, the magnetic moment $\mu=g\,(e/2m)\,\mathbf S$ with $g=2$ , and hydrogen fine structure, for free.
Key idea: forcing a relativistic wave equation to be first order in time and consistent with $E^2=p^2+m^2$ inflates its coefficients into anticommuting $4\times4$ matrices and its wavefunction into a four-component spinor — spin, the moment g = 2, and antiparticles emerge unbidden.

The core insight

Forcing a relativistic wave equation to be first order in time AND consistent with $E^2=p^2c^2+m^2c^4$ inflates the coefficients into anticommuting $4\times4$ matrices, and the wavefunction into a four-component spinor — spin, the electron's magnetic moment $g=2$ , and antiparticles all emerge for free.

Where this lives today

The Dirac equation long ago stopped being only foundational and became a working tool across physics. Its sharpest modern test is still the electron magnetic moment: the 2023 Harvard measurement pins $g/2=1.001\,159\,652\,180\,59$ , and comparing it with the QED prediction — computed to five loops — confirms the Dirac-plus-QED picture to twelve significant figures and yields the most precise value of the fine-structure constant $\alpha$ . The muon's anomaly is the field's most watched frontier: the Fermilab Muon g-2 experiment measures $a_\mu$ to parts per billion, probing whether particles beyond the Standard Model nudge it from theory. In a different direction, the massless Dirac equation describes real materials: the low-energy electrons of graphene behave as two-dimensional massless Dirac fermions, and a whole class of Dirac and Weyl semimetals and topological insulators are classified by the equation's structure — making a 1928 particle-physics equation the daily language of condensed-matter labs. Relativistic quantum chemistry, built on the same equation, even explains why gold is yellow and mercury is liquid.

The physical picture

The whole equation is the resolution of a single tension. You want first order in time, for a clean initial-value problem and a positive probability density; but a first-order operator, squared, must reproduce the relativistic $E^2=p^2c^2+m^2c^4$ , because every physical particle obeys it. Squaring reveals the cross terms vanish only if the coefficients anticommute, and the diagonal terms give $p^2+m^2$ only if their squares are the identity. Numbers cannot do both, so the coefficients inflate into matrices — and counting eigenvalues and traces pins the smallest size at four. The first figure follows that logic from demand to matrices.

Figure

Squaring a first-order operator forces the coefficients to anticommute. Demanding that i∂ₜψ = (α·p̂ + βm)ψ also obey E² = p² + m² gives αᵢ² = β² = 1 with αᵢαⱼ + αⱼαᵢ = 0 and αᵢβ + βαᵢ = 0. No numbers anticommute, so the smallest solution is four 4×4 Hermitian traceless matrices — the Dirac–Pauli set with β = diag(I, −I)

The four components are not idle. Take the equation into its non-relativistic limit — an electron moving slowly in an electromagnetic field — and the four-spinor splits into a large upper pair and a small lower pair; eliminating the small pair leaves the Pauli equation with a magnetic-moment term the Schrödinger theory never had. The coefficient is a pure prediction: the electron's spin gyromagnetic ratio is exactly $g=2$ , twice the classical orbital value. Experiment later found $g$ differs from 2 by a little over a tenth of a percent, and that tiny gap — the anomalous moment — is the first thing quantum electrodynamics was built to compute. The second figure is that number, the sharpest confrontation between the equation and the laboratory.

Figure

The crown jewel: the Dirac equation predicts the electron gyromagnetic ratio g = 2 exactly, with no free parameter, where the classical orbital value is g = 1. The measured value g = 2.00231930436 sits a hair above 2; the excess, the anomaly aₑ = (g − 2)/2 = 0.00116, is the QED loop correction whose leading term is α/2π = 0.00116

The core equations

Dirac's starting point is an energy operator linear in the momentum, so the time evolution is first order and the density stays positive. Using natural units $\hbar=c=1$ , it is a Schrödinger-like statement with a matrix-valued Hamiltonian.

i\frac{\partial\psi}{\partial t}=\hat H\psi=(\boldsymbol\alpha\cdot\hat{\mathbf p}+\beta m)\psi \qquad (1)

where: $\psi$ — four-component Dirac spinor; $\boldsymbol\alpha=(\alpha_x,\alpha_y,\alpha_z)$ and $\beta$ — constant $4\times4$ Hermitian matrices; $\hat{\mathbf p}=-i\nabla$ — momentum operator; $m$ — rest mass. Reading it: the Hamiltonian is linear in $\hat{\mathbf p}$ , unlike the non-relativistic $\hat p^2/2m$ , and that linearity is the whole point — it makes the time derivative first order. The coefficients are not free. Demanding that each component of $\psi$ also satisfy Klein–Gordon — the algebraic content of $E^2=p^2+m^2$ — squares the operator and fixes the entire algebra of $\boldsymbol\alpha$ and $\beta$ .

