Chapter4_17

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

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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”
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