On two approaches to quantization in strong background fields

There are two ways to quantize free (gaussian) theory in strong background fields. In one of them, which we refer to as the Heisenberg approach, the mode functions are defined once and for entire space-time. In this approach there is no any apparent presence of an initial Cauchy surface. Another method, which we refer to as the Schrodinger approach, assumes the presence of the initial Cauchy surface on which one defines a spatial basis of modes and only then considers the evolution of the creation and annihilation operators in time with the use of the (free) Hamiltonian. The first method usually used to respect the symmetries of the problem (e.g. isometry of the de Sitter space-time), while in the second method the isometry is apparently broken by the presence of the initial Cauchy surface. In this paper we compare the two methods of quantization and find conditions under which they give the same result.