Adiabatic Invariants and Scalar Fields in a de Sitter Space-Time

The method of adiabatic invariants for time dependent Hamiltonians is applied to a massive scalar field in a de Sitter space-time. The scalar field ground state, its Fock space and coherent states are constructed and related to the particle states. Diverse quantities of physical interest are illustrated, such as particle creation and the way a classical probability distribution emerges for the system at late times. e–mail: [email protected] e–mail: [email protected] e–mail: [email protected] 0 In a previous paper [1] we illustrated the Born-Oppenheimer (BO) approach to the matter-gravity system in a simple minisuperspace model and in such a context the simplifications associated with the use of adiabatic invariants [2] were pointed out. Indeed through the use of such invariants one can improve on the adiabatic approximation and obtain better results for the evaluation of fluctuations which, in a quantum gravitational context, are associated with the creation of matter. The purpose of this note is to illustrate, within the context of field theory in a curved space-time (de Sitter with flat spatial section in our simplified model), the use of adiabatic invariants with a particular emphasis on the description of the vacuum and the space of physical states. The usefulness of the method of invariants for the calculation of the geometrical (adiabatic) and dynamical phases has been previously noticed [3] and the method of invariants itself applied to quantum cosmology [4], although not in a BO context. Here we shall apply the method to the equation for matter obtained [1] in the BO approach, on neglecting fluctuations and in the semiclassical limit for gravity. This novel application allows for the time variation of the metric in a second quantized (Schrödinger functional approach [5] [6]) scalar field theory thus improving on the adiabatic approximation (static metric) and including some matter creation through the use of the vacuum and the Fock space associated with quantum adiabatic invariants. Let us consider a Friedmann-Robertson-Walker line element: ds = −dτ + a(τ)gijdxdx (1) where gij is the three metric for a flat three-space. We further introduce a real massive scalar field Φ(~x, τ) whose lagrangian density is given by: L = − 2 g∂μΦ∂νΦ− 1 2 ξRΦ − 1 2 μΦ (2) with R the Ricci scalar and decompose the scalar field on a complete basis uk :