\alpha_i^2=\beta^2=\mathbb 1,\qquad \{\alpha_i,\alpha_j\}=2\delta_{ij}\,\mathbb 1,\qquad \{\alpha_i,\beta\}=0 \qquad (2)

where: $\{A,B\}=AB+BA$ — the anticommutator; $\delta_{ij}$ — Kronecker delta; $\mathbb 1$ — the $4\times4$ identity. Reading it: the objects square to one but anticommute — the signature of a Clifford algebra, and why $\boldsymbol\alpha,\beta$ cannot be numbers. One convenient realisation is the Dirac–Pauli (standard) representation, written in $2\times2$ blocks built from the Pauli matrices $\sigma_i$ . It is only a choice — the physics resides in equation (2), not in any particular set of matrices.

\beta=\begin{pmatrix}\mathbb 1_2&0\\[2pt]0&-\mathbb 1_2\end{pmatrix},\qquad \alpha_i=\begin{pmatrix}0&\sigma_i\\[2pt]\sigma_i&0\end{pmatrix} \qquad (3)

where: $\mathbb 1_2$ — the $2\times2$ identity; $\sigma_i$ — Pauli matrices, obeying $\{\sigma_i,\sigma_j\}=2\delta_{ij}$ . Reading it: $\beta$ is diagonal and splits the spinor into upper and lower two-component blocks, while $\alpha_i$ is off-diagonal and couples them — the coupling that becomes the "small component" in the non-relativistic limit. Multiplying (1) by $\beta$ and defining $\gamma^0=\beta$ , $\gamma^i=\beta\alpha_i$ recasts everything in manifestly Lorentz-covariant form — the equation as it is almost always written. Griffiths reaches it directly by factoring the mass-shell condition $p^\mu p_\mu-m^2$ .

(i\gamma^\mu\partial_\mu-m)\psi=0,\qquad \{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}\mathbb 1 \qquad (4)

where: $\gamma^\mu=(\gamma^0,\boldsymbol\gamma)$ — the gamma (Dirac) matrices; $\partial_\mu=(\partial_t,\nabla)$ — four-gradient; $g^{\mu\nu}=\mathrm{diag}(+1,-1,-1,-1)$ — Minkowski metric. Reading it: $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}$ is the Clifford algebra; it contains all of (2), and every property of Dirac theory follows from it. Because the equation is first order and Hermitian, the conserved four-current is bilinear in $\psi$ and its time component is a sum of squares — positive definite, exactly what Klein–Gordon could not deliver.

j^\mu=\bar\psi\gamma^\mu\psi,\qquad \partial_\mu j^\mu=0,\qquad \rho=j^0=\psi^\dagger\psi\ge0 \qquad (5)

where: $\bar\psi=\psi^\dagger\gamma^0$ — the Dirac adjoint; $\rho$ — probability density; $\mathbf j=\psi^\dagger\boldsymbol\alpha\psi$ — probability current. Reading it: $\rho=\psi^\dagger\psi=\sum_a|\psi_a|^2$ can never go negative, so Born's statistical interpretation survives. Finally, minimally coupling to an electromagnetic field and taking the slow-particle limit produces the electron's intrinsic magnetic moment — the equation's most celebrated by-product.

\boldsymbol\mu=g\,\frac{e}{2m}\,\mathbf S,\qquad g=2 \qquad (6)

where: $\boldsymbol\mu$ — magnetic moment; $e$ — elementary charge; $\mathbf S$ — spin operator (eigenvalues $\pm\tfrac12\hbar$ ); $g$ — gyromagnetic ratio. Reading it: the Dirac electron is twice as magnetic per unit spin as a classical charge is per unit orbital angular momentum — a number confirmed to twelve figures once QED corrections are added.

Derivation ALinearizing the energy relation forces anticommuting matrices

The goal of this derivation is to show, with nothing assumed beyond "first order in time" and "consistent with special relativity", that the coefficients in Dirac's equation cannot be numbers. We start from the linear ansatz, impose the single physical requirement that every solution also solve the Klein–Gordon equation — the operator statement of $E^2=p^2+m^2$ — square the Dirac operator, and read off what $\boldsymbol\alpha$ and $\beta$ must satisfy. The anticommutation relations that fall out are the entire origin of the spinor structure.

