Weak Energy: Form and Function
The equation of motion for a time-dependent 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 quantum 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) geometric phase for cyclic evolutions. Pointed weak energy gauge transformations and the PFS 1-form are defined and discussed and the relationship between the PFS 1-form and the AA connection 1-form is established. [Editors note: for a video of the talk given by Prof. Parks at the Aharonov-80 conference in 2012 at Chapman University, see quantum.chapman.edu/talk-25.]
The preparation of this contribution to the Festschrift commemorating Yakir Aharonov’s eightieth birthday was supported by a grant from the Naval Innovation in Science and Engineering program sponsored by the Naval Surface Warfare Center Dahlgren Division.
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