creation operators for time-dependent oscillators, quantum invariants linear in momentum and position operators, may be used to find the Fock space of all the number states, which are the exact solutions for the time-dependent Schrödinger equation [@MMT70; @kim-lee00]. Furthermore, it can be readily used in constructing thermal states and coherent thermal states, which are also the exact quantum states [@kim-lee00; @kim-page13]. In scalar quantum electrodynamics (QED) in time-dependent electric fields, the time-dependent vacuum state leads to the pair-production rate [@kim-schubert11; @KLR12] and also to the in-out scattering matrix between the remote past and the remote future, which yields the renormalized one-loop effective action [@KLY08]. The quantum invariant method can also be used to describe the quantum motion of charged particles in strong electromagnetic fields. In the case of the adiabatically changing magnetic fields, the invariants may be found for coupled, time-dependent oscillators by treating the off-diagonal Hamiltonian as a perturbation for the diagonal Hamiltonian, decoupled time-dependent oscillators. We advance a time-dependent perturbation theory to find the improved states. The method is valid for the time-dependent magnetic fields parallel to electric fields.

The organization of this paper is as follows. In Sec. \[sec2\], in scalar QED we formulate the second quantized 