## 2 Kinetic Equation
### The Low-Energy Approximation
From Equations (2), (6) and (11) it follows the equations of motion of the Heisenberg type for the description of the unitary evolution of the creation and annihilation operators, e.g.,  
\[\dot{a}(\vec{p},t) = \frac{i}{\hbar}\left[H(t),a(\vec{p},t)\right]-\frac{i}{2}\lambda \left(\vec{p},t\right)b^{+}(-\vec{p},t)=\frac{i}{\hbar}\left[H_{tot}(t),a(\vec {p},t)\right], \tag{14}\] \[b(\vec{p},t) = \frac{i}{\hbar}\left[H(t),b(-\vec{p},t)\right]+\frac{i}{2}\lambda \left(\vec{p},t\right)a^{+}(\vec{p},t)=\frac{i}{\hbar}\left[H_{tot}(t),b(-\vec {p},t)\right], \tag{15}\]  
where the amplitude of the transitions between states with the positive and negative energies of the quasiparticles is defined by Equation (10). From Equations (9), (14) and (15) it follows that evolution of the system is unitary. The Fock space is constructed on the time dependent vacuum state. In Equations (14) and (15) \(H_{tot}=H+H_{pol}\), where  
\[H_{pol}(t) = \frac{\hbar}{2}\int[dp]\lambda(\vec{p},t)[a^{\dagger}(\vec{p},t) b^{\dagger}(-\vec{p},t)-b(-\vec{p},t)a(\vec{p},t)] \tag{16}\]  
describes the dynamics of vacuum polarization. Now one can obtain the KE. Let us introduce the distribution functions for the electrons and the holes,  
\[f^{e}(\vec{p},t) = \langle\mathrm{in}|a^{+}(\vec{p},t)a(\vec{p},t)|\mathrm{in}\rangle, \tag{17}\] \[f^{h}(\vec{p},t) = \langle\mathrm{in}|b^{+}(-\vec{p},t)b(-\vec{p},t)|\mathrm{in}\rangle. \tag{18}\]The averaging procedure here is carried out under the in-vacuum state \(|{\rm in}\rangle\). Differentiation with respect to time and taking into account Equations (14) and (15) results in  
\[f^{e}(\vec{p},t)=\frac{i\lambda}{2}\left(\vec{p},t\right)\left\{f^{(+)}(\vec{p},t)-f^{(-)}(\vec{p},t)\right\}, \tag{19}\]  
where anomalous averages have been introduced  
\[f^{(+)}(\vec{p},t) = \langle{\rm in}|a^{+}(\vec{p},t)b^{+}(-\vec{p},t)|{\rm in}\rangle, \tag{20}\] \[= \int[dp]\varepsilon(\vec{p},t)\left[a^{\dagger}(\vec{p},t)a( \vec{p},t)-b(-\vec{p},t)b^{\dagger}(-\vec{p},t)\right],\]0  
The equations of motion for these functions have the form  
\[= \int[dp]\varepsilon(\vec{p},t)\left[a^{\dagger}(\vec{p},t)a( \vec{p},t)-b(-\vec{p},t)b^{\dagger}(-\vec{p},t)\right],\]1 \[= \int[dp]\varepsilon(\vec{p},t)\left[a^{\dagger}(\vec{p},t)a( \vec{p},t)-b(-\vec{p},t)b^{\dagger}(-\vec{p},t)\right],\]2  
Here it was assumed that \(f^{e}=f^{h}=f\) holds as a consequence of the electroneutrality condition. Let us rewrite Equations (22) and (23) in integral form. Substitution of this result in Equation (19) leads to a KE of non-Markovian type  
\[= \int[dp]\varepsilon(\vec{p},t)\left[a^{\dagger}(\vec{p},t)a( \vec{p},t)-b(-\vec{p},t)b^{\dagger}(-\vec{p},t)\right],\]3  
where  
\[= \int[dp]\varepsilon(\vec{p},t)\left[a^{\dagger}(\vec{p},t)a( \vec{p},t)-b(-\vec{p},t)b^{\dagger}(-\vec{p},t)\right],\]4  
is the dynamical phase. In the present work the KE (24) and its reformulation in the form of an equivalent system of ordinary differential equations (ODE), shown below in Equation (27), are considered only for zero initial conditions, \(f_{0}=f(t_{0})=0\). For the first time a KE of such type was obtained in the works [6, 7, 19] in \(D=3+1\) QED for the description of vacuum creation of electron-positron pairs under the action of a time dependent spatially homogeneous linearly polarized electric field. This method is based on the usage of unitary nonequivalent canonical transformations for the transition to the quasiparticle representation [6]. In the considered situation this approach is applicable and leads to the KE (24) that has the same mathematical structure as in the \(D=3+1\) QED case [7, 8, 18]. However, in the massless \(D=2+1\) QED case the transition to the quasiparticle representation is possible in the framework of a unitary transformation [15] (see, e.g., Equation (6)). An advantage of the unitary approach is also the possibility of a generalization of this method [15] to the case of a two-dimensional electric field with the vector potential \(A^{k}(t)(k=1,2)\).