 Solution

In a globally static region, choose the time coordinate along the future timelike Killing vector field $K={\partial }_t$. A positive-frequency solution of the Klein-Gordon equation has $i{\mathcal{L}}_K{p}_i={\omega }_i{p}_i$, with ${\omega }_i>0$. Assume appropriate boundary conditions and a Cauchy surface so that the conserved Klein-Gordon inner product isFor a real scalar, take the negative-frequency modes to be $n_i=p_i^{\ast }$. Choose a complete normalized basis withThe negative norm of the conjugate modes is crucial; this is not a positive-definite inner product on all classical solutions.

Promote the real Klein-Gordon field to the operatorThese canonical commutation relations implement the canonical field and conjugate-momentum commutator. The static vacuum state in a stationary spacetime satisfies $a_i|0⟩=0$, and the Hamiltonian is a sum of oscillators, ${\sum }_i{\omega }_i(a_i^{†}{a}_i+1/2)$ before zero-point regularization. Continuous labels replace sums by correctly normalized integrals. For a complex charged scalar there are separate particle and antiparticle oscillators, with the same mode-mixing argument in each sector.

Choose independently normalized past and future bases, and use the matrix indices of the question. Complex conjugation of the future positive-frequency expansion immediately givesConservation of the Klein-Gordon inner product gives $A^{†}A-B^{†}B=I$ and $A^{†}{B}^{\ast }-B^{†}{A}^{\ast }=0$ in this column-index convention. These ensure preservation of the canonical commutation relations.

Let $a_j$ annihilate past modes and $b_i$ annihilate future modes. Extracting the future coefficient from the field gives the Bogoliubov transformationThe minus sign follows from $(n_j^{-},n_k^{-})=-{\delta }_{jk}$. In the in-vacuum, only $⟨0_{\mathrm{in}}|a_j{a}_k^{†}|0_{\mathrm{in}}⟩={\delta }_{jk}$ contributes to the expectation of the future number operator. ThereforeThis is particle number from Bogoliubov coefficients: the intervening time-dependent geometry makes a past positive-frequency mode contain a future negative-frequency component. In infinite volume, localized wave packets avoid delta-function normalization artifacts; a common Fock-space implementation needs the usual square-summability condition on the creation-mixing coefficients.

For gravitational collapse, the initial vacuum can be defined before the hole forms, while late outgoing modes are defined using the asymptotic stationary time at future null infinity. The complete future basis also includes modes falling through the event horizon; exterior outgoing modes alone are not a complete basis of all propagated data. Tracing an outgoing mode backwards near a nonextremal horizon gives the exponential ray relation $v=v_H-C{e}^{-\kappa u}$, where $u$ is late retarded time and $\kappa$ is the surface gravity. An outgoing factor $e^{-i\omega u}$ becomes proportional to $(v_H-v{)}^{i\omega /\kappa }$ in the early advanced coordinate, and contains both frequency signs when Fourier decomposed there. Their squared Bogoliubov-coefficient ratio is $e^{-2\pi \omega /\kappa }$; the canonical normalization then gives a Bose factor $(e^{2\pi \omega /\kappa }-1{)}^{-1}$, multiplied by the greybody factor, the transmission probability through the exterior potential. Thus Hawking radiation has temperature $T_H=\kappa /(2\pi )$ in units $\hslash =k_B=c=1$, with greybody factor modifications to the spectrum. This is a free-field calculation on a fixed or slowly evolving background; eventual backreaction is an additional problem, and the nonextremal exponential map must not be blindly used when $\kappa =0$.