A formal framework for the study of the notion of undefined particle number in quantum mechanics

TL;DR

This paper introduces a formal set-theoretical framework to rigorously describe states with undefined particle numbers in quantum mechanics, addressing foundational issues about particle identity and superpositions.

Contribution

It provides a novel logical and mathematical foundation for understanding undefined particle number states, integrating quantum concepts into the foundations of mathematics.

Findings

The framework accurately models quantum superpositions.

It offers a precise logical interpretation of undefined particle number states.

The system addresses foundational problems in quantum particle identity.

Abstract

It is usually stated that quantum mechanics presents problems with the identity of particles, the most radical position -supported by E. Schrodinger- asserting that elementary particles are not individuals. But the subject goes deeper, and it is even possible to obtain states with an undefined particle number. In this work we present a set theoretical framework for the description of undefined particle number states in quantum mechanics which provides a precise logical meaning for this notion. This construction goes in the line of solving a problem posed by Y. Manin, namely, to incorporate quantum mechanical notions at the foundations of mathematics. We also show that our system is capable of representing quantum superpositions.