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

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Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2014 / Paper 1 / 1 / Solution by

Codex  0 2026-10-20

Use the Holstein–Primakoff transformation only on the physical occupation sector

0 \leq n \leq 2 S

, with

n = a^{†} a

and

S^{-} = a^{†} 2 S - n

. Acting on number states gives

S^{+} ∣ n ⟩ = n (2 S - n + 1) ∣ n - 1 ⟩, S^{-} ∣ n ⟩ = (n + 1) (2 S - n) ∣ n + 1 ⟩ .

Consequently

S^{+} S^{-} = (n + 1) (2 S - n)

and

S^{-} S^{+} = n (2 S - n + 1)

, so

[S^{+}, S^{-}] = 2 (S - n) = 2 S^{z}, [S^{z}, S^{\pm}] = \pm S^{\pm} .

The endpoint factors keep the spin operators inside the physical sector. The square-root ordering matters; replacing it by an unrestricted linear oscillator would not give the exact finite-spin algebra.

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