Spin Excitations and Sum Rules in the Heisenberg Antiferromagnet

Various bounds for the energy of collective excitations in the Heisenberg antiferromagnet are presented and discussed using the formalism of sum rules. We show that the Feynman approximation significantly overestimates (by about 30\% in the $S={1\over2}$ square lattice) the spin velocity due to the non negligible contribution of multi magnons to the energy weighted sum rule. We also discuss a different, Goldstone type bound depending explicitly on the order parameter (staggered magnetization). This bound is shown to be proportional to the dispersion of classical spin wave theory with a q-independent normalization factor. Rigorous bounds for the excitation energies in the anisotropic Heisenberg model are also presented.