Dirac equation ($(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$)

The Dirac equation is the relativistic equation of motion for a spin-one-half Dirac field, $(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi =0$, in units $\hslash =c=1$. The gamma matrices obey the Clifford algebra, ensuring that each component also obeys the free Klein-Gordon equation.

Being first order in time allows an initial spinor to determine its subsequent free evolution. The components encode spin as well as particle and antiparticle solutions; quantization interprets both kinds of excitation with positive physical energy. Coupling through the gauge covariant derivative describes an electron interacting with an electromagnetic field. The nonrelativistic limit contains the coupling of spin to a magnetic field, explaining why a scalar relativistic wave equation alone cannot describe an electron.

Matrices that factor the relativistic energy-momentum relation. In Hamiltonian form, ${\alpha }_i^2={\beta }^2=I$, $\{{\alpha }_i,{\alpha }_j\}=2{\delta }_{ij}I$ and $\{{\alpha }_i,\beta \}=0$. A physical Hamiltonian uses Hermitian matrices; in three spatial dimensions the smallest complex representation has dimension four.

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.

The two-component model $H=c(p_x{\sigma }_x+p_y{\sigma }_y)+V{\sigma }_z$ has energies $E_s=s\sqrt{c^2{p}^2+V^2}$. The sign of $V$ changes eigenstates and spin polarization but not the uniform energy dispersion at fixed $|V|$. The Pauli matrices may represent actual spin or an effective pseudospin, depending on the physical realization.

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.

With $\{{\gamma }^{\mu },{\gamma }^{\nu }\}=2g^{\mu \nu }$ and spin curvature $[{\nabla }_{\mu },{\nabla }_{\nu }]=R_{\mu \nu ab}{\gamma }^{ab}/4$, the Dirac equation operator squares to the rough Laplacian minus $R/4$. In the curvature contraction the four-gamma term vanishes by the first Bianchi identity, the two-gamma terms vanish by Ricci symmetry, and the scalar term is $-2R$. In Riemannian signature $i/\nabla$ is the self-adjoint convention and its square is $-{\nabla }^2+R/4$.

The four-gamma product decomposes into fully antisymmetric, two-gamma and scalar parts. The first part vanishes on contraction by the first Bianchi identity, and the two-gamma parts vanish because they contract symmetric Ricci curvature against antisymmetric gamma matrices. The scalar contractions are $-g^{ac}{g}^{bd}{R}_{abcd}+g^{ad}{g}^{bc}{R}_{abcd}=-2R$. Here $R=g^{ac}{g}^{bd}{R}_{abcd}$ fixes the curvature-slot convention.

// Target: second-fundamental-form.bigb

For a metric-compatible connection with torsion tensor $T^{\rho }{}_{\mu \nu }={\Gamma }_{\mu \nu }^{\rho }-{\Gamma }_{\nu \mu }^{\rho }$, expand the square using Clifford multiplication and the full affine second derivative. Its antisymmetric part is spin curvature minus $T^{\rho }{}_{\mu \nu }{\nabla }_{\rho }$. This gives the displayed formula. The curvature need not have the torsion-free pair symmetry or first Bianchi identity, so its Clifford contraction need not reduce to a scalar $-R/4$. General torsion produces additional first-order and nonscalar curvature couplings.

On a compact boundaryless Riemannian spin manifold with $R\ge 0$, integrating the Lichnerowicz spinor-square formula against a harmonic spinor gives $\int (|\nabla \psi {|}^2+R|\psi {|}^2/4)=0$. Positivity forces the spinor to be parallel. Its norm is constant on each connected component, so a nontrivial solution is nowhere zero there. Strictly positive scalar curvature at any point on such a component excludes nonzero harmonic spinors.

A parallel spinor has zero spin curvature. Contracting the spinorial Ricci identity with a gamma matrix gives $R_{\mu \nu }{\gamma }^{\nu }\psi =0$. For each fixed $\mu$, squaring Clifford multiplication by that Ricci row gives $|R_{\mu \cdot }{|}^2\psi =0$. Riemannian positivity and a nonzero spinor force every row to vanish. This implication is local and does not itself require compactness; the compact hypothesis is used to obtain parallelism from harmonicity.

A spinor Lorentz matrix satisfies $S^{-1}{\gamma }^{\mu }S={\Lambda }^{\mu }{}_{\nu }{\gamma }^{\nu }$. Combining this identity with the transformed derivative shows that $(i{\gamma }^{\mu }{\partial }_{\mu }^{\prime }-m){\Psi }^{\prime }=S(i{\gamma }^{\mu }{\partial }_{\mu }-m)\Psi$. Thus a Lorentz transformation carries every solution of the Dirac equation to another.

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.

The massless Dirac equation is ${\gamma }^m{\partial }_m\Psi =0$. The Clifford algebra makes its square the massless wave equation. Quantization of a worldline spin variable as ${\hat{\psi }}^m={\gamma }^m/\sqrt{2}$ turns its fermionic constraint into this equation.

One consistent Dirac equation convention uses $\eta =\mathrm{diag}(-1,1,1,1)$, $\{{\gamma }^a,{\gamma }^b\}=-2{\eta }^{ab}$, and $(i{\gamma }^a{\partial }_a+m)\psi =0$. Taking the negatives of the standard mostly-minus gamma matrices, and defining the Dirac adjoint with the new ${\gamma }^0$, gives the same physical Dirac action. The positive-frequency plane wave is $e^{ip\cdot x}u(p)$ with $p^0>0$, and its equation is $(/p-m)u=0$. The mass shell is $p^2=-m^2$. These signs must be translated together when using a formula stated in another convention.