Local spin-connection Clifford compatibility

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $(M,g,N)$ be a smooth time-oriented Lorentzian four-manifold, let $U\subseteq M$ be a nonempty open set, and let $E=(E_0,E_1,E_2,E_3)$ be a smooth future-directed tetrad on $U$. Let $\mathrm{\nabla }$ be a smooth affine Lorentz connection on $TM{\mathrm{\mid }}_U$ with smooth connection matrix ${\omega }^I{}_J$ in $E$, meaning ${\mathrm{\nabla }}_X{E}_J={\sum }_I{\omega }^I{}_J(X)E_I$ as in Local Lorentz tetrad and connection. Write ${\omega }_{IJ}:={\eta }_{IK}{\omega }^K{}_J=-{\omega }_{JI}$ for $\eta =\mathrm{diag}(-1,1,1,1)$, so ${\eta }^{IJ}$ denotes the entries of ${\eta }^{-1}=\eta$. Fix an integer $n\ge 1$ and constant matrices ${\gamma }^I\in M_n(\mathbb{C})$ satisfying

${\gamma }^I{\gamma }^J+{\gamma }^J{\gamma }^I=2{\eta }^{IJ}{I}_n\mathrm{.}$

Let $V\in {\mathrm{\Gamma }}^{\mathrm{\infty }}(TM{\mathrm{\mid }}_U)$ be a smooth vector field and let $\psi \in C^{\mathrm{\infty }}(U,{\mathbb{C}}^n)$ be smooth. With $V_I:={\eta }_{IJ}{V}^J$, define $c(V):=V_I{\gamma }^I$ and, on ${\mathbb{C}}^n$-valued fields and forms,

$D\psi :=d\psi +\mathrm{\Gamma }\psi ,\mathrm{\Gamma }:=\frac{1}{4}{\omega }_{IJ}{\gamma }^I{\gamma }^J,$

where $d$ acts componentwise. Then

$D(c(V)\psi )=c(\mathrm{\nabla }V)\psi +c(V)D\psi ,$

where $(\mathrm{\nabla }V{)}_I:=d{V}_I-{\omega }^J{}_I{V}_J$ are the covector components of the induced Lorentz connection and $c(\mathrm{\nabla }V):=(\mathrm{\nabla }V{)}_I{\gamma }^I$. This is precisely a local matrix identity under the stated hypotheses; it does not assert that the matrices ${\gamma }^I$ or the operator $D$ are compatible with, or induced by, a separately supplied $SL(2,\mathbb{C})$ representation or principal-connection lift. Such compatibility is an additional condition when this identity is used to describe an associated spinor bundle.

Local Lorentz tetrad and connection

Lorentzian spin-lift interface