electric field is added in parallel to the magnetic field and, thus, the same form of invariants. In Sec. \[sec6\], we discuss the physical implications of the result of this paper.

Second Quantized Scalar Field {#sec2}
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In scalar QED, we formulate the second quantization of a spinless, charged scalar field in homogeneous, time-dependent electromagnetic fields, first by expressing the Hamiltonian from the field action through a spectral method, and then by finding quantum states via the functional Schröinger equation. For that purpose, we consider the action for a spinless scalar with charge $q$ and mass $m$ \[in units of $\hbar = c =1$ and the spacetime signature $\eta^{\mu \nu} = (+, -, -, -)$\] $$\begin{aligned}
S = \int dt d^3 {\bf x} \bigl[ \eta^{\mu \nu} (\partial_{\mu} + iqA_{\mu} ) \phi^{*} (\partial_{\mu} - iqA_{\mu} ) \phi - m^2 \phi^{*} \phi \bigr]\end{aligned}$$ in an electromagnetic field, which is given by the four-vector as $$\begin{aligned}
{\bf E} = \nabla A_0 - \frac{\partial {\bf A}}{\partial t}, \quad {\bf B} = \nabla \times {\bf A}.\end{aligned}$$ Introducing the conjugate momenta for each $\phi$ and $\phi^*$ $$\begin{aligned}
\pi = \frac{\partial {\cal L}}{\partial \dot{\phi}} = \dot{\phi}^{*} + iqA^{0} \phi^{*}, \quad
\pi^{*} = \frac{\partial {\cal L}}{\partial \dot{\phi}^{*}} = 