Massive Particles with Spin 1. Massless Spin 1 Particle: Photon Wave Functions. Particles with Higher Spins (3/2, 2, …)

A nonrelativistic particle with spin 1 can be described by a three-component wave function, V(r, t) or, in momentum space, V(p, t). Under Lorentz transformations a three-vector will develop a fourth component; therefore, to describe a relativistic particle with spin 1 (and mass m ≠ 0) we will need a four-vector, V _μ (x). This wave function has one component too many, so we will have to subject it to a supplementary condition. As we shall see in a moment, the one leading to correct interpretation is that of (four-) transversality, ∂ · V(x) = 0. V(x) will also have to verify the Klein—Gordon equation, so that we have, in natural units ћ = c = 1, 5.1.1$$\left( {\partial \cdot \partial + m^2 } \right)V_\mu(x) = 0,\,\,\partial \cdot V_\mu = 0.$$