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Quantum theory of the optical stochastic cooling is presented. Results include full quantum analysis of the interaction of the beam with radiation in the undulators and quantum amplifier. The density matrix of the whole system is constructed. Evolution of the density matrix in the quantum amplifier includes non-diagonal components. The description of a bunch in the undulators follows Bambini-Renieri-Dattoli approach (1D regime) and reproduces Becker-McIver results but formalism is differentdoi:10.1103/physreve.65.016507 pmid:11800804 fatcat:lxbn3puavffu5gspuz7xdyeawu