Evolution of physical systems and nonlinear realized symmetries

Abstract

In this work, a new method to determine and to describe the evolution of physical systems (in classical or quantum regimes) is proposed. The main idea is based in the observation that the complete determination of dynamics of the system is connected to the nonlinear realization (NLR) of the symmetry group associated with the symmetry of the Hamiltonian [Formula: see text] of the system. The Maurer–Cartan forms, when they are parametrically associated to the projection on the hypersurface induced by the group in question via pullback, of the Hamiltonian components (based on the generators of the symmetry group by which it is defined) give rise to a fundamental system of equations by which the precise evolution can be determined. Such a system of equations under certain conditions that we will specify in this paper coincides with the system obtained through the Wei–Norman method, thus showing explicitly that our proposal is more general due the universal character of equations from the NLRs approach. As example, the important problem of the Caldirola–Kanai inverted Hamiltonian is completely solved, showing at the same time a new interpretation of what NLRs refer to.