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## 2. Quasiclassical Green’s function 

Now we discuss the Green’s function for the Dirac equation, 

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Here Σ = (Σ$_{x}$ , Σ$_{y}$) are the pseudospin matrices in the sublattice space, 1 is the unit matrix, and γ$_{ϵ}$ > 0 is the half of the electron or hole inelastic scattering rate, introduced phenomenologically (see Sec. IVC of Ref. 29 for the discussion of its role in Raman scattering). The factor sgn ϵ corresponds to the chronologically ordered Green’s function. 

The quasiclassical Green’s function is represented as a sum over all trajectories of the classical motion from r ′ to r , i. e., sum over the short and long arcs, “ ± ”, and over the number of rotations j = 0 , 1 , . . . : 

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The imaginary part of the argument of the exponential is just the classical action discussed in the previous subsec- tion. Replacement ϵ → | ϵ | is required by the analytical properties of the chronologically ordered Green’s function (otherwise it would correspond to the retarded Green’s function). The damping factor e − γ$_{ϵ}$ϑR/v is determined by the total length of the corresponding arc ϑR , includ- ing 2 πjR from j full rotations. The eigenvector ψ$_{n}$ of ( n · Σ ) ψ$_{n}$ = ψ$_{n}$ sgn ϵ , | n | = 1, can be written in terms of 

the polar angle ϕ$_{n}$ of the unit vector n (we use the basis where Σ$_{x}$ , Σ$_{y}$ are represented by Pauli matrices), 

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Thus defined, ψ$_{n}$ is not a single-valued function: when n is rotated by 2 π , ψ$_{n}$ acquires a minus sign. We can fix the sign by requiring that at r → r $^{′}$, ψ$_{n}$$_{+}$ = − ψ$_{−}$$_{n}$$_{−}$ = ψ$_{r}$$_{−}$$_{r}$ ′ , ψ$_{n}$$_{−}$ = ψ$_{−}$$_{n}$$_{+}$ = ψ$_{r}$ $^{′}$− $_{r}$, and they evolve continu- ously during the motion along the circle. 

We make several further remarks concerning the above expression for G ( r , r $^{′}$), Eqs. (A13), (A14). ′ 

(i) For r → r G ( r , r $^{′}$; ϵ ) reduces to the expression for the quasiclassical Green’s function for B = 0, 

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valid at | ϵ || r − r $^{′}$| / ( ℏ v ) ≫ 1. 

(ii) The difference between the exact Green’s function and G ( r , r $^{′}$; ϵ ) given by Eqs. (A13), (A14) is of higher order in ℏ . This can be checked explicitly by calculating the gradients: 

so that at r ̸ = r ′ 

It is proportional to the eigenvector corresponding to the energy − ϵ ; thus, it can be compensated by a correction to the pre-exponential factor which does not vanish when acted upon by ϵ + iv Σ · ( ∇ S − ie A /c ), and thus is of the next order in ℏ : 

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δG ( j ) + is smaller than G ( j ) + by the dimensionless param- eter ℏ v/ ( ϵR ). However, it has a stronger divergence at 

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