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Chapter4_17 - PHY4604 R D Field Special Unitary 22 Matrices...

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PHY4604 R. D. Field Department of Physics Chapter4_17.doc University of Florida Special Unitary 2×2 Matrices Consider the unitary matrix = * * ) , ( a b b a b a U with |a| 2 + |b| 2 = 1 . Such a matrix is said to be “unimodular” since det(U) = |a| 2 + |b| 2 = 1 . Two Component Spinors: This “special” ( i.e. unimodular) unitary 2×2 matrix operates on two component “spinors” = 2 1 ~ v v v as follows = 2 1 * * 2 1 ' ' v v a b b a v v . The transformation leaves the norm of states invariant since > < = + = + + = + = >= < v v v v v v v v v v b a v v v v v v v v | ) ( | | | | ) | | | )(| | | | (| | ' | | ' | ' ' ) ' ' ( ' | ' 2 1 * 2 * 1 2 2 2 1 2 2 2 1 2 2 2 2 2 1 2 1 * 2 * 1 Definition of a Group: A collection of objects is said to form a “group” if (1) There exists “group multiplication” and the product of any two group members produces another member of the group ( i.e. G 3 = G 1 G 2 “closed” under group multiplication). (2) There exists an “identity”, I , such that GI = IG = G . (3) There exists an “inverse”, G -1 , such that G -1 G = GG -1 = 1 . (4) The group multiplication is “associative”:
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