HW9 - determine 5 X s =1 ~ k,s * ~ k,s in terms of the...

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PHY5667 Problem Set #9 (due Tue Oct 26) (1) Let’s consider a massive spin-2 particle. Analogous to the polarization vectors ± μ ~ k,s for a spin-1 field, we have polarization tensors ± μν ~ k,s for a spin-2 field. (i) In the rest frame ( k = ( m, 0 , 0 , 0)), we should have ± 0 ν = ± μ 0 = 0 while ± ij 6 = 0 corresponding to the five polarizations of the spin-2 particle. Of course, a generic ± ij would have 9 independent components, so we need to impose some constraints on ± ij to reduce 9 to 5. Identify these constraints. [Hint: there are two constraints.] (ii) Still staying in the rest frame, find five polarization tensors ( ± ij s with s = 1 , ··· , 5) normalized as X i,j ± ij s ± ij s 0 = δ ss 0 . (iii) Still staying in the rest frame, determine an expression for 5 X s =1 ± ij ~ k,s ± k‘ * ~ k,s in terms of the metric δ ij . [Hint: Constrain the possible form of the expression as much as possible using the conditions identified in (i).] (iv) Now promote the expression found in (iii) to the form valid in all frames, that is,
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Unformatted text preview: determine 5 X s =1 ~ k,s * ~ k,s in terms of the metric , the 4-momentum k , and the mass m . (The point of this problem is to nd the spin-2 version of the spin-1 formula s ~ k,s * ~ k,s =- + k k /m 2 .) (2) Consider a real scalar with mass m and a real massless vector A . Suppose they interact via an interaction L int = 4 F F , where F = A - A . Calculate the decay rate for AA , with the all polarizations of A summed over. (If you dont see what the Feynman rules should be, do perturbation theory algebraically (that is, using contractions) to obtain the amplitude, then interpret the result diagrammatically.) 1...
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This note was uploaded on 12/14/2011 for the course PHY 5667 taught by Professor Okui during the Fall '10 term at FSU.

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