Step 1. Write the most general Hamiltonian linear in momentum: $\hat H=\boldsymbol\alpha\cdot\hat{\mathbf p}+\beta m$ , with as-yet-unknown constant coefficients $\alpha_i$ and $\beta$ , so that $i\partial_t\psi=\hat H\psi$ . Why: linearity in $\hat{\mathbf p}$ is the one structural choice that makes the equation first order in time and hence gives a positive, single-particle probability density.

Step 2. Apply $i\partial_t$ once more, i.e. act with $\hat H$ twice: $-\partial_t^2\psi=\hat H^2\psi=(\alpha_i\hat p_i+\beta m)(\alpha_j\hat p_j+\beta m)\psi$ . Expanding and grouping gives $\hat H^2=\alpha_i\alpha_j\hat p_i\hat p_j+m(\alpha_i\beta+\beta\alpha_i)\hat p_i+\beta^2m^2$ . Why: squaring is exactly how the first-order equation must reproduce the relativistic mass-shell relation; whatever the coefficients are, $\hat H^2$ has to collapse to $\hat{\mathbf p}^2+m^2$ .

Step 3. Because $\hat p_i\hat p_j$ is symmetric in $i,j$ , only the symmetric part of $\alpha_i\alpha_j$ survives the first sum: $\alpha_i\alpha_j\hat p_i\hat p_j=\tfrac12\{\alpha_i,\alpha_j\}\hat p_i\hat p_j$ . Matching $\hat H^2\psi=(\hat{\mathbf p}^2+m^2)\psi$ term by term then demands $\tfrac12\{\alpha_i,\alpha_j\}=\delta_{ij}$ , $\{\alpha_i,\beta\}=0$ , and $\beta^2=\mathbb 1$ . Why: the cross term $\propto\hat p_i$ must vanish (there is no such term in $p^2+m^2$ ), which is only possible if $\alpha_i$ and $\beta$ anticommute — the decisive step.

Step 4. The relations $\alpha_i^2=\beta^2=\mathbb 1$ with $\alpha_i\beta=-\beta\alpha_i$ and $\alpha_i\alpha_j=-\alpha_j\alpha_i$ cannot hold for scalars, since two numbers always commute. They can hold for matrices, and the objects that satisfy them are the generators of a Clifford algebra. Why: this is the punchline — relativity plus a first-order equation forces the wavefunction's coefficients to be matrices, and therefore forces $\psi$ to have several components. *Verified: $\hat H^2=\hat{\mathbf p}^2+m^2$ requires $\{\alpha_i,\alpha_j\}=2\delta_{ij}$ , $\{\alpha_i,\beta\}=0$ , $\beta^2=\mathbb 1$ ; scalars cannot satisfy anticommutation, so $\alpha_i,\beta$ are matrices.*

Go deeper · Derivation B · 2 minWhy the matrices must be four by four

Derivation BWhy the matrices must be four by four

The aim here is to nail the size of the matrices — to prove the smallest ones satisfying Dirac's algebra are $4\times4$ , so the Dirac spinor has exactly four components, no fewer. We use four short arguments: Hermiticity for real energies, the eigenvalue argument, the trace argument, and the fact that only three mutually anticommuting $2\times2$ matrices exist. Together they exclude every dimension below four.

Step 1. For the energy eigenvalues to be real, $\hat H=\boldsymbol\alpha\cdot\hat{\mathbf p}+\beta m$ must be Hermitian, so each $\alpha_i$ and $\beta$ is Hermitian. Combined with $\alpha_i^2=\beta^2=\mathbb 1$ , this forces every eigenvalue to be real and to square to one — hence each is $+1$ or $-1$ . Why: real observable energies are non-negotiable, and they pin the spectrum of the coefficient matrices to $\pm1$ .

Step 2. Show the matrices are traceless. From $\{\alpha_i,\beta\}=0$ we have $\alpha_i=-\beta\alpha_i\beta^{-1}$ , so using the cyclic property of the trace, $\mathrm{Tr}\,\alpha_i=-\mathrm{Tr}(\beta\alpha_i\beta^{-1})=-\mathrm{Tr}\,\alpha_i$ , giving $\mathrm{Tr}\,\alpha_i=0$ ; the same argument with $\alpha_i^2=\mathbb 1$ gives $\mathrm{Tr}\,\beta=0$ . Why: a vanishing trace plus eigenvalues $\pm1$ means there must be equal numbers of $+1$ and $-1$ eigenvalues, so the dimension is even.

