How to count polarization states of massive particles?

[arroy_0205]

I understand that a massless photon has two polarization states. But I do not understand why a massive spin=1 particle has three polarization states. Can anybody explain? Does the answer depend on the number of spacetime?

 

[blechman]

You need the following conditions:

$$\epsilon_\mu p^\mu = 0$$
$$\epsilon^2=-1$$

It is especially the first equation that matters. For a massless photon, the momentum can only be reduced to the form (E;E,0,0) - so there are only 2 polarization states (if you can't see it right away, work it out - it's very easy). However, if the photon is massive, you can go to its rest frame, where p = (m;0,0,0) - now there are three polarizations that are allowed.

Basically, the idea is to find the rotation group that leaves the momentum invariant, which is to say, only rotates the vanishing components around. This is called the "Little Group" (i swear I didn't make it up!). Knowing the Little Group tells you all you need to know about polarizations. Generally, massive particles have a little group SO(D-1), and massless particles have little group SO(D-2), where D is the number of spacetime dimensions (D=4 for us).

As you can plainly see, the answer is very sensitive to the number of dimensions you are in.

For a field theory explanation, check out my post on the thread:
https://www.physicsforums.com/showthread.php?t=192572