Von Neumann algebras, affiliated operators and representations of the Heisenberg relation
type: CreativeWork
description: Von Neumann algebras are self-adjoint, strong-operator closed subalgebras (containing the identity operator) of the algebra of all bounded operators on a Hilbert space. Factors are von Neumann algebras whose centers consist of scalar multiples of the identity operator. In this thesis, we study unbounded operators affiliated with finite von Neumann algebras, in particular, factors of Type II1. Such unbounded operators permit all the formal algebraic manipulations used by the founders of quantum mechanics in their mathematical formulation, and surprisingly, they form an algebra. The operators affiliated with an infinite von Neumann algebra never form such an algebra. The Heisenberg commutation relation, QP -- PQ = --ihI , is the most fundamental relation of quantum mechanics. Heisenberg's encoding of the ad-hoc quantum rules in this simple relation embodies the characteristic indeterminacy and uncertainty of quantum theory. Representations of the Heisenberg relation in various mathematical structures are discussed. In particular, we answer the question --- whether the Heisenberg relation can be realized with unbounded operators in the algebra of operators affiliated with a factor of type II1.
abstract: Von Neumann algebras are self-adjoint, strong-operator closed subalgebras (containing the identity operator) of the algebra of all bounded operators on a Hilbert space. Factors are von Neumann algebras whose centers consist of scalar multiples of the identity operator. In this thesis, we study unbounded operators affiliated with finite von Neumann algebras, in particular, factors of Type II1. Such unbounded operators permit all the formal algebraic manipulations used by the founders of quantum mechanics in their mathematical formulation, and surprisingly, they form an algebra. The operators affiliated with an infinite von Neumann algebra never form such an algebra. The Heisenberg commutation relation, QP -- PQ = --ihI , is the most fundamental relation of quantum mechanics. Heisenberg's encoding of the ad-hoc quantum rules in this simple relation embodies the characteristic indeterminacy and uncertainty of quantum theory. Representations of the Heisenberg relation in various mathematical structures are discussed. In particular, we answer the question --- whether the Heisenberg relation can be realized with unbounded operators in the algebra of operators affiliated with a factor of type II1.
author: Liu, Zhe
keywords: Theoretical Mathematics, Mathematics, Physics, Quantum
date published: 2010-01-01T08:00:00Z
date modified: 2026-07-24T05:22:31Z
provider: University of New Hampshire
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of New Hampshire