What really matters in Hilbert-space stochastic processes

TL;DR

This paper explores the connection between continuous and discontinuous stochastic processes in Hilbert space, emphasizing the importance of operator choice, especially position, in quantum measurement problems.

Contribution

It demonstrates that every continuous process has a discontinuous parent process converging to it, highlighting the significance of operator selection in quantum measurement.

Findings

Discontinuous processes can approximate continuous ones in Hilbert space.

Position operator sharpness plays a crucial role in quantum measurement.

The relationship between stochastic processes informs quantum measurement solutions.

Abstract

The relationship between discontinuous and continuous stochastic processes in Hilbert space is investigated. It is shown that for any continuos process there is a parent discontinuous process, that becomes the continuous one in the proper infinite frequency limit. From the point of view of solving the quantum measurement problem, what really matters is the choice of the set of operators whose value distributions are made sharp. In particular, the key role of position sharping is emphasized.