Wigner's theorem for an infinite set

TL;DR

This paper extends Wigner's theorem to infinite sets by characterizing the automorphism group of the orthomodular poset of set decompositions, linking it to the permutation group of the set.

Contribution

It proves a version of Wigner's theorem for infinite sets, showing the automorphism group of the poset of set decompositions is isomorphic to the permutation group.

Findings

Automorphism group of Fact(X) is isomorphic to permutation group of X

Fact(X) is atomistic in a strong sense

Provides foundational insights into set decompositions and their symmetries

Abstract

It is well known that the closed subspaces of a Hilbert space form an orthomodular lattice. Elements of this orthomodular lattice are the propositions of a quantum mechanical system represented by the Hilbert space, and by Gleason's theorem atoms of this orthomodular lattice correspond to pure states of the system. Wigner's theorem says that the automorphism group of this orthomodular lattice corresponds to the group of unitary and anti-unitary operators of the Hilbert space. This result is of basic importance in the use of group representations in quantum mechanics. The closed subspaces $A$ of a Hilbert space $\mathcal{H}$ correspond to direct product decompositions $\mathcal{H}\simeq A\times A^\perp$ of the Hilbert space, a result that lies at the heart of the superposition principle. It has been shown that the direct product decompositions of any set, group, vector space, and topological space form an orthomodular poset. This is the basis for a line of study in foundational quantum mechanics replacing Hilbert spaces with other types of structures. It is the purpose of this note to prove a version of Wigner's theorem: for an infinite set $X$, the automorphism group of the orthomodular poset Fact $(X)$ of direct product decompositions of $X$ is isomorphic to the permutation group of $X$. The structure Fact $(X)$ plays the role for direct product decompositions of a set that the lattice of equivalence relations plays for surjective images of a set. So determining its automorphism group is of interest independent of its application to quantum mechanics. Other properties of Fact $(X)$ are determined in proving our version of Wigner's theorem, namely that Fact $(X)$ is atomistic in a very strong way.