Nonprobabilistic typicality with application to quantum mechanics

TL;DR

This paper introduces a nonprobabilistic framework called typicality spaces to model phenomena where probabilities are undefined, applying it to quantum mechanics to propose an alternative formulation that avoids the measurement problem.

Contribution

It develops the concept of typicality spaces and applies them to quantum mechanics, offering a new formulation that sidesteps the measurement problem and differs from Bohmian mechanics.

Findings

Introduces typicality spaces replacing probability measures

Models quantum evolution as a typicalistic phenomenon

Proposes an alternative quantum mechanics formulation

Abstract

In this paper two hypotheses are developed. The first hypothesis is the existence of random phenomena/experiments in which the events cannot generally be assigned a definite probability but that nevertheless admit a class of nearly certain events. These experiments are referred to as \textit{typicalistic} (instead of probabilistic) experiments. As probabilistic experiments are represented by probability spaces, typicalistic experiments can be represented by \textit{typicality spaces}, where a typicality space is basically a probability space in which the probability measure has been replaced by a much less structured typicality measure $T$. The condition $T(A) \approx 1$ defines the typical sets, and a typicality space is related to a typicalistic experiment by associating the typical sets of the former with the nearly certain events of the latter. Some elements of a theory of typicality, including the definition of typicality spaces, are developed in the first part of the paper. The second hypothesis is that the evolution of a quantum particle (or of a system of quantum particles) can be considered a typicalistic phenomenon, so that it can be represented by the combination of typicality theory and quantum mechanics. The result is a novel formulation of quantum mechanics that does not present the measurement problem and that could be a viable alternative to Bohmian mechanics. This subject is developed in the second part of the paper.