A new quantum operator for distance

Abstract

We introduce a new semi-relativistic quantum operator for the length of the worldline a particle traces out as it moves. In this article the operator is constructed in a heuristic way and some of its elementary properties are explored. The operator ends up depending in a very complicated way on the potential of the system it is to act on so as a proof of concept we use it to analyze the expected distance traveled by a free Gaussian wavepacket with some initial momentum. It is shown in this case that the distance such a particle travels becomes light-like as its mass vanishes and agrees with the classical result for macroscopic masses. This preliminary result has minor implications for the Weak Equivalence Principle (WEP) in quantum mechanics. In particular it shows that the logical relationship between two formulations of the WEP in classical mechanics extends to quantum mechanics. That our result is qualitatively consistent with the work of others emboldens us to start the task of evaluating the new operator in non-zero potentials. However, we readily acknowledge that the looseness in the definition of our operator means that all of our so-called results are highly speculative. Plans for future work with the new operator are discussed in the last section.

Proposal to observe paths superpositions in a double-slit setup

“…as a sum over classical paths [7]. In more complex situations [8][9][10][11][12][13], several Feynman paths compatible with pre-and post-selection can interfere at the position of a weakly coupled probe, and may even result in a vanishing weak value (e.g., [14]). In a generic situation, Feynman paths interfere as the system evolves, so that reconstructing a weak trajectory from observed pointers is expected to be a difficult task.…”