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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/iii/paper-64/1/v/solution

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Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 64 / 1 / v / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-14

The spatial momentum operator acts only on the plane wave:

\frac{ℏ}{i} \nabla [a^{- 1} A_{k} e^{i k \cdot x - ik t}] = ℏ k [a^{- 1} A_{k} e^{i k \cdot x - ik t}] .

(1)

Thus the modes have fixed canonical/comoving wavevector momentum, not fixed physical momentum. In the usual positive-energy field-mode convention the local physical momentum and energy are

ℏ k / a

and

ℏ k / a

. This agrees with conserved spatial comoving momentum and cosmological redshift proportional to

a^{- 1}

The overall particle-action sign noted in (i) reverses the literal worldline canonical label relative to this usual mechanical momentum: it is

P^{c} = - a P

for a future-directed particle with that printed action. The conserved plane-wave label therefore has the opposite sign if it is identified with that literal worldline label. This sign convention does not change the wavenumber magnitude or either conservation/redshift conclusion.

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