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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 \begin{equation} \epsilon_\mu p^\mu = 0 \, . \end{equation} 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.

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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.

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  • $\begingroup$ Thanks for your answer, although I don't find it entirely satisfying. But this may be a chicken-egg problem. If we start with the known properties of spin 1 particles and then argue that we want a field description of them, we end up with the conclusion that one component of our four-vector field can be eliminated. I would love to know if it is also possible to start with our field description and use it to understand why there are only three physical DOF. $\endgroup$ – jak Oct 1 at 7:57
  • $\begingroup$ 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. $\endgroup$ – jak Oct 1 at 7:57
  • $\begingroup$ @jak As said, a generic four-vector field will contain four independent DOFs. In order to reduce the number of DOFs, you have to impose constraints, either explicitly (e.g. by demanding gauge invariance) or implicitly (e.g. by picking a specific action). For the massive vector (Proca) field, such a constraint is built in through the fact that the temporal component of the field is not dynamical. So there is a correspondence between the number of independent DOFs of the field and the number of spin states of the corresponding particle. It's a matter of taste what you prefer to start with. $\endgroup$ – Tomáš Brauner Oct 1 at 8:09
  • $\begingroup$ Yes exactly. I know that many people are perfectly happy with the answer that a physical spin 1 particle has 3 dof and therefore we get a constraint on the four-vector field. But I was hoping for some physical way to understand what it means that the temporal component of the field is not dynamical. PS: I didn't downvote your answer. $\endgroup$ – jak Oct 1 at 8:13
  • $\begingroup$ @jak So are you actually asking why a concrete theory of a vector field (say the free Proca field) only has three independent degrees of freedom, or why it is the temporal component of the field that is special? I believe that the former question is answered by my comments above. $\endgroup$ – Tomáš Brauner Oct 1 at 8:54

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