Gravitational amplification and attenuation as part of the mutual interaction of quantum fields in curved space-times

We continue our study of quantum field theory of mutually interacting fields in a given unquantized curved background space-time. The mean value of the particle number operator is discussed in the interaction picture using an in-out scheme (S-matrix approach). For free fields there is a gravitational amplification (for bosons) or attenuation (for fermions) of the ingoing particle content which adds in the out region to the particles created out of the vacuum. In case there is an additional mutual interaction between fields, a similar amplification or attenuation of the particles coming out of the mutual interaction is found. A massive conformally coupled scalar field φ in a Robertson-Walker universe shows amplification; a massless field ψ does not. As case studies we give for a particular expansion law a mathematically rigorous calculation of the ‘‘Compton effect’’ in lowest order of the ${\phi }^2$ ${\psi }^2$ interaction and of the particle creation out of the vacuum in the φψ model. The ‘‘Compton effect’’ in exact energy-momentum conservation goes solely back to the amplification term, thus giving an operational physical meaning to this concept: The number of outgoing massive φ particles is not equal to but greater than the corresponding number of massless ψ particles. The case of vac- uum creation on the other hand represents an example in which the amplification term is accompanied by an additional term of comparable magnitude.