  The equation of motion for a time-independent weak value of a quantum
mechanical observable contains a complex valued energy factor - the weak energy
of evolution. This quantity is defined by the dynamics of the pre-selected and
post-selected states which specify the observable's weak value. It is shown
that this energy: (i) is manifested as dynamical and geometric phases that
govern the evolution of the weak value during the measurement process; (ii)
satisfies the Euler-Lagrange equations when expressed in terms of Pancharatnam
(P) phase and Fubini-Study (FS) metric distance; (iii) provides for a PFS
stationary action principle for quantum state evolution; (iv) time translates
correlation amplitudes; (v) generalizes the temporal persistence of state
normalization; and (vi) obeys a time-energy uncertainty relation. A similar
complex valued quantity - the pointed weak energy of an evolving state - is
also defined and several of its properties in PFS-coordinates are discussed. It
is shown that the imaginary part of the pointed weak energy governs the state's
survival probability and its real part is - to within a sign - the
Mukunda-Simon geometric phase for arbitrary evolutions or the Aharonov-Anandan
(AA) phase for cyclic evolutions. Pointed weak energy gauge transformations and
the PFS 1-form are discussed and the relationship between the PFS 1-form and
the AA connection 1-form is established.
