Quantum Field Theory and the Standard Model by Schwartz M.D. z-lib.org · Problem 8.6

Spin 1 and gauge invariance — Schwartz M.D. z-lib.org

Physics — QFT · tensor-polarization-sums-spin-2

Problem

Tensor polarization sums. A spin-2 particle can be embedded in a 2-index tensor hμνh_{\mu\nu}. Therefore, its polarizations are tensors too, ϵμνi\epsilon^i_{\mu\nu}. These should be orthonormal, ϵμνiϵ⋆j μν=δij\epsilon^i_{\mu\nu}\epsilon^{\star j\,\mu\nu} = \delta^{ij}, where the sum is over μ\mu and ν\nu contracted with the Minkowski metric. (a) The polarizations should be transverse, kμϵμνi=0k_\mu \epsilon^i_{\mu\nu} = 0, and symmetric, ϵμνi=ϵνμi\epsilon^i_{\mu\nu} = \epsilon^i_{\nu\mu}. How many degrees of freedom do these conditions remove? (b) For a massive spin-2 particle, choose a frame in which the momentum kμk_\mu is simple. How many orthonormal ϵμνi\epsilon^i_{\mu\nu} can you find? Write your basis out explicitly, as 4×44 \times 4 matrices. (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 take kμk_\mu to 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 to get the right number of degrees of freedom?

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