Step 3. Rule out dimension two. A $2\times2$ Hermitian traceless matrix space is spanned by the three Pauli matrices, and any two distinct Pauli matrices already anticommute — but there are only three of them, while Dirac's algebra needs four mutually anticommuting matrices ( $\alpha_x,\alpha_y,\alpha_z,\beta$ ). Three slots cannot hold four objects, so $2\times2$ is impossible; dimension zero and the odd dimensions are excluded by Steps 1–2. Why: the count "four anticommuting matrices" against "only three available at $2\times2$ " is precisely what pushes the size up.

Step 4. Dimension four works. The Dirac–Pauli set $\beta=\mathrm{diag}(\mathbb 1_2,-\mathbb 1_2)$ , $\alpha_i=\left(\begin{smallmatrix}0&\sigma_i\\\sigma_i&0\end{smallmatrix}\right)$ is Hermitian, traceless, squares to $\mathbb 1$ , and satisfies every anticommutator — an explicit existence proof. Hence the smallest spinor has four components. Why: four is both necessary (Steps 1–3) and sufficient (this explicit construction), so the Dirac spinor is four-dimensional and no smaller theory exists. *Verified: Hermiticity ⟹ eigenvalues $\pm1$ ; $\mathrm{Tr}\,\alpha_i=\mathrm{Tr}\,\beta=0$ ⟹ even dimension; only three anticommuting $2\times2$ matrices exist ⟹ dimension $>2$ ; Dirac–Pauli set realises the algebra at $4\times4$ .*

Go deeper · Derivation C · 1 minThe non-relativistic limit gives spin and a magnetic moment with g = 2

Derivation CThe non-relativistic limit gives spin and a magnetic moment with g = 2

This derivation extracts the single result that convinced physicists the four components were physical, not formal: coupling the Dirac equation to a magnetic field and letting the electron move slowly reproduces the Pauli equation and fixes the electron's spin gyromagnetic ratio at exactly $g=2$ , with no adjustable parameter. We minimally couple, split the spinor into upper and lower blocks, eliminate the small block, and identify the magnetic-moment term. Nowhere is spin assumed — it is output.

Step 1. Minimally couple by $\hat{\mathbf p}\to\boldsymbol\pi=\hat{\mathbf p}-e\mathbf A$ , and write the four-spinor in the Dirac–Pauli basis as an upper block $\varphi$ and lower block $\chi$ . Subtracting the rest energy $m$ and working to leading order in $v/c$ , the coupled equations read $(\boldsymbol\sigma\cdot\boldsymbol\pi)\chi=(E_{\rm NR}-e\phi)\varphi$ and $(\boldsymbol\sigma\cdot\boldsymbol\pi)\varphi\approx2m\chi$ . Why: the off-diagonal $\alpha_i$ ties $\varphi$ to $\chi$ , and in the slow limit $\chi$ is smaller than $\varphi$ by $\sim v/c$ — the "small component".

Step 2. Solve the second equation for the small component, $\chi\approx(\boldsymbol\sigma\cdot\boldsymbol\pi)\varphi/2m$ , and substitute into the first to get a closed equation for $\varphi$ alone: $\dfrac{(\boldsymbol\sigma\cdot\boldsymbol\pi)^2}{2m}\varphi+e\phi\,\varphi=E_{\rm NR}\varphi$ . Why: eliminating $\chi$ collapses the four-component equation to a two-component one — the natural home of a spin-½ wavefunction.

Step 3. Reduce the operator $(\boldsymbol\sigma\cdot\boldsymbol\pi)^2$ with the Pauli identity $(\boldsymbol\sigma\cdot\mathbf a)(\boldsymbol\sigma\cdot\mathbf b)=\mathbf a\cdot\mathbf b+i\boldsymbol\sigma\cdot(\mathbf a\times\mathbf b)$ . Because $\boldsymbol\pi\times\boldsymbol\pi=-ie\,\mathbf B$ (the momenta fail to commute through $\mathbf A$ ), this gives $(\boldsymbol\sigma\cdot\boldsymbol\pi)^2=\boldsymbol\pi^2-e\,\boldsymbol\sigma\cdot\mathbf B$ . Why: the cross-product term is the entire origin of the spin–field coupling; it appears only because $\boldsymbol\pi$ is a gauge-covariant momentum, not because spin was inserted.

