Complex germ method in the Fock space. I. Asymptotics like wave packets

In this paper, we establish a new method of constructing approximate solutions to secondary-quantized equations, for instance, for many-particle Schrödinger and Liouville equations written in terms of the creation and annihilation operators, and also for equations of quantum field theory. The method is based on transformation of these equations to an infinite-dimensional Schrödinger equation, which is investigated by semiclassical methods. We use, and generalize to the infinite-dimensional case, the complex germ method, which yields wave packet type asymptotics in the Schrödinger representation. We find the corresponding asymptotics in the Fock space and show that the state vectors obtained are actually asymptotic solutions to secondary-quantized equations with an accuracy $O(\varepsilon ^{M/2})$, $M\in \mathbb N$, with respect to the parameter $\varepsilon$ of the semiclassical expansion.