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When doing QFT in curved spacetime we do Bogoliubov transformation to find one set of annihilation and creation operators in terms of others. In the paper Particle creation by black holes Hawking states following:

One could still require that the $\{f_i\}$ and $\{\bar{f_i}\}$ together formed a complete basis for solutions of the wave equations with $$\frac{i}{2}\int_S(f_i\bar{f}_{j;a}-f_{i;a}\bar{f}_j)d\Sigma^a=\delta_{ij}$$ where S is a suitable surface. However (above) condition does not uniquely fix the subspace of the space of all solutions which is spanned by the $\{f_i\}$ and therefore does not determine the splitting of the operator $\phi$ into annihilation and creation parts.

Can someone explain and point me to why the above condition doesn't uniquely decompose the annihilation and creation operator in curved spacetime but does it in flat Minkowski spacetime where we do our usual QFT calculation at early and late time. Since this mixing is basic of Unruh, Hawking radiation.

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I'm going to approach the issue from another point of view than Hawking's quote, though it is essentially the same thing.

To be able to discuss creation and annihilation operators, one must first be working in a Fock space $\mathcal{F}$. To construct such a Fock space, one needs to know what the one-particle Hilbert space, $\mathcal{H}$, is, for then $$\mathcal{F} = \mathbb{C} \oplus \mathcal{H} \oplus (\mathcal{H} \otimes_S \mathcal{H}) \oplus \cdots,$$ where $\otimes_S$ denotes a symmetrized tensor product (to comply with Bose–Einstein statistics). To build $\mathcal{H}$, you would need to know what are the positive-frequency solutions to the Klein--Gordon equation, so you can associate them with particles. However, "positive frequency" is a statement that must refer to some notion of time, and in general, one doesn't know what "time" means.

In Minkowski spacetime, one can usually get away with this problem by exploiting the privilege of Poincaré symmetry. You have a timelike Killing field coming from the time translation symmetry of the Poincaré group and can use it to determine which solutions are of positive frequency, and hence perform the construction I outlined in the previous paragraph. Nevertheless, this isn't your only choice. If you restrict your attention to the region of spacetime with $t < |x|$ (the right Rindler wedge), the Killing field associated with boosts in the $x$ direction is also timelike and you could just as well choose to work with them instead and build a different Fock space starting from this positive-frequency notion. Naturally, this will lead to a different notion of particles, and, in particular, the Minkowski vacuum will now be seen as a thermal state—this is the essence of the Unruh effect.

In curved spacetimes, this can get even wilder. If you have some timelike Killing field, this notion works and you get a preferred notion of particles to work with. In Minkowski, we pick the preferred notion of "inertial time" for inertial observers (a pretty good approximation for standard QFT). In contrast, for accelerated observers, we would rather pick the "boost time" since the worldlines of accelerated observers are exactly the curves parallel to the boost Killing field. On the other hand, your spacetime could simply not have any particular timelike Killing field, which leaves you without any preferred notion of particles. In this situation, one might need to drop the concept of particle and describe the theory in other terms (such as the algebraic approach described in Wald's Section 4.5).

In summary, the condition doesn't uniquely specify the annihilation and creation operators because it needs you to have a preferred time direction, so to speak, which is not provided in general curved spacetime. Even in Minkowski spacetime we could argue that it isn't either, since in the right Rindler wedge we might pick "boost time" as well, but we get away with it by choosing the time which is preferred for inertial observers.

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