Weak values, 'negative probability' and the uncertainty principle

Abstract

A quantum transition can be seen as a result of interference between various pathways(e.g. Feynman paths) which can be labelled by a variable $f$. An attempt to determine the value of f without destroying the coherence between the pathways produces a weak value of $\bar{f}$. We show $\bar{f}$ to be an average obtained with amplitude distribution which can, in general, take negative values which, in accordance with the uncertainty principle, need not contain information about the actual range of the values $f$ which contribute to the transition. It is also demonstrated that the moments of such alternating distributions have a number of unusual properties which may lead to misinterpretation of the weak measurement results.We provide a detailed analysis of weak measurements with and without post-selection. Examples include the double slit diffraction experiment,weak von Neumann and von Neumann-like measurements, traversal time for an elastic collision, the phase time, the local angular momentum(LAM) and the 'three-box case' of {\it Aharonov et al}