# Bargmann–Wigner equations in NP formalism

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?