Finite-Size Corrections to Fermi’s Golden Rule

Abstract

Decay rates and scattering cross sections in the situation where the wave functions overlap in wide area have been unclarified long time. We solve this problem with the scattering matricies that satisfy the boundary conditions of experiments of finite time interval T between the initial and final states, S[T]. S[T] is different from the standard S[∞] that satisfies them at T = ∞. The transition rates computed using Fermi’s golden rules need no corrections normaly. However they are subject to corrections of non-negligible magnitudes and necessary to compare the theory with experiments in the present case, particuraly for light particles. The wave functions that evolve according to Schrodinger equation conserve the total energy but not the kinetic energy at a finite t. Thus S[T] has variouis unusual properties which are caused by non-conservation of the kinetic-energy, different from those of S[∞]. The corrections can be computed rigorously and have universal properties in relativisticaly invariant systems. Derivations, origins, and unusual properties of the finite-size corrections of processes on neutrinos and gamma rays are presented. Especially applications to the determination of neutrino absolute mass and to phenomena of anomolous light emittion are presented.