Salecker-Wigner-Peres clock, Feynman paths, and a tunneling time that should not exist

Abstract

The Salecker-Wigner-Peres (SWP) clock is often used to determine the duration a quantum particle is supposed to spend is a specified region of space $\Om$. By construction, the result is a real positive number, and the method seems to avoid the difficulty of introducing complex time parameters, which arises in the Feynman paths approach. However, it tells little about about the particle's motion. We investigate this matter further, and show that the SWP clock, like any other Larmor clock, correlates the rotation of its angular momentum with the durations, $\tau$, which the Feynman paths spend in $\Om$, thereby destroying interference between different durations. An inaccurate weakly coupled clock leaves the interference almost intact, and the need to resolve the resulting "which way?" problem is one of the main difficulties at the centre of the "tunnelling time" controversy. In the absence of a probability distribution for the values of $\tau$, the SWP results are expressed in terms of moduli of the "complex times", given by the weighted sums of the corresponding probability amplitudes. It is shown that over-interpretation of these results, by treating the SWP times as physical time intervals, leads to paradoxes and should be avoided. We also analyse various settings of the SWP clock, different calibration procedures, and the relation between the SWP results and the quantum dwell time. The cases of stationary tunnelling and tunnel ionisation are considered in some detail. Although our detailed analysis addresses only one particular definition of the duration of a tunnelling process, it also points towards the impossibility of uniting various time parameters, which may occur in quantum theory, within the concept of a single "tunnelling time".