Bargmann-Wigner equations describe free particles of arbitrary spin $j$, namely

$$(-\gamma^{\mu}\partial_{\mu}+m)_{\alpha_r \alpha_{r’}}\Psi_{\alpha_1,..,\alpha_{r’},...,\alpha_{2j}}=0$$ where we have set $c=\hbar=1$ and $m$ the mass of the particle.

In curved spacetime $(M,g_{\mu\nu})$, given a minkowskian vierbein $b_i^{\mu}$ such that $g^{\mu\nu}=\eta^{ij}b^{\mu}_{i}b^{\nu}_{j}$. $\gamma_{\mu}$ and $\partial_{\mu}$ in BW equations should be replaced by the gamma matrices in curved spacetime $\gamma’^{\mu}$ and covariant derivative $\mathcal{D}_{\mu}$ respectively, that is $$\gamma^{\mu} \rightarrow \gamma’^{\mu}=b_i^{\mu}\gamma^i$$ and $$\partial_{\mu} \rightarrow \mathcal{D}_{\mu} = \partial_{\mu}+\Omega_{\mu}$$ where $$\Omega_{\mu}=\frac{1}{4}\partial_{\mu}\omega^{ij}(\gamma_i \gamma_j-\gamma_j \gamma_i)$$ are connection 1-form assoicated with the spin connection $\omega$.

The BW equations in curved spacetime could be written as

$$(-i\gamma’^{\mu}\mathcal{D}_{\mu}+m)_{\alpha_r \alpha_{r’}}\Psi_{\alpha_1,..,\alpha_{r’},...,\alpha_{2j}}=0$$

So how could I reformulate it into Newman-Penrose formalism, to be specific, similar to rewrite EM tensor $F_{\mu\nu}$ into three complex scalars $\Phi_{0},\Phi_{1},\Phi_{2}$,or Dirac spinor $\psi$ into four scalar $F_1 ,F_2,G_1,G_2$, how should I rewrite the rank $2j$ 4 component spinor wave function $\Psi_{\alpha_1 \alpha_2 ,..., \alpha_{2j}}$ (with $4^{2j}$ componets entirely) in BW equations into scalars so as to make it adaptable to null tetrad formalism?



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