In the book "Introduction to Quantum Effects in Gravity" by Mukhanov and Winitzki, chapters 6.6 and 6.7, a comparison of the amplitude of fluctuations of a scalar field between vacuum and an excited state is presented. To calculate the amplitude in an excited state, a Bogolyubov transformation is used to related the operator that annihilates the vacuum state to that which annihilates the excited state. However, the main point of these transformations, as presented in the book, ,is because of the ambiguity in defining the vacuum for non Minkowski spaces. Since the calculations are being done in Minkowski, why the use of these transformations.

Moreover, if an operator annihilates an excited state, shouldn't it be equivalent to the product of operators that annihilate the vacuum state?