Quantum mechanics of confinement and chiral symmetry breaking in two-dimensional QCD

The system of light quark and heavy anti-quark source is studied in 1+1 QCD in the large $N_C$ limit. Making use of the modified Fock-Schwinger gauge allows to consider simultaneously the spectroscopical problem of the q\bar Q bound states and the problem of the light quark Green function. The Dirac-type equation for the spectrum of the system is proved to be equivalent to the well-known 't Hooft one in the one body limit. The unitary transformation from the Dirac-Pauli representation to the Foldy-Wouthuysen one is carried out explicitly, and it is shown that the equation in the Foldy-Wouthuysen representation can be treated as a gap equation which defines the light quark self-energy in the modified Fock-Schwinger gauge. The Foldy--Wouthuysen angle is found to play the role of the Bogoliubov-Valatin one and to give the standard value of the chiral condensate. Connections of the given formalism to the standard four-dimensional QCD are outlined and discussed.