The geometry and significance of weak energy

Abstract

The theory of weak values for quantum mechanical observables has come to serve as a useful basis for contemporary discussions concerning such varied topics as the tunnelling-time controversy and quantum stochastic processes. An intrinsic complex-valued weak energy has recently been observed experimentally and reported in the literature. In this paper it is shown that: (a) the real and imaginary valued parts of this weak energy have geometric interpretations related to a phase acquired from parallel transport in Hilbert space and the variational dynamics occurring in the associated projective Hilbert space, respectively; (b) the weak energy defines functions which translate correlation amplitudes and probabilities in time; (c) correlation probabilities can be controlled by manipulating the weak energy and there exists a condition of weak stationarity that guarantees their time invariance; and (d) a time-weak energy uncertainty relation of the usual form prevails when a suitable set of dynamical constraints are imposed.