Past exam of the theoretical physics major option course of the University of Oxford / 2022 / 4 / b / Solution

Treat $B_{\mu }$ and $B_{\mu }^{\ast }$ as independent variables in the Euler-Lagrange equation. Varying $B_{\nu }^{\ast }$ gives the Proca equationThe divergence of its first term vanishes by antisymmetry and commuting partial derivatives. Since $M\ne 0$, it follows that ${\partial }_{\nu }{B}^{\nu }=0$. Substitution into the original equation givesThe complex-conjugate variation gives ${\partial }_{\mu }{G}^{\ast \mu \nu }+M^2{B}^{\ast \nu }=0$, hence $(\square +M^2)B^{\ast \nu }=0$ and ${\partial }_{\nu }{B}^{\ast \nu }=0$.

For an on-shell plane wave $B^{\mu }={\epsilon }^{\mu }{e}^{-ipx}$, the Lorenz constraint in Proca theory is $p_{\mu }{\epsilon }^{\mu }=0$. In the rest frame it sets ${\epsilon }^0=0$, leaving three complex particle polarizations. Thus the complex Proca field has three independent complex dynamical components, or six real ones; its quantized particle and antiparticle each have three physical particle polarizations. The conjugate field is not another unrelated classical field.