W02L05_Dirac Algibra

Gamma matrix traces a using the identities tr a b

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Unformatted text preview: xσ x + myσ y + mzσ z . Expanding the claimed identity given: 1 Physics 637 2013F Name___________________________ Pauli and Dirac Algebra 1 2 (Tr ( M )12 + σ ⋅ Tr ( M σ )) = 1 (Tr ( m0 12 + mxσ x + myσ y + mzσ z ) 2 +σ xTr (σ x ( m0 12 + mxσ x + myσ y + mzσ z )) +σ yTr (σ y ( m0 12 + mxσ x + myσ y + mzσ z )) +σ zTr (σ x ( m0 12 + mxσ x + myσ y + mzσ z ))) = m0 12 + mxσ x + myσ y + mzσ z =M Here we have used the results of part (a) 2. Gamma matrix traces: (a) Using the identities Tr ( a b ) = 4 a ⋅ b and a b + b a = 2 a ⋅ b to obtain an expansion for Tr ( a b c d ) in terms of dot products. Same method as 1(c): Tr ( a b c d ) = 4 a ⋅ b Tr ( c d ) − Tr ( b a c d ) = 8(a ⋅ b )(c ⋅ d ) − 8(a ⋅ c)Tr ( b d ) + Tr ( b c a d ) = 8(a ⋅ b )(c ⋅ d ) − 8(a ⋅ c)(b ⋅ d ) + 4 (a ⋅ d )Tr ( b c ) − Tr ( b c d a ) = 8(a ⋅ b )(c ⋅ d ) − 8(a ⋅ c)(b ⋅ d ) + 8(a ⋅ d )(b ⋅ c) − Tr ( a b c d ) ∴ 2Tr ( a b c...
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This document was uploaded on 11/12/2013.

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