the Minkowski positive frequency solution, while in the magnetic field it coincides with the standard Landau states at $t = - \infty$. In the generic, time-dependent electromagnetic field without any asymptotic limit, the quantum invariants cannot be uniquely selected, but one may employ other principle, such as the minimum uncertainty, to choose the complex solution [@kim-kim99].

Note that the quantum invariants (\[E-in1\]) and (\[E-in2\]) $$\begin{aligned}
\hat{A}_{\alpha} (t) = \frac{1}{\sqrt{2}} \bigl( \hat{a}_{\alpha} (t) + \hat{b}_{\alpha} (t) \bigr), \nonumber\\
\hat{A}^{\dagger}_{\alpha} (t) = \frac{1}{\sqrt{2}} \bigl(\hat{a}^{\dagger}_{\alpha} (t) + \hat{b}^{\dagger}_{\alpha} (t) \bigr), \label{E-inv}\end{aligned}$$ can be expressed in terms of the annihilation and the creation operators for particles $$\begin{aligned}
\hat{a}_{\alpha} (t) &=& i \bigl( \varphi_{\alpha}^{*} (t) \hat{\pi}_{\alpha}^{*}  - \dot{\varphi}_{\alpha}^{*} (t) \hat{\phi}_{\alpha} \bigr), \nonumber\\
\hat{a}_{\alpha}^{\dagger} (t) &=& -i \bigl( \varphi_{\alpha} (t) \hat{\pi}_{\alpha}  - \dot{\varphi}_{\alpha} (t) \hat{\phi}_{\alpha}^{*} \bigr), \label{pa op}\end{aligned}$$ and those for antiparticles $$\begin{aligned}
\hat{b}_{\alpha} (t) &=& i \bigl( \varphi_{\alpha}^{*}(t) \hat{\pi}_{\alpha}  - \dot{\varphi}_{\alpha}^{*} (t) \hat{\phi}_{\alpha}^{*} \bigr),
 \nonumber\\
\hat{b}_{\alpha}^{\dagger} (t) &=& - i \bigl( \varphi_{\alpha} (t) \hat{\pi}_{\alpha}^{*}  - \dot{\varphi}_{\alpha} (t) \hat{\phi}_{\alpha} \bigr). \label{an op}\end{aligned}$$ The commutators hold $$\begin{aligned}
[\hat{A}_{\alpha} (t), \hat{A}^{\dagger}_{\beta} (t) ] = [\hat{a}_{\alpha} (t), \hat{a}^{\dagger}_{\beta} (t) ] =  [\hat{b}_{\alpha} (t), \hat{b}^{\dagger}_{\beta}  (t)]= \delta_{\alpha \beta}, \label{com}\end{aligned}$$ while other commutators vanish. Thus, $\hat{A}_{\alpha} (t)$ annihilates one particle-antiparticle pair, 