Algebraic aspects of quantum indiscernibility

Abstract

We show that using quasi-set theory, or the theory of collections of indistinguishable objects, we can define an algebra that has most of the standard properties of an orthocomplete orthomodular lattice, which is the lattice of all closed subspaces of a Hilbert space. We call the mathematical structure so obtained $\mathfrak{I}$-lattice. After discussing some aspects of such a structure, we indicate the next problem of axiomatizing the corresponding logic, that is, a logic which has $\mathfrak{I}$-lattices as its Lindembaum algebra, which we postpone to a future work. Thus we conclude that the initial intuitions by Birkhoff and von Neumann that the ``logic of quantum mechanics" would be not classical logic, is consonant with the idea of considering indistinguishability right from the start, that is, as a primitive concept. In the first sections, we present the main motivations and a ``classical'' situation which mirrors that one we focus on the last part of the paper. This paper is our first analysis of the algebraic structure of indiscernibility.

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Décio Krause
Federal University of Santa Catarina

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References found in this work

Identity in physics: a historical, philosophical, and formal analysis.Steven French & Décio Krause - 2006 - New York: Oxford University Press. Edited by Decio Krause.
The problem of indistinguishable particles.Bas van Fraassen - 1984 - In James T. Cushing, Cornelius F. Delaney & Gary Gutting (eds.), Science and Reality: Recent Work in the Philosophy of Science. University of Notre Dame Press.
An axiom system for the modular logic.Jerzy Kotas - 1967 - Studia Logica 21 (1):17 - 38.
Axioms for Collections of Indistinguishable Objects.Décio Krause - 1996 - Logique Et Analyse 39 (153/154):69–93.

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