The classically conformal invariant scalar theory with a non-positive potential

We review the conformal symmetry group and investigate the isomorphism between the conformal group and O( D,2 ) . We study the classically conformal invariant scalar theory in D -dimensions with a non-positive potential . We solve the equations of motion by assigning O(D-1, 2)symmetry to the classical solutions with broken translational symmetry in all directions. Then we consider a six dimensional theory and obtain a set of classical solutions, which are invariant under translation in four out of six directions. We make use of them to construct an effective four dimensional scalar theory induced by the fundamental six dimensional theory. Finally we elaborate the most general solutions of the 4 massless model in four dimensions.