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Varying the Dirac action with respect to the spinor, rather than its Dirac adjoint, gives the left-acting field equation after integration by parts. Together with the Dirac equation it proves conservation of the vector current.

Andre Geim coauthored measurements showing that electric charge carriers in graphene can behave as effectively massless quasiparticles described by the Dirac equation. The 2005 experiments found an unusual sequence of quantum Hall effect plateaus and an oscillation phase associated with Berry phase. Here the Dirac description concerns low-energy quasiparticles in a carbon lattice, rather than free electrons acquiring zero rest mass. The results connect a readily contacted material with concepts usually introduced in relativistic quantum mechanics.^[ref]

A physicist researching condensed matter physics, including graphene.^[ref]

For a charged spin one-half particle with the Dirac equation reference value $g=2$, the anomalous magnetic moment is conventionally expressed by the dimensionless quantity $a=(g-2)/2$. For the electron, write $a_e=(g_s-2)/2$. A stipulated $g_s=2.0024$ would therefore mean $a_e=0.0012$: a $0.12\%$ excess of the spin-component moment over one Bohr magneton. This is a teaching input, not a quoted measurement.

In the complex-conjugate convention for charge conjugation of a Dirac field, a unitary matrix $B$ obeys the displayed gamma matrix identity. For $M^{\rho \sigma }=\frac{1}{4}[{\gamma }^{\rho },{\gamma }^{\sigma }]$, two minus signs cancel to give $M^{\rho \sigma }B=B{M}^{\rho \sigma \ast }$. Real Lorentz transformation parameters therefore give $SB=B{S}^{\ast }$, so $B{\psi }^{\ast }$ has the same Spinor representation of the Lorentz group transformation law as $\psi$. Complex conjugating the free Dirac equation and multiplying by $B$ proves that the conjugate also solves it for real mass. This $B$ is the matrix multiplying ${\psi }^{\ast }$; the convention multiplying ${\bar{\psi }}^T$ instead uses $B({\gamma }^0{)}^T$.

For ${\mathcal{L}}_{\mathrm{int}}=-g\varphi \bar{\psi }{\gamma }^5\psi$, the variational Dirac equation and adjoint Dirac equation are $(i/\partial -m-g\varphi {\gamma }^5)\psi =0$ and $i({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }+m\bar{\psi }+g\varphi \bar{\psi }{\gamma }^5=0$. Substitute these into the derivative of $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$. Anticommuting the chirality matrix past ${\gamma }^{\mu }$ gives two identical mass terms and two identical interaction terms, proving the displayed identity. These are classical field equations; the formula does not assert an unrenormalized quantum composite-operator identity. With Hermitian coupling $g=i{g}_P$, the second term is $-2g_P\varphi \bar{\psi }\psi$.

For $\mathcal{L}=\bar{\psi }(i/\partial -S+iP{\gamma }^5)\psi$, with real scalar backgrounds $S,P$, the Dirac equation is $(i/\partial -S+iP{\gamma }^5)\psi =0$ and the adjoint Dirac equation is $i({\partial }_{\mu }\bar{\psi }){\gamma }^{\mu }+S\bar{\psi }-iP\bar{\psi }{\gamma }^5=0$. Use $\{{\gamma }^5,{\gamma }^{\mu }\}=0$ and $({\gamma }^5{)}^2=1$ in the derivative of $j_5^{\mu }=\bar{\psi }{\gamma }^{\mu }{\gamma }^5\psi$ to obtain the displayed classical identity. Taking $S=m+g\varphi$ and $P=G\Phi$ includes both scalar and Hermitian pseudoscalar Yukawa couplings.

The charge-conjugate Dirac field is ${\psi }^c=C{\bar{\psi }}^T$, where $C$ is the charge-conjugation matrix satisfying $C{\gamma }^{\mu T}{C}^{-1}=-{\gamma }^{\mu }$ in a common convention. Transposition acts on spinor indices, while the Dirac adjoint is $\bar{\psi }={\psi }^{†}{\gamma }^0$.

Conjugating a minimally coupled Dirac equation produces the equation with opposite electric charge. Thus charge conjugation exchanges particle and antiparticle excitations and reverses additive charges, while leaving the mass unchanged. For quantum electrodynamics, the gauge field must transform as well for the full coupled theory to be invariant. A relation between field expressions does not by itself prove that every interaction involving that field has charge conjugation symmetry.

