Paramagnons and magnons of the quantum Heisenberg model and the modular Hilbert algebras

The problem of the Fourier transform of generalized operators is considered for the case where the operators grow with respect to the transforming parameter n as O(nsup(N)). These considerations allow to discuss linear response theory of the quantum Heisenberg model by modular Hilbert algebras. The response and relaxation functions are defined for any vectorstate to discuss them by the perturbation theory for nearly degenerate spectra. It is shown for the quantum Heisenberg model that the dynamical longitudinal and transversal magnetic susceptibilities possess at least a two pole structure for temperatures T above and below the transition temperature Tsub(c). The longitudinal ones are overdamped for all temperatures; on the contrary the transversal ones are not for T<Tsub(c); that is the longitudinal and transverse paramagnons and the longitudinal magnons are overdamped, but the transverse magnons are not and they possess a gap at the center of the Brillouin zone. (Auth.)