Quantum scattering in gauge fields of adiabatic representations

A geometric approach to the method of adiabatic representations is developed for a class of relativistic Hamiltonians. The theory is used to analyze the associated dynamical equations with effective nonabelian interactions that can be regarded as gauge fields, induced by dimensional reduction of the initial problem in a special representation. It is shown that the approach can be used to study $2\to(2,3)$ quantum scattering processes in a three-body system, and a one-to-one relation between the complete and the effective $S$-matrices is derived. Asymptotic expressions are found for the solutions of the effective dynamical equation and for the gauge fields in the adiabatic representations. The method is illustrated for systems admitting adiabatic representations with a one-dimensional base; in several cases the field operator is proved to be Hilbert–Schmidt.