Step 4. The equation for $\varphi$ becomes the Pauli equation $\left[\dfrac{\boldsymbol\pi^2}{2m}-\dfrac{e}{2m}\boldsymbol\sigma\cdot\mathbf B+e\phi\right]\varphi=E_{\rm NR}\varphi$ . Writing the spin operator $\mathbf S=\tfrac12\boldsymbol\sigma$ , the field term is $-\boldsymbol\mu\cdot\mathbf B$ with $\boldsymbol\mu=(e/m)\mathbf S=g\,(e/2m)\mathbf S$ and $g=2$ . Why: comparing coefficients reads off $g=2$ directly — twice the classical orbital value $g=1$ — which is the Dirac equation's signature triumph and the launch point for the QED anomaly. *Verified: elimination of the small component yields the Pauli equation; $(\boldsymbol\sigma\cdot\boldsymbol\pi)^2=\boldsymbol\pi^2-e\boldsymbol\sigma\cdot\mathbf B$ ; magnetic term $-\tfrac{e}{2m}\boldsymbol\sigma\cdot\mathbf B$ gives $g=2$ with no free parameter.*

Limiting cases

Case	Result	Meaning
Rest frame ( $\mathbf p=0$ )	$\hat H=\beta m$ , energies $\pm m$	spinor splits into upper (E = +m) and lower (E = −m) blocks — particle and antiparticle at rest
Free particle	four plane waves, $E=\pm\sqrt{p^2+m^2}$	two spin states at each energy sign; the negative root is the seed of antimatter
Non-relativistic ( $p\ll m$ )	Pauli equation, $g=2$	spin, the magnetic moment, spin–orbit and Darwin terms emerge; ordinary spin-½ QM recovered
Massless ( $m\to0$ )	decouples into two Weyl equations	left- and right-handed two-component fields; chirality becomes exact, relevant to neutrinos
Minimal EM coupling	hydrogen fine structure	exact relativistic levels $E_{nj}$ , correct to order $\alpha^2$ , matching the observed fine structure

Common trapThe Dirac spinor is a four-vector, or the Dirac–Pauli matrices are "the" definition

Both are misconceptions. The four components of $\psi$ do not transform like a spacetime four-vector $(ct,\mathbf x)$ : under a Lorentz boost or rotation, $\psi$ transforms by a special $4\times4$ matrix $S(\Lambda)$ built from the gamma matrices, $\psi'(x')=S(\Lambda)\psi(x)$ , and $S$ is not the vector transformation $\Lambda$ — a $2\pi$ rotation even flips the sign of $\psi$ , the hallmark of a spinor. So a Dirac spinor is a genuinely new object, a bi-spinor, not a four-vector and not four scalars. Equally, the Dirac–Pauli matrices in (3) are only one representation: any $\alpha_i'=U\alpha_iU^{-1}$ , $\beta'=U\beta U^{-1}$ with unitary $U$ satisfies the same algebra and yields identical physics — the Weyl and Majorana representations are common alternatives. The content is the Clifford algebra $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}$ , never a particular array of numbers.

Worked examples

1 · The electron's magnetic moment from g = 2 (fully worked)

The Dirac equation predicts an electron spin magnetic moment $\boldsymbol\mu=g\,(e/2m_e)\,\mathbf S$ with $g=2$ . Compute the size of one Bohr magneton, give the predicted $z$ -component of the electron's spin moment, and compare with the measured $g$ to extract the anomaly.

Step 1. The natural unit of magnetic moment is the Bohr magneton $\mu_B=e\hbar/2m_e$ . Insert $e=1.602\times10^{-19}\ \mathrm C$ , $\hbar=1.055\times10^{-34}\ \mathrm{J\,s}$ , $m_e=9.109\times10^{-31}\ \mathrm{kg}$ : $\mu_B=(1.602\times10^{-19})(1.055\times10^{-34})/(2\times9.109\times10^{-31})=\mathbf{9.274\times10^{-24}\ J/T}$ .

Step 2. With $S_z=\pm\tfrac12\hbar$ and $g=2$ , the moment's $z$ -component is $\mu_z=g\,(e/2m_e)\,S_z=2\cdot(e/2m_e)\cdot(\tfrac12\hbar)=e\hbar/2m_e=\mu_B$ . So the Dirac electron carries a spin moment of exactly one Bohr magneton, $9.274\times10^{-24}\ \mathrm{J/T}$ — twice what a naive "spinning charge" model gives.

