Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 54 / 4 / b / Solution

Quantize the real massless scalar field using the normalized in-modes:Use the Klein-Gordon inner product convention that is antilinear in its first argument. Mode normalization givesThe out annihilation operator is $b_i=(p_i,\hat{\varphi })$. Applying the Bogoliubov transformation and the negative norm of conjugate modes givesBoth the complex conjugation and the minus sign follow from the inner-product convention. These operators annihilate the out-vacuum, but need not annihilate the in-vacuum. The particle number operator in out-mode $i$ is $N_i^{\mathrm{out}}=b_i^{†}{b}_i$. In the in-vacuum, the only nonzero contraction is $⟨0_{\mathrm{in}}|a_j{a}_k^{†}|0_{\mathrm{in}}⟩={\delta }_{jk}$, obtained from the canonical commutation relation. ThusThis is particle number from Bogoliubov coefficients: mixing with negative-frequency in-modes produces out-particles even though no in-particles were present. For continuous mode labels the sum becomes an integral; normalized wave packets make an individual occupation number well-defined. No explicit collapse geometry or thermal spectrum is needed for this conclusion.