Dimensional Reduction and Dynamical Chiral Symmetry Breaking by a Magnetic Field in 3 + 1 Dimensions

It is shown that in 3 + 1 dimensions, a constant magnetic field is a catalyst of dynamical chiral symmetry breaking, leading to generating a fermion mass even at the weakest attractive interaction between fermions. The essence of this effect is the dimensional reduction D → D − 2 (3 + 1 → 1 + 1) in the dynamics of fermion pairing in a magnetic field. The effect is illustrated in the Nambu-Jona-Lasinio model. Possible applications of this effect are briefly discussed.