Step 3. Experiment finds $g=2.00231930436$ , not exactly 2. The fractional excess is the anomaly $a_e=(g-2)/2=(0.00231930436)/2=\mathbf{0.00115965}$ . It is small — about one part in a thousand — but nonzero, and the Dirac equation alone cannot produce it.

Step 4. Quantum electrodynamics supplies the anomaly through loop corrections; the leading (Schwinger) term $a_e=\alpha/2\pi=(1/137.036)/(2\pi)=\mathbf{0.00116}$ matches the measured excess to three figures on the first term alone. The Dirac value $g=2$ is exact at tree level; $\mu_z=\mu_B=9.274\times10^{-24}$ J/T; the residual anomaly $a_e\approx0.00116$ is pure QED.

2 · Verifying the Dirac–Pauli matrices satisfy the algebra (fully worked)

Confirm that the standard matrices $\beta=\mathrm{diag}(\mathbb 1_2,-\mathbb 1_2)$ and $\alpha_i=\left(\begin{smallmatrix}0&\sigma_i\\\sigma_i&0\end{smallmatrix}\right)$ obey $\beta^2=\mathbb 1$ , $\alpha_i^2=\mathbb 1$ , $\{\alpha_i,\beta\}=0$ , and $\{\alpha_i,\alpha_j\}=2\delta_{ij}$ , and check they are traceless.

Step 1. Square $\beta$ : $\beta^2=\mathrm{diag}(\mathbb 1_2,-\mathbb 1_2)\,\mathrm{diag}(\mathbb 1_2,-\mathbb 1_2)=\mathrm{diag}(\mathbb 1_2,\mathbb 1_2)=\mathbb 1_4$ . ✓ Square $\alpha_i$ : block multiplication gives $\alpha_i^2=\left(\begin{smallmatrix}\sigma_i^2&0\\0&\sigma_i^2\end{smallmatrix}\right)$ , and since $\sigma_i^2=\mathbb 1_2$ , this is $\mathbb 1_4$ . ✓

Step 2. Anticommute $\alpha_i$ with $\beta$ . Compute $\alpha_i\beta=\left(\begin{smallmatrix}0&\sigma_i\\\sigma_i&0\end{smallmatrix}\right)\left(\begin{smallmatrix}\mathbb 1&0\\0&-\mathbb 1\end{smallmatrix}\right)=\left(\begin{smallmatrix}0&-\sigma_i\\\sigma_i&0\end{smallmatrix}\right)$ and $\beta\alpha_i=\left(\begin{smallmatrix}0&\sigma_i\\-\sigma_i&0\end{smallmatrix}\right)$ . Their sum is the zero matrix, so $\{\alpha_i,\beta\}=0$ . ✓

Step 3. Anticommute $\alpha_i$ with $\alpha_j$ . Block multiplication gives $\alpha_i\alpha_j=\left(\begin{smallmatrix}\sigma_i\sigma_j&0\\0&\sigma_i\sigma_j\end{smallmatrix}\right)$ , so $\{\alpha_i,\alpha_j\}=\mathrm{diag}(\{\sigma_i,\sigma_j\},\{\sigma_i,\sigma_j\})$ . Using the Pauli relation $\{\sigma_i,\sigma_j\}=2\delta_{ij}\mathbb 1_2$ , this equals $2\delta_{ij}\mathbb 1_4$ . ✓ The Dirac algebra reduces entirely to the Pauli algebra of the blocks.

Step 4. Traces: $\mathrm{Tr}\,\beta=\mathrm{Tr}\,\mathbb 1_2-\mathrm{Tr}\,\mathbb 1_2=2-2=0$ , and $\alpha_i$ has only off-diagonal blocks, so $\mathrm{Tr}\,\alpha_i=0$ . Both are Hermitian ( $\beta$ is real diagonal; $\alpha_i$ is Hermitian because $\sigma_i$ is). All four defining relations hold, both matrices are Hermitian and traceless, confirming the Dirac–Pauli set is a valid $4\times4$ realisation of Dirac's algebra.

Physics of the portrait

The Dirac equation is the rarest of things — a piece of physics a single mind produced almost whole, from an aesthetic demand rather than a puzzle in the data. Its author trusted the mathematics so completely that he accepted its every consequence, including ones he could not yet explain.

