In this paper, we consider the occupation number of induced quasi-particles which are produced during a time-dependent process using three different methods: Instantaneous diagonalization, the usual Bogolyubov transformation between two different vacua (more precisely the instantaneous vacuum and the so-called adiabatic vacuum), and the Unruh–DeWitt detector methods. Here we consider the Hamiltonian for a time-dependent Harmonic oscillator, where both the mass and frequency are taken to be time-dependent. From the Hamiltonian we derive the occupation number of the induced quasi-particles using the invariant operator method. In deriving the occupation number we also point out and make the connection between the Functional Schrödinger formalism, quantum kinetic equation, and Bogolyubov transformation between two different Fock space basis at equal times and explain the role in which the invariant operator method plays. As a concrete example, we consider particle production in the flat FRW chart of de Sitter spacetime. Here we show that the different methods lead to different results: The instantaneous diagonalization method leads to a power law distribution, while the usual Bogolyubov transformation and Unruh–DeWitt detector methods both lead to thermal distributions (however the dimensionality of the results are not consistent with the dimensionality of the problem; the usual Bogolyubov transformation method implies that the dimensionality is 3D while the Unruh–DeWitt detector method implies that the dimensionality is 7D/2). It is shown that the source of the descrepency between the instantaneous diagonalization and usual Bogolyubov methods is the fact that there is no notion of well-defined particles in the out vacuum due to a divergent term. In the usual Bogolyubov method, this divergent term cancels leading to the thermal distribution, while in the instantaneous diagonalization method there is no such cancelation leading to the power law distribution. However, to obtain the thermal distribution in the usual Bogolyubov method, one must use the large mass limit. On physical grounds, one should expect that only the modes which have been allowed to sample the horizon would be thermal, thus in the large mass limit these modes are well within the horizon and, even though they do grow, they remain well within the horizon due to the mass. Thus, one should not expect a thermal distribution since the modes will not have a chance to thermalize.
Time-dependent particle production and particle number in cosmological de Sitter space
