Derivation. Starting with a three-component wave function $\phi_{i}$ describing a massive spin-1 free particle in its rest frame, two possible rest-frame covariant forms exist: a covariant four-vector $\phi_{\mu}=\left(0, \phi_{i}\right)$ and a rank-two antisymmetric (field) tensor $\phi_{\mu \nu}$ given by (recall the angular-momentum tensor operators $L_{\mu v}$ and $J_{\mu v}$ in Chapter 3) $\phi_{0 i}=-\phi_{i 0}=\partial_{t} \phi_{i}$ and $\phi_{00}=\phi_{i j}=0$. In a general frame, the boosted form of $\phi_{\mu v}$ can be obtained from $\phi_{m}$ as
$$\phi_{\mu v}=\partial_{\mu} \phi_{v}-\partial_{v} \phi_{\mu^{}}$$ The free-particle dynamical relation between $\phi_{\mu}$ and $\phi_{\mu v}$ is called the Proca equation: $$\partial^{v} \phi_{\mu v}=m^{2} \phi_{\mu^{}}$$
Owing to the antisymmetric structure of $\phi_{\mu \nu}$, the derivative of (4.38) implies the subsidiary condition
$$\partial^{\mu} \phi_{\mu}=0$$

## PH40084 COURSE NOTES ：

Furthermore, since $H_{0}$ must be an observable hermitian operator, so must $\alpha_{i}$ and $\beta$ be hermitian:
$$\alpha_{i}^{\dagger}=\alpha_{i}, \quad \beta^{\dagger}=\beta .$$
The adjoint row bispinor $\psi^{\dagger}$ can be combined with the column bispinor $\psi$ to form a positive definite probability density
$$\rho=\psi^{\dagger} \psi=\sum_{\sigma=1}^{4} \psi_{\sigma}^{*} \psi_{\sigma},$$
now naturally linked with the hermitian probability current density
$$\mathbf{j}=\psi^{\dagger} \boldsymbol{\alpha} \psi .$$