Adiabatic vacua from linear complex structures

Adiabatic vacua play a central role in quantum fields in cosmological spacetimes, where they serve as distinguished initial conditions and as reference states for the renormalization of observables. In this paper we introduce new methods based on linear complex structures which provide a powerful tool for determining adiabatic vacua. The new methods generalize both the standard WKB appoach and the Lewis-Riesenfeld invariants, and allow us to study the problem of many coupled bosonic degrees of freedom with general quadratic time-dependent Hamiltonian. We show that the adiabatic number operator and the adiabatic vacuum of finite order can be expressed in terms of the adiabatic complex structure of the same order. We compare our results to standard techniques which apply only to a single degree of freedom, and comment on its applicability to problems in quantum fields in cosmological spacetimes, many-body systems and quantum thermodynamics, where the Hamiltonian is time dependent with slowly-changing parameters.