Bogoliubov coefficients between inertial frames The Klein–Gordon Equation in Minkowski space says $\partial_\mu \partial^\mu \phi+ m^2 \phi = 0$.
The solution has modes $e^{ikx}$, $e^{-ikx}$ scaled by creation and annihilation operators $a^{\dagger}(k)$, $a(k)$.
When using an arbitrary diffeomorphism $(x,t) \rightarrow (x’,t’)$, we will get some sort of “modified” Klein Gordon equation, with new creation and annihilation operators $b^{\dagger}(k’)$, $b(k’)$ which are related to the Minkowski operators via the Bogoliubov transformation
$$a_{i} = \sum_{j}(\alpha_{ji}b_{j} + \beta^{*} _{ji}b^{\dagger}_j).$$
From this one can derive that
$$\langle{0_A}|N_{b}|0_A\rangle = \sum_{j}|\beta_{ij}|^2.$$
My question is: for the case where the diffeomorphism is just a Lorentz transformation $x^{\mu}$ = $\Lambda^{\mu}{}_{\nu}x^{\nu}$, how do I show that $\beta_{ij} = 0$? I’ve seen some heuristic arguments but I’m looking for how to do the mathematical calculation.
 A: I'll sketch a proof so that you can fill in the gaps later. The language I'll use might be similar to the one found in Wald's QFTCS textbook, but also on Chap. 14 of his GR book and on his 1975 paper.
The situation we are interested in is relating the creation and annihilation operators of two different Fock spaces $\mathcal{F}$ and $\mathcal{F}'$. As you said, these ladder operators are defined on each of these Fock spaces according to the solutions to the Klein–Gordon equation. Namely, they are associated with positive or negative–energy solutions. This is because the $k^0$ components of $k^\mu$ are taken to be positive in the exponentials $e^{i kx}$ and $e^{-i kx}$ that you wrote.
Formally, the way this split happens has to do with how one defines the one-particle Hilbert space of the Fock space. Skipping through some details, one starts with the solutions to the KG equation, complexifies this vector space, equips it with the KG inner product (which is not positive-definite yet), and then selects the subspace of positive-energy solutions. The KG inner product is positive definite in this subspace and hence this is a perfectly good Hilbert space $\mathcal{H}$. The Hilbert space for the quantum field theory is the Fock space constructed out of symmetric tensor products of $\mathcal{H}$. Notice that the same procedure happens with $\mathcal{F}'$ and some one-particle Hilbert space $\mathcal{H}'$.
Notice that the creation and annihilation operators are then intimately connected with whether the two observers considered have the same "notion of positive energy". In other words, if a given diffeomorphism preserves the sign of the time component of all timeline four-vectors (all possible momenta), then the Bogoliubov transformation will have $\beta_{ij} = 0$. This is not a heuristic argument but rather follows directly from the very definition of these creation and annihilation operators, regardless of whether you define them as Fourier coefficients in a plane wave expansion for the field or as the ladder operators on a Fock space. In both cases, the ladder operators are defined from the notion of positive-energy solutions. Hence, if the diffeomorphism preserves the notion of positive energy (by preserving the time component of timelike four-vectors), then $\beta_{ij} = 0$.
This reduces the problem to an exercise in Special Relativity: prove that any ortochronous Lorentz transformation (i.e., any Lorentz transformation with $\Lambda^0{}_0 > 0$) preserves the sign of the time component of timelike four-vectors. In other words, $\mathrm{sign}\ \Lambda^0{}_\nu p^\nu = \mathrm{sign}\ p^0$ for any timelike four-vector $p^\mu$.
Notice that I restricted your statement: there are Lorentz transformations with $\beta_{ij} \neq 0$. However, those involve a time reversal transformation.
Wald's discussions on relating the in and out Fock spacetimes for asymptotically stationary spacetimes give a more detailed proof that you only get particle creation when positive-energy modes in the in-space can gain a negative-energy component on the out-space classically. I suggest his references in case you want some more detail than I provided here.
Of course, an alternative approach is to simply write the Bogoliubov coefficients in terms of KG inner products of the modes the creation and annihilation coefficients represent. This is a bit more straightforward, but I think it might be more valuable to notice that the key point is in the notion of positive energy.
