 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-48/4/b/solution

Promote the canonical field and momentum

π = f^{'}

to operators satisfying the canonical commutation relation,

[\hat{f} (τ, x), \overset{π}{^} (τ, y)] = i δ^{(3)} (x - y)

. A real field has Fourier reality condition

\hat{f}_{k}^{†} = \hat{f}_{- k}

. Its oscillator expansion is

\hat{f}_{k} = u_{k} \overset{a}{^}_{k} + u_{k}^{*} \overset{a}{^}_{- k}^{†} .

The creation and annihilation operators add or remove excitations in the selected mode basis, with

[\overset{a}{^}_{k}, \overset{a}{^}_{k^{'}}^{†}] = δ^{(3)} (k - k^{'}), [\overset{a}{^}_{k}, \overset{a}{^}_{k^{'}}] = [\overset{a}{^}_{k}^{†}, \overset{a}{^}_{k^{'}}^{†}] = 0, \overset{a}{^}_{k} ∣0 ⟩ = 0.

Canonical normalization requires the Wronskian

u_{k} u_{k}^{*'} - u_{k}^{*} u_{k}^{'} = i

For the Bunch-Davies vacuum, choose the positive-frequency short-wavelength behavior

u_{k} \sim e^{- ik τ} / 2 k

k ∣ τ ∣ \to \infty

. For the proposed solution, direct differentiation gives

u_{k}^{'} = \frac{e ^{- ik τ}}{2 k} [- ik - \frac{1}{τ} + \frac{i}{k τ ^{2}}], u_{k}^{''} = \frac{e ^{- ik τ}}{2 k} [- k^{2} + \frac{ik}{τ} + \frac{2}{τ ^{2}} - \frac{2 i}{k τ ^{3}}] .

Substitution cancels every term in

u_{k}^{''} + (k^{2} - 2/ τ^{2}) u_{k}

. Its Wronskian is

i

, and its large-

∣ k τ ∣

limit is the required flat-spacetime positive-frequency mode. Thus

u_{k} = \frac{e ^{- ik τ}}{2 k} (1 - \frac{i}{k τ}) .

A term with the conjugate frequency is another allowed classical solution, but would describe a different quantum state, through a Bogoliubov transformation. Its coefficient is set to zero by the early-time vacuum condition, not by absence of a second solution to the differential equation.

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