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:

[tex]\epsilon_\mu p^\mu = 0[/tex]
[tex]\epsilon^2=-1[/tex]

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

 

[sir-pinski]

Yes, the number of polarization states for a massive spin=1 particle does depend on the number of spacetime dimensions. In general, a massive particle with spin S in N-dimensional spacetime will have 2S+1 polarization states. This can be understood by considering the representation of the particle's spin in terms of the Lorentz group.

In a 3-dimensional space, a massive spin=1 particle can be described by a vector representation, which has three components. Each component corresponds to a different polarization state. However, in 4-dimensional spacetime, the Lorentz group has two additional generators, resulting in two additional polarization states. This is why a massive spin=1 particle in 4-dimensional spacetime has three polarization states.

In general, the number of polarization states for a massive particle will increase as the number of spacetime dimensions increases. This is because the Lorentz group has more generators in higher dimensions, allowing for more possible spin representations.

It is important to note that this is a purely theoretical concept and does not have any physical significance in terms of the properties or behavior of the particle. The number of polarization states simply reflects the mathematical description of the particle's spin in a particular spacetime.

In conclusion, the number of polarization states for a massive particle does depend on the number of spacetime dimensions, with the formula being 2S+1, where S is the spin of the particle.