Origins · 1 minPaul Dirac and how this was discovered

Paul Dirac

1902–1984 · relativistic quantum mechanics and the equation of the electron

Paul Adrien Maurice Dirac trained as an electrical engineer in Bristol before turning to mathematics and physics at Cambridge, and he brought an engineer's economy and a mathematician's faith in beauty to everything he did. Famously reticent — colleagues joked that a "dirac" was one word per hour — he worked alone and thought in equations. In 1928, dissatisfied with the Klein–Gordon equation's negative probability density, he sought a relativistic wave equation first order in time. The demand that its solutions still obey $E^2=p^2c^2+m^2c^4$ led him, by an argument he called almost forced, to coefficients that had to be $4\times4$ matrices and a four-component wavefunction. The equation immediately paid dividends he had not sought: the electron's spin and magnetic moment $g=2$ came out automatically, and the hydrogen fine structure emerged to the right order. It also carried negative-energy solutions he could not discard, interpreted through his "hole theory" and, in 1931, as a new particle of the same mass and opposite charge — the positron, found by Anderson the next year. Dirac shared the 1933 Nobel Prize with Schrödinger. His conviction that "it is more important to have beauty in one's equations than to have them fit experiment" was here spectacularly vindicated: an equation demanded by consistency alone contained spin, antimatter, and the most precisely tested prediction in physics.

History

1926

— Schrödinger and, independently, Klein and Gordon write the relativistic scalar wave equation, plagued by negative probabilities

1927

— Pauli introduces spin into quantum mechanics by hand, with two-component spinors and the σ matrices

1928

— Dirac publishes the first-order relativistic equation of the electron; spin and g = 2 emerge automatically

1928

— the equation's hydrogen spectrum reproduces the fine structure to order α²

1931

— Dirac interprets the negative-energy solutions and predicts the anti-electron

1947

— Kusch and Foley measure g slightly above 2, and Schwinger computes the anomaly α/2π

Going further

Texts. Thomson, Modern Particle Physics, §4.1–4.4 (the Dirac equation, its solutions, and antiparticles) · Griffiths, Introduction to Elementary Particles, 2nd ed., §7.1 (the Dirac equation by factorization) · Halzen & Martin, Quarks and Leptons, Ch. 5 (the Dirac equation and its non-relativistic limit) · Peskin & Schroeder, An Introduction to Quantum Field Theory, §3.2–3.4 (the Dirac field and its algebra) · Schwartz, Quantum Field Theory and the Standard Model, Ch. 10–11 (spinors and the Dirac equation).

Provenance. The negative-probability failure of the scalar theory and the relation $E^2=p^2+m^2$ are recalled by name from aq-pp-klein-gordon-equation; the plane-wave spinors, spin content, antiparticle interpretation, and covariant bilinears are developed in the articles this one leads to. A verifier should confirm: (1) squaring $\hat H=\boldsymbol\alpha\cdot\hat{\mathbf p}+\beta m$ and matching to $p^2+m^2$ gives $\{\alpha_i,\alpha_j\}=2\delta_{ij}$ , $\{\alpha_i,\beta\}=0$ , $\beta^2=\mathbb 1$ ; (2) Hermiticity, tracelessness, and the count of anticommuting matrices force dimension four; (3) the Dirac–Pauli matrices satisfy the algebra and are traceless; (4) the non-relativistic limit gives $(\boldsymbol\sigma\cdot\boldsymbol\pi)^2=\boldsymbol\pi^2-e\boldsymbol\sigma\cdot\mathbf B$ and hence $g=2$ ; (5) numeric anchors $\mu_B=9.274\times10^{-24}$ J/T, $g_{\rm meas}=2.00231930436$ , $a_e=(g-2)/2\approx0.00116=\alpha/2\pi$ .

Prerequisites. The Klein–Gordon Equation. Leads to. Solutions of the Dirac Equation · Spin from the Dirac Equation · Antiparticles · Covariant Form and Bilinear Covariants.

Ask the tutor: Ask me to square the Dirac operator and watch the anticommutation relations appear, or to walk through the non-relativistic limit that hands you spin and g = 2.

Why did Dirac insist on an equation first order in time?

Because being second order in time is what wrecked the Klein–Gordon equation as a single-particle theory. A second-order equation needs both $\psi$ and $\dot\psi$ as initial data, and its conserved density $\rho\propto E$ is proportional to energy — so for negative-energy solutions it goes negative, and a negative probability is meaningless. A first-order equation, like Schrödinger's, has $\psi$ now determine its whole future, and its density $\rho=\psi^\dagger\psi$ is a sum of squares, never negative. Dirac wanted Born's statistical interpretation intact, and first order in time is what makes that possible. The price — matrices and multiple components — turned out to be a bargain.

