  We formulate the second quantization of a charged scalar field in
homogeneous, time-dependent electromagnetic fields, in which the Hamiltonian is
an infinite system of decoupled, time-dependent oscillators for electric
fields, but it is another infinite system of coupled, time-dependent
oscillators for magnetic fields. We then employ the quantum invariant method to
find various quantum states for the charged field. For time-dependent electric
fields, a pair of quantum invariant operators for each oscillator with the
given momentum plays the role of the time-dependent annihilation and the
creation operators, constructs the exact quantum states, and gives the vacuum
persistence amplitude as well as the pair-production rate. We also find the
quantum invariants for the coupled oscillators for the charged field in
time-dependent magnetic fields and advance a perturbation method when the
magnetic fields change adiabatically. Finally, the quantum state and the pair
production are discussed when a time-dependent electric field is present in
parallel to the magnetic field.
