Algebraic aspects of quantum indiscernibility

Krause, Decio and Feitosa, Hercules de Araujo (2008) 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 (in a preliminary form)
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 (a Boolean algebra), 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.