The geometry and significance of weak energy

Thr theory of weal; values for quantum mechanical observables has come to s erve 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 experim entally 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 int erpretations 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 detines functions which tr anslate correlation amplitudes and probabilities in time: (c) correlation p robabilities can be controlled by manipulating the weak energy and there ex ists a condition of weak stationarity that guarantees their time invariance ; and (d) a time-weak energy uncertainty relation of the usual form prevail s when a suitable set of dynamical constraints are imposed.