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W02L05_Dirac Algibra

W02L05_Dirac Algibra - Physics 637 2013F Name Pauli and...

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Physics 637 2013F Name___________________________ Pauli and Dirac Algebra 1 1. Pauli Trace Algebra: Here I will use the notation that for a 3-­‐vector v , The bold face indicates the combination with sigma matrices: v = v σ and 1 2 is the identity 2 × 2 matrix. a) Evaluate Tr ( a ) , Tr ( ab ) and Tr ( abc ) in terms of dot products, cross products and/or epsilon tensors. Tr ( a ) = 0 because the pauli matrices are traceless. Tr ( ab ) = Tr ( a b 1 2 + i ( a × b ) σ ) = 2 a b Tr ( abc ) = Tr ( a b c + i ( a × b ) σ c ) = Tr ( i ( a × b ) σ c ) = 2 i ( a × b ) c ( ) [using previous result] b) Show that ab + ba = 2 a b 1 2 ab + ba = a b 1 2 + i ( a × b ) σ ( ) + a b 1 2 + i ( b × a ) σ ( ) = 2 a b 1 2 c) Use this to obtain an expansion for Tr ( abcd ) in terms of dot products using the result of (b). (use the identity to commute a thru to the right side using (b) and then bring it back to the front).
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