# What's the physical reason that a massive vector field has only three linearly-independent physical polarizations?

While a four-vector field $$A_\mu$$ has four components, for a massive field there are only three linearly independent combinations of these components that correspond to physical situations. This follows from Maxwell's equations which yield the condition $$$$\epsilon_\mu p^\mu = 0 \, .$$$$ What's the physical meaning of this condition? In other words, what's the physical reason that a massive vector field only possesses three polarization degrees of freedom?

In a previous related question of mine, @ACuriousMind mentioned that

The other degree of freedom (the one also having to be eliminated for a massive vector field with three degrees of freedom) is eliminated by a constraint setting the timelike Hamiltonian canonical momentum to zero, which has nothing to do with Maxwell's equations (it's simply the almost trivial statement that $$F^{00} =0$$).

However, I'm not really able to understand the physical content of this comments.

A generic four-vector field contains four degrees of freedom that upon quantization describe a spin-1 and a spin-0 particle. In the action for the Proca field, the spin-0 component is projected out by using an antisymmetric kinetic term. So the answer to your question really is: there are only three degrees of freedom in the Proca field since we want it to describe a spin-1 particle. Why spin one corresponds to three spin polarizations is ordinary quantum mechanics. Your equation $$\epsilon_\mu p^\mu=0$$ is just a Lorent-covariant form of the statement that in the rest frame, the polarization vector of a spin-1 particle is purely spatial.
• Analogously, it is commonly argued that we only get gauge symmetry because a photon has only two possible polarizations. If we want to describe a photon (2DOF) with a four-vector, we get redundancy. However, this argument seems backwards to me if we compare it to the situation for a marble on a circle. We can describe the location of the marble using just one variable $\phi$ because of rotational symmetry and a description in terms of $(x,y)$ is redundant. But origin of this redundancy is the physical rotational symmetry and not that the marble has just one DOF. – jak Oct 1 at 7:57