# Tensor polarization sums a spin 2 field can be

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5. * Tensor polarization sums. A spin 2 field can be embedded in a 2-index tensorhμν. Thereforethe polarizations are tensors too,ǫμνi. These should be orthonormal,ǫμνiǫμνj=δijwhere the sumis overμandνcontracted withηαβ.a) The polarizations should be transverse,kμǫμνi= 0and symmetricǫμνi=ǫνμi. How manydegrees of freedom do these conditions remove (ǫμνstarted with 16 dof)?b) For a massive spin 2 field, choose a frame in which the momentumkμis simple. Howmany orthonormalǫμνican you find? Write your basis out explicitly, as4×4matrices.c) Guess which of these correspond to spin 0, spin 1, or spin 2. What kind of Lorentz-invariant condition can you impose so that you just get the spin 2 polarizations?d) If you use the same conditions but takekμto be the momentum of a massless tensor,what are the polarizations? Do you get the right number?e) What would you embed a massive spin 3 field in? What conditions could you impose toget the right number of degrees of freedom?Page 2/2