HW8 - PHY5667 Problem Set#8(due Tue Oct 19(1 The right-handed spinors uk,s and vk,s(s = 1 2 are given as follows in a frame where k =(E 0 0 p uk,1

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(1) The right-handed spinors u ~ k,s and v ~ k,s ( s = 1 , 2) are given as follows in a frame where k = ( E, 0 , 0 ,p ): u ~ k, 1 = ± E + p 0 ² , v ~ k, 1 = ± 0 E - p ² , u ~ k, 2 = ± 0 E - p ² , v ~ k, 2 = ± - E + p 0 ² . Prove the identity 2 X s =1 u ~ k,s u ~ k,s = 2 X s =1 v ~ k,s v ~ k,s = k · ¯ σ for ~ k in a general direction in two steps: (i) Show that the identity holds for k = ( E, 0 , 0 ,p ). (ii) Derive the identity for general k by performing an appropriate rotation that brings k = ( E, 0 , 0 ,p ) to k = ( E,p sin θ cos φ,p sin θ sin φ,p cos θ ). (2) The left-handed spinors u ~ k,s and v ~ k,s ( s = 1 , 2) are given as follows in a frame where k = ( E, 0 , 0 ,p ): u ~ k, 1 = ± E - p 0 ² , v ~ k, 1 = ± 0 E + p ² , u ~ k, 2 = ± 0 E + p ² , v ~ k, 2 = ± - E - p 0 ² . Prove the identities (i) 2 X s =1 u ~ k,s v T ~ k,s = m i σ 2 (ii) 2 X s =1 u ~ k,s u ~ k,s = 2 X s =1 v ~ k,s v ~ k,s = k · σ for ~ k in a general direction. 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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HW8 - PHY5667 Problem Set#8(due Tue Oct 19(1 The right-handed spinors uk,s and vk,s(s = 1 2 are given as follows in a frame where k =(E 0 0 p uk,1

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