ar X iv : 0 80 3 . 12 93 v 2 [ he p - th ] 1 5 M ar 2 00 8 Lobachevsky geometry of ( super ) conformal mechanics

We give a simple geometric explanation for the similarity transformation mapping one-dimensional conformal mechanics to free-particle system. Namely, we show that this transformation corresponds to the inversion of the Klein model of Lobachevsky space (non-compact complex projective plane) fI CP 1 . We also extend this picture to the N = 2k superconformal mechanics described in terms of Lobachevsky superspace fI CP 1|k . Introduction Conformal symmetry plays an important role in modern field theory. So, the study of various aspects of simple (super)conformal invariant models could be useful for more complicated systems. Since the middle of seventies, after [1], it was realized that even the one-dimensional one-particle mechanics given by the Hamiltonian