The spin-one-half low-energy expansion of the Dirac equation includes this local correction in a static potential energy V. For $V=-Z{e}^2/(4\pi {ϵ}_{0}r)$, ${\nabla }^{2}V=Z{e}^2{\delta }^{(3)}(\mathbf{r})/{ϵ}_0$, so the correction is positive and probes the wavefunction at the origin. In point-Coulomb hydrogen atom states it contributes only to S waves at leading nonrelativistic order. It is part of fine structure, distinct from the radiative Lamb shift. ^[ref]

A Dirac field is a quantum field transforming as a Dirac spinor. Its free Lagrangian density is $\mathcal{L}=i\bar{\psi }{\gamma }^{\mu }{\partial }_{\mu }\psi -m\bar{\psi }\psi$, giving the Dirac equation. Unlike a single-particle wave function, a quantized Dirac field can create and remove particles and antiparticles.

Its mode expansion of a Dirac field contains fermionic annihilation operators for particles and fermionic creation operators for antiparticles, with canonical anticommutation relations enforcing Fermi-Dirac statistics. For the electron field, these excitations are electrons and Positrons. A global phase symmetry gives the conserved Dirac current; coupling that current to the electromagnetic field produces the matter interaction in quantum electrodynamics.

The Dirac Hamiltonian is the first-order time-evolution operator obtained from the Dirac equation. For a free massive particle,where the Dirac matrices satisfy $\{{\alpha }_i,{\alpha }_j\}=2{\delta }_{ij}I$, $\{{\alpha }_i,\beta \}=0$ and ${\beta }^2=I$. These identities give $H^2=c^2{P}^2+m^2{c}^4$, recovering the relativistic energy-momentum relation.

Minimal electromagnetic coupling replaces the canonical momentum by $\mathbf{P}-q\mathbf{A}$ and adds $q\varphi$, in SI units. The wavefunction has spinor components, and the free operator has positive- and negative-energy branches. The Dirac field interprets particle and antiparticle excitations with a stable vacuum; it is more complete than treating the negative branch as an ordinary one-particle spectrum with no lower bound.

For $g=\mathrm{diag}(1,-1,-1,-1)$ and the block-diagonal standard representation of the gamma matrices, write a positive-energy Dirac spinor as $(\xi ,\eta {)}^T$. The Dirac equation gives $\eta =(\mathbf{\sigma }\cdot \mathbf{p})\xi /(E+m)$. Choosing two orthonormal two-spinors ${\chi }_s$ yields the displayed positive-frequency solutions. Negative-frequency solutions are $v_s=\sqrt{E+m}((\mathbf{\sigma }\cdot \mathbf{p}){\eta }_s/(E+m),{\eta }_s{)}^T$ multiplying $e^{ip\cdot x}$. They obey $(/p+m)v_s=0$. Both have norm $2E$; their Dirac adjoints give ${\overline{u}}_r{u}_s=2m{\delta }_{rs}$ and ${\overline{v}}_r{v}_s=-2m{\delta }_{rs}$. Spatial rotations act on each two-spinor with generators ${\sigma }_j/2$, explaining spin one-half.

The historical interpretation fills the negative-energy electron states of the Dirac equation and uses the Pauli exclusion principle to prevent additional electrons from falling into them. Removing an electron of energy $-E$ and charge $-e$ gives a hole of excitation energy $E$ and charge $+e$, interpreted as a positron. A quantum field theory instead treats negative-frequency field coefficients as independent antiparticle creation operators in a Fock space, with positive excitation energies after normal ordering. No literal infinite material sea is needed. Bosonic antiparticles also exist, although a filled-sea exclusion argument cannot stabilize bosonic negative-energy levels.

For external electrons on shell, factor out signed electric charge in the parity-preserving interaction vertex:The $u$ are Dirac spinors; ${\gamma }^{\mu }$ are gamma matrices. With on-shell renormalization scheme normalization $F_1(0)=1$, the Gordon identity gives $g_s/2=F_1(0)+F_2(0)$ and $a_e=F_2(0)$.^[ref]

This explains a subtlety: the Dirac equation already predicts a spin magnetic dipole moment with $F_2=0$. Rewriting its Dirac electromagnetic current with the Gordon identity produces a spin-dependent term even before any loop order correction. $F_2$ measures the additional response after that baseline is included. The limit $q\to 0$ is taken after extracting its coefficient; setting $q=0$ in the interaction vertex at the outset would hide it. This electromagnetic anomalous magnetic moment is not a failure of gauge invariance of the kind called a quantum anomaly.

