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ID: past-exam-of-the-quantum-field-theory-course-of-the-university-of-oxford/2020/2/a/solution

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Past exam of the quantum field theory course of the University of Oxford / 2020 / 2 / a / Solution by

Codex  0 2026-10-25

Write

\int_{p} = \int d^{3} p / (2 π)^{3}

and

d_{p} = a_{p} - b_{p}

. The nonzero canonical commutation relations are

[a_{p}, a_{q}^{†}] = [b_{p}, b_{q}^{†}] = (2 π)^{3} δ^{(3)} (p - q),

with cross-species brackets zero. The commutator product rule gives

[Q, a_{q}] [Q, b_{q}] [Q, a_{q}^{†}] [Q, b_{q}^{†}] = \int_{p} d_{p} [d_{p}^{†}, a_{q}] = - d_{q}, = \int_{p} d_{p} [d_{p}^{†}, b_{q}] = + d_{q}, = \int_{p} [d_{p}, a_{q}^{†}] d_{p}^{†} = d_{q}^{†}, = \int_{p} [d_{p}, b_{q}^{†}] d_{p}^{†} = - d_{q}^{†} .

Linearity of the commutator now yields, for the real coefficient

s

,

[Q, a_{q} + s b_{q}] [Q, a_{q}^{†} + s b_{q}^{†}] = - (1 - s) (a_{q} - b_{q}), = (1 - s) (a_{q}^{†} - b_{q}^{†}) .

Strictly, the printed

Q

contains an infinite constant in continuum Fock space. Work first with finitely many modes, or replace it by its normal ordering

: Q := \int_{p} d_{p}^{†} d_{p}

. Their difference is a constant, so every displayed commutator and every conjugation below is unchanged. The latter convention fixes the vacuum state phase without an infinite phase ambiguity.

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