On Reductions and Real Hamiltonian Forms of Affine Toda Field Theories

A family of real Hamiltonian forms (RHF) for the special class of affine 1+1 - dimensional Toda field theories is constructed. Thus the method, proposed in [Mikhailov;1981] for systems with finite number of degrees of freedom is generalized to infinite-dimensional Hamiltonian systems. We show that each of these RHF is related to a special Z_2-symmetry of the system of roots for the relevant Kac-Moody algebra. A number of explicit nontrivial examples of RHF of ATFT are presented.