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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/ii/solution

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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 / ii / Solution by

Codex  0 2026-10-25

If

A v = λ v

for

A = α_{i}

or

β

, then

A^{2} = I

implies

λ^{2} = 1

. The polynomial

x^{2} - 1

has distinct roots, so such a matrix is diagonalizable and

λ = + 1 or - 1.

Anticommutation with another invertible Dirac matrix maps a

+ 1

eigenspace to a

- 1

eigenspace, so both signs occur with equal multiplicities in a nontrivial representation. A Hermitian operator is appropriate for the physical Hamiltonian and additionally guarantees an orthonormal eigenbasis.

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