Tetrads, Lorentz connections and spin interfaces — model 1

Let $J$ be the stated generator, so $\mathrm{\Lambda }=\exp (\theta J)$. Its spatial block is an ordinary rotation, hence ${\mathrm{\Lambda }}^{\mathsf{T}}\eta \mathrm{\Lambda }=\eta$, $\det \mathrm{\Lambda }=1$, and ${\mathrm{\Lambda }}^0{}_0=1$; thus $\mathrm{\Lambda }(x)\in S{O}^{+}(1,3)$. Since $d\mathrm{\Lambda }{\mathrm{\Lambda }}^{-1}=Jd\theta$, [F1] with initial $\omega =0$ gives ${\omega }^{\mathrm{\prime }}=-Jd\theta$. Because $d\theta =d{x}^1\mathrm{\ne }0$ and $J\mathrm{\ne }0$, this connection matrix is nonzero. The metric is unchanged because ${\mathrm{\Lambda }}^{\mathsf{T}}\eta \mathrm{\Lambda }=\eta$.

Initially $T=0$ and $F=0$. By [F2], $T^{\mathrm{\prime }}=\mathrm{\Lambda }T=0$ and $F^{\mathrm{\prime }}=\mathrm{\Lambda }F{\mathrm{\Lambda }}^{-1}=0$. The connection is $\eta$-skew since $J^{\mathsf{T}}\eta +\eta J=0$; the torsion calculation and [F2] therefore identify it as the Levi-Civita connection in the rotated tetrad.

The matrix $A=\mathrm{diag}(e^{-i\theta \mathrm{/}2},e^{i\theta \mathrm{/}2})$ has determinant one. Conjugating ${\sigma }_1,{\sigma }_2$ by $A$ gives $A{\sigma }_1{A}^{†}=\cos \theta {\sigma }_1+\sin \theta {\sigma }_2$ and $A{\sigma }_2{A}^{†}=-\sin \theta {\sigma }_1+\cos \theta {\sigma }_2$, while $I$ and ${\sigma }_3$ are fixed. Thus [F3] sends $A(x)$ to $\mathrm{\Lambda }(x)$, so $A$ is a local lift of this Lorentz-valued map; this asserts no global spin structure. In any chosen local ${\mathbb{C}}^n$ trivialization with constant gamma matrices satisfying the Clifford relation, [F5] gives the local matrix identity for $D=d+\frac{1}{4}{\omega }_{IJ}^{\mathrm{\prime }}{\gamma }^I{\gamma }^J$, without asserting group-representation compatibility.