## 3 Observables  
It is straightforward to write expressions for the pair number density  
\[n(t) = N\int[dp]f(\vec{p},t). \tag{42}\]  
The factor \(N\) corresponds to number of species (or flavors) of quasiparticles in graphene [3, 14, 25]: \(N=4\) in the low energy model and \(N=2\) in the tight binding model. For exact solutions of the ODE system (27) and correct nonperturbative solutions of the type (33) and (34) it follows from the normalization integral (42) that the distribution function is limited everywhere, \(f(\vec{p},t)\leq 1\). Then both polarization functions \(u(\vec{p},t)\) and \(v(\vec{p},t)\) are limited also everywhere under the integral of motion (29). This conclusion relates to the neighborhood of the Dirac lines (26) and to the ultraviolet behavior of these functions as well. The current density consists of two components, the conduction and polarization current densities,  
\[j_{k}(t)=j_{k}^{\rm cond}(t)+j_{k}^{\rm pol}(t). \tag{43}\]  
These currents are defined by the distribution function \(f(\vec{p},t)\) and the polarization function \(u(\vec{p},t)\), correspondingly [26]. Firstly we consider the currents in the low-energy model. On the basis of the standard definition of the current density [27] (\(k=1,2\))  
\[j_{k}(t) = -e\frac{\delta H(t)}{\delta A_{k}(t)} \tag{44}\]  
one can obtain for the theory with the Hamiltonian (4) taking into account the flavor number  
\[j_{k}(t) = 4 ev_{F}\int d^{2}x\Psi^{*}\left(\vec{x},t\right)\sigma_{k}\Psi \left(\vec{x},t\right). \tag{45}\]  
Going over to the quasiparticle representation with the help of the unitary operator (7) we obtain  
\[j_{k}(t)=4 ev_{F}\int[dp]\Phi^{\dagger}\left(\vec{p},t\right)U^{\dagger}(t) \sigma_{k}U(t)\Phi\left(\vec{p},t\right). \tag{46}\]  
Taking into account the spinor (11) and the definition (43), one can separate the conduction and polarization currents,  
\[j_{l}^{\rm cond}(t) = 8\int[dp]v_{q}^{i}(\vec{p},t)f(\vec{p},t), \tag{47}\] \[j_{l}^{\rm pol}(t) = 4\int[dp]\varepsilon(\vec{p},t)l_{i}(\vec{p},t)u(\vec{p},t),\]  
where \(v_{q}^{i}(\vec{p},t)=P_{l}/\varepsilon(\vec{p},t)\) and the vector \(l_{i}(\vec{p},t)=\delta\lambda(\vec{p},t)/\delta E^{i}(t)\) is defined by the components  
\[l_{1}(\vec{p},t)=\frac{ev_{F}^{2}P_{2}}{\varepsilon^{2}},\quad l_{2}(\vec{p}, t)=-\frac{ev_{F}^{2}P_{1}}{\varepsilon^{2}}. \tag{48}\]  
One can see from the system (27) and its nonperturbative solutions (33) and (34), that the polarization effects dominate in the leading approximation for weak fields, \(\alpha=E/E_{0}\ll 1\), i.e., \(f\sim\alpha^{2}\), \(u\sim\alpha\) and so it follows that  
\[|j^{\rm pol}(t)|\gg|j^{\rm cond}(t)|. \tag{49}\]This conclusion is supported also by direct numerical calculations. Let us note, that the conduction and polarization currents are not collinear in the general case. In the case of the linearly polarized electric field collinearity of the currents (47) rebuilds. In order to ascertain this fact, let us consider the situation when the electric field acts along the axis \(x_{1}\), \(\vec{E}(E_{1}(t),0)\). Then \(P_{2}\to p_{2}\) and the functions \(f(\vec{p},t)\) and \(u(\vec{p},t)\) are even and odd under reflection \(p_{2}\to-p_{2}\), respectively, as it can be seen from the structure of the amplitude (10) and Equation (27). This makes the integrals for \(j_{2}^{\rm cond}(t)\) and \(j_{2}^{\rm pol}(t)\) in Equation (47) vanish. In order to investigate the theory we calculate the currents (47) in the framework of the perturbation theory in the minimal leading approximation for relatively small external field, see Appendix A. From Equation (47) it follows that the function \(u(\vec{p},t)\) determines the vacuum polarization current. The physical meaning of the other polarization function \(v(\vec{p},t)\) is revealed if one considers the total energy density of the quasiparticles including the polarization energy.