A weak energy stationary action principle for quantum state evolution

Abstract

It is shown that the actual paths in Hilbert space followed by a finite set of n ⩾ 2 quantum states evolving between initial and final end point configurations are such that an associated weak energy functional defined by Pancharatnam phases and state separation distances in projective Hilbert space determined by the generalized Fubini-Study metric is stationary for all variations of these phases, separations and time which vanish at the end points. Noether's theorem is used to identify two weak energy conservation laws which are shown to be the analogues of the momentum and energy conservation laws of Langrangian mechanics.