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HW8 - PHY5667 Problem Set#8(due Tue Oct 19(1 The...

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PHY5667 Problem Set #8 (due Tue Oct 19) (1) The right-handed spinors ~ k,s and ~ k,s ( s = 1 , 2) are given as follows in a frame where k = ( E, 0 , 0 , p ): ~ k, 1 = E + p 0 , ~ k, 1 = 0 E - p , ~ k, 2 = 0 E - p , ~ k, 2 = - E + p 0 . Prove the identity 2 X s =1 ~ k,s ~ k,s = 2 X s =1 ~ k,s ~ 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 ~ k,s and ~ k,s ( s = 1 , 2) are given as follows in a frame where k = ( E, 0 , 0 , p ): ~ k, 1 = E - p 0 , ~ k, 1 = 0 E + p , ~ k, 2 = 0 E + p , ~ k, 2 = - E - p 0 . Prove the identities (i) 2 X s =1 ~ k,s T ~ k,s = m i σ 2 (ii) 2 X s =1 ~ k,s ~ k,s = 2 X s =1 ~ k,s ~ k,s = k · σ for ~ k in a general direction. 1
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(3) The Dirac spinors ~ k,s and ~ k,s ( s = 1 , 2) are given as follows in a frame where k = ( E, 0 , 0 , p ): ~ k, 1 = E - p 0 E + p 0 , ~ k, 2 = 0 E + p 0 E - p , ~ k, 1 = E - p 0 - E + p 0 , ~ k, 2 =
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