A discussion on particle number and quantum indistinguishability

TL;DR

This paper examines the limitations of quasiset theory in representing quantum indistinguishability, especially regarding systems with undefined particle number, and proposes an axiomatic variant to address this issue.

Contribution

It demonstrates that standard quasiset theory cannot fully capture quantum non-individuality, and introduces an axiomatic variant where quasicardinal is not primitive.

Findings

Q cannot represent systems with undefined particle number

A variant of Q without primitive quasicardinal is proposed

Results suggest the need for alternative theories for quantum collections

Abstract

The concept of individuality in quantum mechanics shows radical differences from the concept of individuality in classical physics, as E. Schroedinger pointed out in the early steps of the theory. Regarding this fact, some authors suggested that quantum mechanics does not possess its own language, and therefore, quantum indistinguishability is not incorporated in the theory from the beginning. Nevertheless, it is possible to represent the idea of quantum indistinguishability with a first order language using quasiset theory (Q). In this work, we show that Q cannot capture one of the most important features of quantum non individuality, which is the fact that there are quantum systems for which particle number is not well defined. An axiomatic variant of Q, in which quasicardinal is not a primitive concept (for a kind of quasisets called finite quasisets), is also given. This result encourages the searching of theories in which the quasicardinal, being a secondary concept, stands undefined for some quasisets, besides showing explicitly that in a set theory about collections of truly indistinguishable entities, the quasicardinal needs not necessarily be a primitive concept.