A formal framework for quantum non-individuality

Graciela Domenech, Christian DE RONDE

Jonas R. Becker Arenhart

Décio Krause

In this article we try to justify a way of looking for an alternative approach to quantum mechanics based on a non-classical logic. The main motivation is both to accommodate a metaphysical view according to which quantum objects are non-individuals and to introduce, so to say, these objects within the formalism, for as we know, the usual formalism treats of states of quantum systems and not (directly) of quantum systems properly speaking. We consider two specific questions related to quantum theory, namely, entanglement and the indiscernibility of quanta. Our way of looking to these concepts motivate us to construct a metaphysical view according to which quantum objects do not present identity criterion (they are ‘non-individuals’), and then we try to construct a quantum theory based on such a view, which is grounded on a non-classical logic, termed non-reflexive. From the formal point of view, the main motivation comes from the fact that in using classical logic and classical set theory (such as ZFC), we are necessarily committed to individuals, and non-individuals can be considered only by assuming ‘strong’ symmetry principles, which mask the basic idea involving non-individuality. In the core of the article, due to questions of space, only the main philosophical arguments are presented, but in two appendixes we outline, firstly, quasi-set theory, the non-reflexive mathematical framework that sustain our view, and a bottom-top construction of a quantum mechanics based on quasi-set theory.

Décio Krause

Steven French, Décio Krause

Abstract This chapter presents the theory of quasi-sets, and argues that this offers an appropriate set-theoretic formalism for non-individual quantum objects. The basic idea is that in a quasi-set, there may exist elements for which the traditional concept of identity does not apply — these elements are called the ‘m-atoms’. Thus, a quasi-set may have a cardinal but not an associated ordinal. The other elements can be regarded as standard elements of a set. Hence, the theory encompasses a copy of Zermelo-Frankel set theory with Urelement. A number of alternatives are also presented with a view to possible physical applications, such as allowing the cardinality of a quasi-set to vary in time and thereby accommodating aspects of relativistic quantum theory. The formal framework is also applied to the development of quantum statistics, which can be naturally captured by the theory.

Décio Krause, Jonas R. Becker Arenhart