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We present a general derivation of semi-fermionic representation for spinoperators in terms of a bilinear combination of fermions in real and imaginarytime formalisms. The constraint on fermionic occupation numbers is fulfilled bymeans of imaginary Lagrange multipliers resulting in special shape ofquasiparticle distribution functions. We show how Schwinger-Keldysh techniquefor spin operators is constructed with the help of semi-fermions. Wedemonstrate how the idea of semi-fermionic representation might be extended tothe groups possessing dynamic symmetries (e.g. singlet/triplet transitions inquantum dots). We illustrate the application of semi-fermionic representationsfor various problems of strongly correlated and mesoscopic physics.Comment: Review article, 40 pages, 11 figure

SEMI-FERMIONIC REPRESENTATION FOR SPIN SYSTEMS UNDER EQUILIBRIUM AND NON-EQUILIBRIUM CONDITIONS