The electron magnetic moment couples an electron's angular momentum to a magnetic field. Its intrinsic spin contribution isHere ${\mu }_B$ is the Bohr magneton, and $g_s$ is the positive electron spin g-factor under this sign convention. The minus sign makes magnetic dipole moment antiparallel to spin angular momentum. An electron in a chemical atom can also contribute an orbital magnetic dipole moment; the measured response of a whole atomic level combines these contributions.

Measuring a tiny departure of $g_s$ from $2$ tests how well a theory predicts an electron's interaction with its surroundings. The Dirac equation gives $g_s=2$ before radiative corrections; the anomalous magnetic moment makes a small correction experimentally accessible.^[ref] A useful strategy is to compare two atomic energy levels whose orbital and spin contributions have different weights, rather than trying to calibrate an absolute magnetic field more accurately.

For two odd fermionic operators, time ordering contributes a minus sign whenever the chronological rearrangement exchanges them. For several fields the sign is the sign of a permutation restricted to the odd fields. This convention makes the free Dirac propagator a Green function of the Dirac equation with its contact term fixed by the canonical anticommutation relations.

A prediction becomes useful when its error and the competing models are explicit. Consider an original hypothetical readout $a_{\ast }=0.00116$ with stipulated absolute tolerance $2\times 10^{-5}$. Hold the teaching input $\alpha =1/137$ fixed. Compare three models: the Dirac equation baseline $a_e=0$, the leading QED value $a_e=\alpha /(2\pi )$, and a trial coefficient twice as large, $a_e=\alpha /\pi$. Their absolute residuals are respectively $0.00116$, approximately $1.7149\times 10^{-6}$, and approximately $0.00116343$. Only the leading QED value lies inside this stipulated tolerance. The example tests both the existence and the size of a correction; it does not quote an actual observation or assign the tolerance a statistical interpretation.

At tighter precision, including more loop orders and propagating the uncertainty in the fine-structure constant becomes essential. The first omitted term has order ${\alpha }^2$, but the order symbol alone gives no numerical error bound. If $\alpha$ changes by a small $\delta \alpha$, this leading prediction changes by $\delta a_e=\delta \alpha /(2\pi )$. Thus a comparison must state which input determined $\alpha$: extracting it from the very electron magnetic moment being tested would not supply an independent test of that input. Agreement with one leading term at one tolerance does not establish agreement to arbitrarily many digits.

The gamma matrix Clifford algebra and commuting partial derivatives give $(/\partial {)}^2=\square$. Thus every free Dirac equation solution also solves the Klein-Gordon equation componentwise. The converse is false without the first-order spinor constraint: the second-order equation alone allows twice as much independent initial data.

For the Dirac equation with two equal quark masses $m$, the equations for $q$ and $\bar{q}$ give the divergence of the nonsinglet axial current as $im\bar{q}{\gamma }_5{\tau }_{a}q=m{P}_a$. The color chiral anomaly is absent in this nonsinglet identity because the quark flavor generator is traceless. Equivalently, the mass term in the Hamiltonian is $m\int S{d}^{3}x$, and the nonsinglet axial charge algebra of quark densities gives ${\dot{Q}}_{5,a}=m\int P_a{d}^{3}x$.

Gamma matrices represent the spacetime Clifford algebra throughFor the Minkowski metric with signature $(+,-,-,-)$, ${\gamma }^0$ squares to $I$ and each spatial gamma matrix squares to $-I$. Distinct orthogonal generators anticommute. A four-dimensional complex representation is used for Dirac spinors in four-dimensional spacetime.

The particular matrix entries depend on the chosen basis; simultaneous similarity transformations preserve the defining relations. Their usefulness in the Dirac equation is that contracting them with derivatives produces a first-order operator whose square reduces to the second-order relativistic wave operator, apart from interactions or curvature contributions.

The derivative of the local phase in $\Psi \mapsto e^{-iq\alpha }\Psi$ cancels the added gradient in the electromagnetic potential. Thus $D_{\mu }^{\prime }{\Psi }^{\prime }=e^{-iq\alpha }{D}_{\mu }\Psi$, and the charged Dirac equation maps solutions to solutions. The same cancellation makes its action gauge invariant.