Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 342 / 3 / b

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- Solution b

Solution

χ = O c .

Because

O

is real orthogonal, the

χ_{j}

are again self-adjoint and satisfy the Majorana anticommutation relations. Using

A = O^{T} εO

,

H = \frac{i}{4} χ^{T} ε χ .

Each two-dimensional block contributes twice the same ordered bilinear:

χ_{2 j - 1} ϵ_{j} χ_{2 j} + χ_{2 j} (- ϵ_{j}) χ_{2 j - 1} = 2 ϵ_{j} χ_{2 j - 1} χ_{2 j} .

Hence

H = \frac{1}{2} j = 1 \sum M ϵ_{j} i χ_{2 j - 1} χ_{2 j}

up to the original additive constant.

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