: General curved backgrounds lack global Poincaré invariance and time-translation symmetry, making it impossible to define a unique, preferred vacuum state.
bj=∑i(αjiai+βji*ai†)b sub j equals sum over i of open paren alpha sub j i end-sub a sub i plus beta sub j i end-sub raised to the * power a sub i raised to the † power close paren If the "mixing coefficient" βjibeta sub j i end-sub is non-zero, the vacuum of the first observer ( Quantum Field Theory in Curved Spacetime: Quant...
: The concept of a "particle" becomes local and observer-dependent. Different observers (e.g., one inertial and one accelerating) may disagree on whether a state contains particles or is a vacuum. To relate the perspectives of different observers or
To relate the perspectives of different observers or the state of fields at different times in an expanding universe, physicists use : Field Decomposition : A scalar field is expanded into a set of basis modes with creation and annihilation operators The Fundamental Shift: From Particles to Fields
. This approach serves as a robust approximation for environments where gravity is strong but quantum gravitational effects—such as fluctuations of the metric itself—are not yet dominant. 1. The Fundamental Shift: From Particles to Fields