Pointed weak energy and quantum state evolution in Pancharatnam–Fubini-Study configuration space

Abstract

The weak energy for a quantum state evolving relative to its fixed initial state (pointed weak energy) in a Hilbert space of arbitrary dimension is defined and aspects of the classical mechanical, transformational and geometric relationships between this quantity and the motion of the associated quantum state are studied in a two-dimensional configuration space with Pancharatnam (P) phase and Fubini-Study (FS) metric distance as generalized coordinates (PFS configuration space). It is shown that when expressed in these coordinates: (1) the evolution of the quantum state is described by an equation of motion for an associated correlation amplitude (pointed correlation amplitude) in which the dynamics are induced by the interaction of the pointed weak energy with this correlation amplitude; (2) the pointed weak energy defines a 1-form with the transformational properties of a U(1) gauge potential (pointed weak energy gauge potential) which produces infinitesimal changes in the pointed correlation amplitude via its interaction with the amplitude; (3) the pointed correlation amplitude is defined by an exponential function of a line integral of the pointed weak energy gauge potential; (4) pointed weak energy gauge transformations exist that are equivalent to local U(1) gauge transformations of quantum states and-when examined within the context of the pointed probability current and pointed weak energy stationary action principles-offer an interpretation of gauge invariance analogous to that found in classical mechanics; and (5) integral invariants exist which relate line integrals of the weak energy gauge potential to geometric phase and the Aharonov-Anandan connection 1-form in the associated Hilbert space. States which irreversibly decay and Grover's quantum search algorithm serve as simple illustrations of the theory.