For the <plane wave> $A^\mu(x)=\epsilon^\mu(k)e^{-ik\cdot x}$, the equations from part a become
$$
k^2=m^2,
\qquad
k_\mu\epsilon^\mu=0.
$$
In the rest frame $k^\mu=(m,\mathbf0)$, transversality sets $\epsilon^0=0$ while the three spatial components remain independent. Hence a <Proca field> has three polarization states, as required for a massive spin-one particle. A covariant orthonormal choice obeys
$$
\sum_{\lambda=1}^3\epsilon_\mu^{(\lambda)}(k)epsilon_\nu^{(\lambda)}(k)^*
=-\eta_{\mu\nu}+\frac{k_\mu k_\nu}{m^2}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 301 / 1 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-301/1/c/solution