Non-Individuality in the Formal Structure of Quantum Mechanics

D�cio Krause, Steven French

D. A. Slavnov

We describe a scheme for constructing quantum mechanics in which the Hilbert space and linear operators are only secondary structures of the theory, while the primary structures are the elements of a non-commutative algebra (observables) and the functionals on this algebra, associated with the results of a single observation.

Christian DE RONDE, Graciela Domenech, Federico Holik +1 more

The concept of individuality in quantum mechanics shows radical differences from the one used in classical physics. In particular, it is not possible to consider the fundamental particles described by quantum theory as individual distinguishable objects. In this paper we present arguments in favor of quantum non-individuality, which -in addition to those based on quantum statistics- relate to the Kochen-Specker theorem and the principle of superposition. Then, we analyze the possibility of referring to 'possible individuals' instead of 'actual individuals', and show that the Modal-Kochen-Specker theorem precludes this interpretational move.

Jaison Schinaider

In this article, we discuss some questions about the nature of elementary particles treated by quantum mechanics, in particular related to the concepts of identity and individuality of these particles. We started briefly exposing the philosophical and formal concepts of the identity and individuality, and then show how these notions are problematic when applied to elementary particles such as electrons, protons and neutrons. In particular, we emphasize that both philosophy, logic and set theory (and thus the mathematics) admit the usual assumption that things have a ‘type’ of identity and individuality (i.e., are individuals), in the sense that objects which have all the same properties are the same object (are equal). Nevertheless, we show that in the quantum universe is possible to find objects that share all their properties, but are not just one, constituting in something like “nonindividuals” (thesis defended by many physicists and philosophers of science). In sequence, we show how classical mathematics — which, as we said, assumes an individuality to your entities — handle this situation, in particular admitting assumptions external to the physical theories. To avoid this procedure, and look for a formalism more ‘natural’ and appropriate to work with these quantum characteristics, we suggest the use of a non-classical set theory called the quasi-set theory, based on a non-reflective logics, which admits objects devoid of identity and individuality ab initio. Finally, we show another application of this theory: the search to a quasi-set mathematical structure that describes the ‘behaviour’ of non-relativistic quantum theory, and discuss some advantages of this ‘quasi-structure’ to a classical one.