## 2 Kinetic Equation
### The Low-Energy Approximation  
The Dirac-type equation for the low-energy excitations in graphene in a time dependent electric field described above is  
\[i\hbar\dot{\Psi}\left(\vec{x},t\right)=v_{F}\hat{\vec{P}}\vec{\sigma}\Psi\left( \vec{x},t\right), \tag{2}\]  
where \(\hat{P}_{k}=-i\hbar\
abla_{k}-(e/c)A_{k}(t)\) is the quasi-momentum \((k=1,2)\) and \(\sigma_{k}\) are the Pauli matrices corresponding to the pseudospin structure of graphene. The Hamiltonian of the theory,  
\[H(t)=\frac{i\hbar}{2}\int d^{2}x\left[\Psi^{\dagger}(\vec{x},t)\dot{\Psi}(\vec {x},t)-\dot{\Psi}^{\dagger}(\vec{x},t)\Psi(\vec{x},t)\right]\, \tag{3}\]  
is the \(00\) component of the corresponding energy-momentum tensor and it can be transformed with help of the equation of motion (2) to the form  
\[H(t)=v_{F}\int d^{2}x\Psi^{\dagger}(\vec{x},t)\hat{\vec{P}}\vec{\sigma}\Psi( \vec{x},t). \tag{4}\]  
Here we dropped the spin indices. The wave function here is a two-component spinor permitting the decomposition  
\[\Psi^{T}\left(\vec{x},t\right)=\frac{1}{\left(2\pi\hbar\right)^{2}}\int d^{2} p\left(\Psi^{(1)}_{\vec{p}}\left(t\right),\Psi^{(2)}_{-\vec{p}}\left(t \right)\right)e^{i\vec{p}\vec{x}/\hbar}\, \tag{5}\]  
which translates the Hamiltonian function (4) to the momentum representation. For the physical interpretation of the model it is appropriate to go over to the quasiparticle representation, where the Hamiltonian of the theory is diagonal. As it was shown in the work [15], this is achieved with the unitary transformation  
\[U^{\dagger}(t)v_{F}\vec{P}\vec{\sigma}U(t)=\epsilon(\vec{p},t)\sigma_{3}=H_{ \vec{p}}(t), \tag{6}\]  
and \(\Phi=U^{\dagger}\Psi\) with the unitary matrix [15]  
\[U(t)=\frac{1}{\sqrt{2}}\left( \begin{array}{cc}\exp(-i\varkappa/2)&\exp(-i \varkappa/2)\\ \exp(i\varkappa/2)&-\exp(i\varkappa/2)\end{array} \right). \tag{7}\]  
The function \(\varkappa\) is defined by the condition (6) [15], corresponding to \(\tan\varkappa=P^{2}/P^{1}\), where \(P^{k}=p^{k}-(e/c)A^{k}(t)\). The quasienergy \(\epsilon(\vec{p},t)\) in (6) is determined by the dispersion relation in the vicinity of the Dirac points  
\[\epsilon(\vec{p},t)=v_{F}\sqrt{P^{2}}=v_{F}\sqrt{(P^{1})^{2}+(P^{2})^{2}}. \tag{8}\]  
Equation (2) transforms then to the form  
\[i\hbar\dot{\Phi}=H_{\vec{p}}(t)\Phi+\frac{1}{2}\lambda\hbar\sigma_{1}\Phi, \tag{9}\]where \(H_{\vec{p}}(t)\) is defined by Equation (6) and  
\[\lambda\left(\vec{p},t\right)=\dot{\varkappa}=\frac{ev_{F}^{2}[E_{1}P_{2}-E_{2}P_{ 1}]}{\varepsilon^{2}(\vec{p},t)}. \tag{10}\]  
Introducing the notation  
\[\Phi(\vec{p},t)=\left[ \begin{array}{c}a(\vec{p},t)\\ b^{\dagger}(-\vec{p},t)\end{array} \right], \tag{11}\]  
the Hamiltonian function (4) can be rewritten in the quasiparticle form  
\[H(t) = \int[dp]\varepsilon(\vec{p},t)\Phi^{\dagger}\left(\vec{p},t \right)\sigma_{3}\Phi\left(\vec{p},t\right)\] \[= \int[dp]\varepsilon(\vec{p},t)\left[a^{\dagger}(\vec{p},t)a( \vec{p},t)-b(-\vec{p},t)b^{\dagger}(-\vec{p},t)\right],\]  
where the abbreviation \([dp]=d^{2}p(2\pi\hbar)^{-2}\) has been used. Apparently, the realization of the unitary transformation in the explicit form in both the low-energy and the tight-binding (see below Section 7) models is a result of the fact that these models belong to the class of conformal-invariant field theories (see, e.g., Ref. [6]). At this stage one can go over to the occupation number representation and replace the amplitudes \(a^{\dagger}(t)\),\(a(t)\) and \(b^{\dagger}(t)\), \(b(t)\) by the corresponding creation and annihilation operators for electrons and holes considered as quasiparticles. These operators are defined on the in-vacuum state \(|\mathrm{in}\rangle\) with vector potential \(\vec{A}_{\mathrm{in}}\) and satisfy the canonical anti-commutation relations  
\[\left\{a(\vec{p},t),a^{\dagger}(\vec{p}^{\prime},t)\right\}_{+} = \left\{b(\vec{p},t),b^{\dagger}(\vec{p}^{\prime},t)\right\}_{+}= (2\pi)^{2}\delta(\vec{p}-\vec{p}^{\prime}). \tag{13}\]  
Other elementary anti-commutators are equal to zero.