Polarizations of vector/tensor fields

[michael879]

ok this is a really simple question but I've spent 30 minutes searching and can't find a definitive answer. I'm working with a tensor field right now (linearized graviton) and I'm trying to generalize all the vector field stuff to it. One thing I'm confused about is the polarization states.

I understand that vector fields have 4 polarizations, and (symmetric) tensor fields have 10.
Setting the gauge reduces this to 3 and 6
Setting the fields massless reduces this to 2 and 3

My question is in the field operator, which polarization states do you sum over? Is it 3 or 4 for vector fields, and is it 6 or 10 for tensor fields? I understand why its not 2 and 3 respectively, but my QFT book sums over 4 photon polarizations in the definition of A and I just can't understand why.

 

[Bill_K]

Setting the fields massless reduces this to 2 and 3

Actually, the number of polarizations is reduced to 2 and 2. Free massless fields always have two helicity states. For gravity after imposing the covariant gauge condition (four conditions) the remaining freedom is ◻ξ_μ = 0 (four more conditions).

which polarization states do you sum over?

Answer depends on which framework you're using, but in the usual approach the gauge condition is imposed only on the mass shell (external lines only) and internal lines are summed over all 4 (or 10) polarizations.