1211.5632 (Antonio Di Lorenzo)

Inadequacy of Taylor series for perturbation expansion: a lesson from
weak measurement    [PDF]

Antonio Di Lorenzo

We all learnt from calculus textbooks that a Taylor expansion should be made consistently: all first-order terms should be grouped together, then all the second-order terms, etc. However, when applying perturbation theory to estimate probabilities, a common task in quantum mechanics, this automatic procedure can lead to nonpositive-definite probabilities. (Here, we are talking about probabilities that must be positive, as they can be inferred directly from the frequency of observed events, and not about intermediate functions, as Wigner quasiprobabilities, the inference of which from experimental data is a nontrivial task.) We demonstrate how to preserve the nonnegativity of probabilities at the cost of getting a bad grade in Calculus, and we show how the corrected expansion leads to a modification of the commonly accepted expressions for weak measurements, curing unphysical divergences at the same time. We provide the corrected formulas in the trivial case of an instantaneous interaction, the most commonly studied, and in the case of a finite-duration interaction during which the measured observable is not conserved. In the linear regime, the response of the detector is shown to be given by a generalized Kubo formula for a non-Hermitian perturbation.

View original: http://arxiv.org/abs/1211.5632