Dirac–Klein–Gordon factorization

Statement

Let $D_{\mathrm M}=\gamma^\mu\partial_\mu$ be the Minkowski Dirac operator with

\{\gamma^\mu,\gamma^\nu\}=-2\eta^{\mu\nu}I, \qquad \Box_\eta=-\eta^{\mu\nu}\partial_\mu\partial_\nu.

For every constant $m$ ,

(iD_{\mathrm M}-m)(iD_{\mathrm M}+m) =(iD_{\mathrm M}+m)(iD_{\mathrm M}-m) =-(\Box_\eta+m^2)I.

Consequently every solution of either massive free Dirac equation satisfies the Klein–Gordon equation componentwise.

Proof

Commutativity of the constant $m$ with $D_{\mathrm M}$ cancels the cross terms. The Clifford relation and commutativity of partial derivatives give

D_{\mathrm M}^{\,2} =\frac12\{\gamma^\mu,\gamma^\nu\}\partial_\mu\partial_\nu =-\eta^{\mu\nu}\partial_\mu\partial_\nu I =\Box_\eta I.

Therefore

(iD_{\mathrm M})^2-m^2=-D_{\mathrm M}^{\,2}-m^2 =-(\Box_\eta+m^2)I.

What the implication does not say

The converse is false without additional data: a tuple of Klein–Gordon solutions need not satisfy a first-order Dirac equation. The factorization selects spinor solutions obeying an extra linear constraint.

Curvature and gauge warning

On a curved spin manifold, the square of the Dirac operator contains a scalar-curvature term; coupling to a gauge field adds Clifford contraction of its curvature. Thus the free flat identity acquires lower-order corrections. The principal part remains a normally hyperbolic connection operator, but one cannot replace it by the scalar $\Box_g$ componentwise in an arbitrary frame.

References

H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin Geometry, Princeton University Press, 1989. Publisher record. Relevant: Chapter II, §8.
Christian Bär, Nicolas Ginoux, and Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, European Mathematical Society, 2007. Publisher record. Relevant: §§1.3 and 3.4.