Dirac equation - Knowlpedia

Definition

For a spinor field $\psi$ of mass $m\geq0$ on Minkowski spacetime, the free Dirac equation in the convention of this collection is

(iD_{\mathrm M}-m)\psi=0,

where $D_{\mathrm M}=\gamma^\mu\partial_\mu$ is the Minkowski Dirac operator and the Clifford matrices obey

\gamma^\mu\gamma^\nu+\gamma^\nu\gamma^\mu=-2\eta^{\mu\nu}I

for $\eta=\operatorname{diag}(-1,1,1,1)$ .

Mass shell

A plane-wave spinor $\psi(x)=u(p)e^{-ip_\mu x^\mu}$ satisfies an algebraic equation for $u(p)$ . Multiplication by the conjugate first-order factor implies

(\Box_\eta+m^2)\psi=0.

Thus each spinor component satisfies the Klein–Gordon equation and the momentum lies on the relativistic mass shell.

Massless equation

For $m=0$ , the equation is $D_{\mathrm M}\psi=0$ . In even spacetime dimension the complex spin representation has a chirality decomposition, and the massless equation can split into equations for chiral spinors. A nonzero mass term couples the two chiralities.

Curved and coupled forms

On a Lorentzian spinor bundle, the curved equation is

(iD_g-m)\psi=0, \qquad D_g=c\circ\nabla^S.

A gauge potential replaces the spin connection by a coupled connection. Squaring a curved or gauge-coupled equation produces curvature or field-strength terms, so only the flat free equation has the elementary componentwise factorization stated above.

Convention warning

Some sources use the mostly-minus metric and matrices satisfying $\{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu}I$ ; others absorb the factor $i$ into the geometric Dirac operator. Equivalent formulas can therefore look different. The metric, Clifford relation, and differential operator must be compared together.

References

Paul A. M. Dirac, “The Quantum Theory of the Electron,” Proceedings of the Royal Society A 117 (1928), 610–624. Journal record.
Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, 1995. Publisher record. Relevant: Chapters 3–4.
Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: §3.4.