 Solution

ID: past-exam-of-the-quantum-condensed-matter-field-theory-course-of-the-university-of-cambridge/2014/paper-1/1/c/solution

For real even

θ_{k}

[α_{k}, α_{q}^{†}] = (cosh^{2} θ_{k} - sinh^{2} θ_{k}) δ_{k q} = δ_{k q}, [α_{k}, α_{q}] = 0.

Equivalently, its Nambu matrix obeys $T_{k} σ_{z} T_{k}^{†} = σ_{z}$ . The bosonic canonical metric is preserved;

T

need not be unitary in the ordinary positive-definite metric.

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