W02L05_Dirac Algibra

2 the set of matrices 1 x y z are linearly independent

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Unformatted text preview: = 4 (a ⋅ b )(c ⋅ d ) − 4 (a ⋅ c)(b ⋅ d ) + 4 (a ⋅ d )(b ⋅ c) − Tr (abcd ) ∴ 2Tr (abcd ) = 4 (a ⋅ b )(c ⋅ d ) − 4 (a ⋅ c)(b ⋅ d ) + 4 (a ⋅ d )(b ⋅ c) Tr (abcd ) = 2(a ⋅ b )(c ⋅ d ) − 2(a ⋅ c)(b ⋅ d ) + 2(a ⋅ d )(b ⋅ c) d) Show that for an arbitrary 2 × 2 matrix M, M = 1 (Tr ( M )12 + σ ⋅ Tr ( M σ )) . 2 The set of matrices {1, σ x , σ y , σ z } are linearly independent and since the space of such matrices is 4 dimensional, any matrix can be expressed as a linear combination of these matrices. Thus, M = m0 12 + m...
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