Dynamical symmetry breaking in spaces with a constant negative curvature

By using the Nambu-Jona-Lasinio model, we study dynamical symmetry breaking in spaces with a constant negative curvature. We show that the physical reason for the zero value of the critical coupling constant g(c) = 0 in these spaces is connected with the effective reduction of the dimension of spacetime 1 + D-->1 + 1 for fermions in the infrared region, which takes place for any dimension 1+D. Since the Laplace-Beltrami operator has a gap in spaces with a constant negative curvature, such an effective reduction for scalar fields is absent and there are no problems with radiative corrections due to scalar fields. Therefore, dynamical symmetry breaking with the effective reduction of the dimension of spacetime for fermions in the infrared region is consistent with the Mermin-Wagner-Coleman theorem, which forbids spontaneous symmetry breaking in (1 + 1)-dimensional spacetime.