Are the α and β matrices unique?

No, and this matters. Only the algebra they satisfy — $\alpha_i^2=\beta^2=\mathbb 1$ , $\{\alpha_i,\alpha_j\}=2\delta_{ij}$ , $\{\alpha_i,\beta\}=0$ — is fixed by the physics. Any unitary change of basis $\alpha_i\to U\alpha_iU^{-1}$ , $\beta\to U\beta U^{-1}$ gives a different-looking set obeying the same relations and identical predictions for every observable. The Dirac–Pauli representation is convenient for the non-relativistic limit because $\beta$ is diagonal; the Weyl (chiral) representation diagonalises $\gamma^5$ and suits massless problems; the Majorana representation makes the matrices purely imaginary. Choosing a representation is like choosing coordinates — convenience, never physics.

How does spin appear if it was never put into the equation?

It falls out of the four-component structure the moment you couple the equation to a magnetic field. Reducing the Dirac equation to its non-relativistic limit yields the Pauli equation — the Schrödinger equation for a two-component spinor — with a term $-\tfrac{e}{2m}\boldsymbol\sigma\cdot\mathbf B$ that couples the electron's intrinsic angular momentum $\mathbf S=\tfrac12\boldsymbol\sigma$ to the field. Identifying it as $-\boldsymbol\mu\cdot\mathbf B$ gives $\boldsymbol\mu=g\,(e/2m)\mathbf S$ with $g=2$ . Pauli had inserted spin and its two-valued wavefunction by hand; Dirac's equation derives it, with the correct gyromagnetic ratio, from relativistic invariance alone. That spin is a consequence of relativity, not an add-on, is one of the deepest lessons of the equation.

Is a Dirac spinor the same as a four-vector, since both have four components?

No — this is the most common trap. A four-vector like $(ct,\mathbf x)$ transforms under a Lorentz transformation $\Lambda$ by the vector rule $x'^\mu=\Lambda^\mu{}_\nu x^\nu$ . A Dirac spinor transforms by a different matrix, $\psi'(x')=S(\Lambda)\psi(x)$ , where $S(\Lambda)$ is built from the gamma matrices and is not equal to $\Lambda$ . The clearest sign: under a full $2\pi$ rotation a four-vector returns to itself while a spinor changes sign, returning only after $4\pi$ — the defining property of spin-½ objects. So the four components of $\psi$ form a bi-spinor, a genuinely new representation of the Lorentz group, not four scalars and not a four-vector.

What is the difference between the α, β matrices and the γ matrices?

They are two ways of writing the same content. The $\boldsymbol\alpha,\beta$ set appears in the Hamiltonian form $i\partial_t\psi=(\boldsymbol\alpha\cdot\hat{\mathbf p}+\beta m)\psi$ , which singles out time and is handy for the positive density $\psi^\dagger\psi$ and the non-relativistic limit. The $\gamma^\mu$ set appears in the covariant form $(i\gamma^\mu\partial_\mu-m)\psi=0$ , which treats space and time alike and makes Lorentz invariance manifest. They are related by $\gamma^0=\beta$ , $\gamma^i=\beta\alpha_i$ , and the anticommutators $\{\alpha_i,\alpha_j\}=2\delta_{ij}$ , $\{\alpha_i,\beta\}=0$ , $\beta^2=\mathbb 1$ are equivalent to the single Clifford relation $\{\gamma^\mu,\gamma^\nu\}=2g^{\mu\nu}$ . Use $\alpha,\beta$ for the energy picture, $\gamma^\mu$ for covariant calculations.

Does the Dirac equation still have negative-energy solutions like Klein–Gordon?

Yes — linearizing fixed the probability-density problem but not the energy spectrum. Squaring the Dirac operator returns each component to the Klein–Gordon form, so the plane-wave energies are still $E=\pm\sqrt{p^2+m^2}$ , both signs present, with two spin states at each sign — four solutions per momentum. What changed is the interpretation: the density $\psi^\dagger\psi$ is now positive for all four, so there is no negative-probability disaster, and the negative-energy solutions read physically as antiparticles — Dirac's holes, later the Feynman–Stückelberg antiparticle, finally the creation operator of quantum field theory. The negative energies are not a defect to remove; they are the equation telling you antimatter must exist.