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Unformatted text preview: dimensions act on fermions in nature; for example, on electrons. (i) These matrices are given by σ 1 = ± 0 1 1 0 ² , σ 2 = ±i i ² , σ 3 = ± 11 ² . Compute the commutator [ σ i ,σ j ] and the anticommutator { σ i ,σ j } . (ii) Add the identity matrix σ = 1 to these three matrices and show that any 2 × 2 matrix M can be written in the form M = 3 X i =0 α i σ i for some complex α i . Under what conditions on α i is M unitary? Under what conditions is M Hermitian? 1 (iii) Consider the operator P = p σ + X i p i σ i where the 4momentum p = ( p ,p 1 ,p 2 ,p 3 ) is real. What is det( P )? What are the eigenvalues and eigenvectors of P ? 2...
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 Fall '10
 Sghy
 Physics, Linear Algebra, mechanics, Matrices, Hilbert space, Unitary matrix, unitary similarity transformation, normal matrix satisﬁes

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