A New Quantum Operator for Distance

TL;DR

This paper introduces a semi-relativistic quantum operator for measuring the length of a particle's worldline, explores its properties, and demonstrates its behavior for a free Gaussian wavepacket, with implications for quantum versions of the Weak Equivalence Principle.

Contribution

It presents a novel quantum operator for worldline length, constructed heuristically, and explores its properties and preliminary implications for quantum gravity and the Weak Equivalence Principle.

Findings

Distance becomes light-like as mass approaches zero.

Operator agrees with classical results for large masses.

Results are highly speculative due to operator's loose definition.

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.