Functional approach for quantum systems with continuous spectrum

Abstract

Considering quantum states as functionals acting on observables to give their mean values, it is possible to deal with quantum systems with continuous spectrum, generalizing the concept of trace. Generalized observables and states are defined for a quantum oscillator linearly coupled to a scalar field, and the analytic expression for time evolution is obtained. The ``final'' state $(\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty})$ is presented as a weak limit. Finite and infinite numbers of excited modes of the field are considered.