HW5 - x 1-direction as L = e i K 1 L with K 1 =-i 1 2 (and...

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PHY5667 Problem Set #5 (due Tue Sep 28) (1) Find a two-dimensional representation of S 3 , the permutation group of three objects. (Do not just write down the answer — derive or prove!) (2) Suppose ψ and χ are spinors of the rotation group, that is, they transform under rotations as ψ 0 = e i θ i J i ψ with J i = σ i 2 (and similarly for χ ). Show that the combinations v i = ψ σ i χ ( i = 1 , 2 , 3) transform as the components of a 3-vector under rotations. (3) Suppose ψ L and χ L are left-handed spinors, so in particular they transform under boosts in the
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Unformatted text preview: x 1-direction as L = e i K 1 L with K 1 =-i 1 2 (and similarly for L ). Show that 0 L a L = L a L for a = 2 , 3, so these have the right property to be the 2- and 3-components of a 4-vector. (4) Show that, if L and R are left- and right-handed spinors, respectively, then 2 * L and 2 * R transform as right- and left-handed spinors, respectively. 1...
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