Weak covariance and the correlation of an observable with pre-selected and post-selected state energies during its time-dependent weak value measurement

The peculiar weak energy of evolution appears as a factor in the equation of motion A˙ w for a time-dependent weak value of an observable A^. This energy has the mathematical form of the weak value of the difference between the two Hamiltonian operators H^ i and H^ f that describe the evolution of the associated pre- and post-selected states, respectively. Here, the weak covariance cov w(X^ , Y^) for operators X^ and Y^ is introduced and it is shown that |A˙ w| can be expressed entirely in terms of cov w(H^ f, A^) , cov w(A^ , H^ i) , and an angle θ that is governed by the complex valued nature of the terms defining each covariance. Several cases are briefly discussed and an experiment is used to illustrate the H^ i= 0 ^ ≠ H^ f case. It is shown that covw(H^ f, A^) is observed in the associated experimental data. © 2018, This is a U.S. government work and its text is not subject to copyright protection in the United States; however, its text may be subject to foreign copyright protection.