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In-medium Yang-Mills equations: a derivation and canonical quantization 2 Abstract. The equations for a Yang-Mills field in a medium are derived in the approximation of linear response to an external field. Introducing tensors of generalized susceptibilities, the in-medium equations are written in a form similar to the in-medium Maxwell equations. The non-Abelian character of the gauge group G is reflected in the presence of susceptibility tensors with no counterpart in electrodynamics and the necessity to define modified "electric" and "magnetic" fields (apart from the "color" induction vectors). which enter the in-medium equations. The latter reduce to dimG copies of Maxwell in-medium equations in the approximation up to second order in the gauge coupling constant. For a medium uniformly moving, a canonical quantization is performed for a family of Fermi-like gauges in the case of constant and diagonal (in the group indices) tensors of "electric" permittivity and "magnetic" permeability. The physical subspace is defined and the gauge field propagator is evaluated for a particular choice of the gauge. It is applied firstly for evaluation of the cross-section of elastic quark scattering in the Born approximation. Possible applications to Cherenkov-type gluon radiation are commented briefly.

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