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Sometimes they are referred as Keldysh quasiclassical propagator or Eilenberger propagator. How the quasiclassical propagator is derived? How they are implement with a simple case of a Green's function? \begin{equation} G^{K}= \begin{bmatrix} (g^K+\vec{g}^K \cdot \vec{\sigma}) & (f^K+\vec{f}^K \cdot \vec{\sigma})i\sigma_y \\ i\sigma_y(\underline{f}^K+\vec{\underline{f}}^K \cdot \vec{\sigma}) & (\underline{g}^K-\sigma_y\vec{\underline{g}}^K \cdot \vec{\sigma}\sigma_y) \end{bmatrix} \end{equation}

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They come from a quasi-classical expansion of the Dyson's equation (in the context of superconductivity, the Dyson's equations for the normal $G$ and anomalous $F$ Green's function are usually called the Gor'kov equations). They originate from

Larkin, A. I., & Ovchinnikov, Y. N. (1969). Quasiclassical method in the theory of superconductivity. Sov. Phys. JETP, 28, 1200–1205.

Eilenberger, G. (1968). Transformation of Gorkov’s equation for type II superconductors into transport-like equations. Zeitschrift Für Physik, 214(2), 195–213.

Usadel, K. D. (1970). Generalized Diffusion Equation for Superconducting Alloys. Physical Review Letters, 25(8), 507–509.

and they are many reviews of them, including

Serene, J. ., & Rainer, D. (1983). The quasiclassical approach to superfluid 3He. Physics Reports, 101(4), 221–311.

Langenberg, D. N., & Larkin, A. I. (1986). Nonequilibrium superconductivity. Amsterdam: North-Holland.

Wilhelm, F. K., Belzig, W., Bruder, C., Schön, G., & Zaikin, A. D. (1999). Quasiclassical Green’s function approach to mesoscopic superconductivity. Superlattices and Microstructures, 25(5–6), 1251–1288. Mesoscale and Nanoscale Physics; Superconductivity.

Kopnin, N. B. (2001). Theory of nonequilibrium superconductivity. Oxford: Oxford University Press.

Gor’kov, L. P. (2008). Theory of superconducting alloys. In K. H. Bennemann & J. B. Ketterson (Eds.), Superconductivity (in two volumes), vol. 1: Conventional and unconventional superconductors. Berlin, Heidelberg: Springer Berlin Heidelberg.

and many other.

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