How the Weak Variance of Momentum Can Turn Out to be Negative

TL;DR

This paper explores the concept of weak variance of momentum in quantum mechanics, revealing how its negativity relates to the Wigner function's negativity and connecting it to quantum potential and subquantum theories.

Contribution

It introduces a position-postselected weak variance of momentum based on the Wigner function, linking it to measurable quantities and the imaginary part of weak values.

Findings

Weak variance can be negative due to Wigner function negativity.

Negative weak variances relate to subquantum theories.

Connection established between weak variance, quantum potential, and determinism.

Abstract

Weak values are average quantities,therefore investigating their associated variance is crucial in understanding their place in quantum mechanics. We develop the concept of a position-postselected weak variance of momentum as cohesively as possible, building primarily on material from Moyal (Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, Cambridge, 1949) and Sonego (Found Phys 21(10):1135, 1991) . The weak variance is defined in terms of the Wigner function, using a standard construction from probability theory. We show this corresponds to a measurable quantity, which is not itself a weak value. It also leads naturally to a connection between the imaginary part of the weak value of momentum and the quantum potential. We study how the negativity of the Wigner function causes negative weak variances, and the implications this has on a class of `subquantum' theories. We also discuss the role of weak variances in studying determinism, deriving the classical limit from a variational principle.