Infrared singularities in the null-plane bound-state equation when going to 1+1 dimensions

In this paper we first consider the null-plane bound-state equation for a $q \bar q$ pair in 1+3 dimensions and in the lowest-order Tamm-Dancoff approximation. Light-cone gauge is chosen with a causal prescription for the gauge pole in the propagator. Then we show that this equation, when dimensionally reduced to 1+1 dimensions, becomes 't Hooft's bound-state equation, which is characterized by an $x^+$-instantaneous interaction. The deep reasons for this coincidence are carefully discussed.