 Solution

ID: past-exam-of-the-short-options-paper-shared-by-prelims-and-parts-a-and-b-of-the-physics-course-of-the-university-of-oxford/2018/s18/3/a/iii/solution

For any one of the four anticommuting matrices

A

, choose another

B

. Since

B^{- 1} = B

and

B A B = - A

, cyclic invariance of the trace gives

Tr A = Tr (B^{- 1} A B) = Tr (- A) .

Therefore

Tr α_{i} = 0, Tr β = 0.

Equivalently, the paired positive and negative unit eigenvalues cancel. This argument does not depend on a particular matrix representation.

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