On the Weak Measurement of Velocity in Bohmian Mechanics

Detlef Dürr, Sheldon Goldstein, Nino Zanghı̀

C. Richard Leavens

Kazuki Fukuda, Jaeha Lee, Izumi Tsutsui

Weixiang Ye

We systematically analyze Holland's local expectation values within Bohmian mechanics, referring to them as weak actual values to emphasize their connection with weak measurement theory. We derive the exact time evolution equation for these quantities along a Bohmian trajectory and formally establish their correspondence with the real part of the weak value under position postselection. The explanatory power of this framework is demonstrated by revisiting a recent waveguide experiment: we show that the measured quantity corresponds to the imaginary part of the momentum weak value, which reflects the spatial decay of the wave field, not the particle velocity. This cleanly clarifies the distinct physical roles of the real and imaginary parts of weak values.

Travis Norsen, Ward Struyve

Joshua Foo, Estelle Asmodelle, Austin P. Lund +1 more

Bohmian mechanics is a nonlocal hidden-variable interpretation of quantum theory which predicts that particles follow deterministic trajectories in spacetime. Historically, the study of Bohmian trajectories has mainly been restricted to nonrelativistic regimes due to the widely held belief that the theory is incompatible with special relativity. Here, we present an approach for constructing the relativistic Bohmian-type velocity field of single particles. The advantage of our proposal is that it is operational in nature, grounded in weak measurements of the particle's momentum and energy. We apply our weak measurement formalism to obtain the relativistic spacetime trajectories of photons in a Michelson-Sagnac interferometer. The trajectories satisfy quantum-mechanical continuity and the relativistic velocity addition rule. We propose a modified Alcubierre metric which could give rise to these trajectories within the paradigm of general relativity.