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Hindustan Book Agency

Physical Applications of Algebras of Unbounded Operators

Fabio Bagarello

Published Jan 1, 2007• 0 citations
Publisher

Abstract

During the past 20 years a long series of papers concerning algebras of unbounded operators appeared in the literature, papers which, though being originally motivated by physical arguments, contain almost no physics at all. On the contrary the mathematical aspects of these algebras have been analyzed in many details and this analysis produced, up to now, the monographes [32] and [2]. Some physics appeared first in [28] and [31], in the attempt to describe systems with a very large (10 24 ) number of degrees of freedom, following some general ideas originally proposed in the famous Haag and Kastler’s paper, [27], on QM ∞ .

DOI
10.1007/978-93-86279-33-0_4

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During the past 20 years a long series of papers concerning algebras of unbounded operators appeared in the literature, papers which, though being originally motivated by physical arguments, contain almost no physics at all. On the contrary the mathematical aspects of these algebras have been analyzed in many details and this analysis produced, up to now, the monographes [32] and [2]. Some physics appeared first in [28] and [31], in the attempt to describe systems with a very large (10 24 ) number of degrees of freedom, following some general ideas originally proposed in the famous Haag and Kastler’s paper, [27], on QM ∞ .
Forty years ago, Haag and Kastler [27] introduced the algebraic approach to Quantum Field Theory, and soon their method was extended to statistical mechanics. The rationale was to free the theory from particular realizations, linked to specific models, and to focus on the basic structure, namely, the algebra of observables and states on it. This more abstract language, exploiting the deep mathematical theories of C*-algebras and von Neumann algebras, soon became the standard approach to the mathematically rigorous description of physical systems with infinitely many degrees of freedom. Textbooks flourished and, among many authors, Gérard Emch argued forcefully for the new approach [6].
Introduction One of the most influencial contributions of H ermann W eyl to mathematical physics has been his paper Gruppentheorie und Quantenmechanik [W e 27] from 1927 and its extended version, the book [W e 28] which was published a year later and carries the same title. The main topic of this part of H ermann W eyl's work is the mathematics of quantum mechanics. After the fundamental papers by H eisenberg and S chrödinger on the foundations of quantum mechanics had appeared in the twenties of the last century this was the central question studied in mathematical physics at that time and which to a certain degree still is present in all attempts to construct mathematically rigorous theories unifying quantum mechanics and general realtivity. In his article Gruppentheorie und Quantenmechanik , H ermann W eyl essentially introduced two novel aspects to the mathematics of quantum mechanics, namely the following: (i) The representation theory of (compact) Lie groups on Hilbert spaces was applied to mathematically determine atomic spectra. (ii) A conceptually clear quantization method was proposed which associates quantum mechanical operators to classical observables which mathematically are represented by appropriate functions of the space and momentum variables. Nowadays, this quantization scheme is named after his inventor Weyl quantization. In this paper I will elaborate only on the second aspect, since the representation theory of compact Lie groups has already been covered in detail in other contributions to these proceedings.
Quantum theory and functional analysis were created and put into essentially their final form during similar periods ending around 1930. Each was also a key outcome of the major revolutions that both physics and mathematics as a whole underwent at the time. This paper studies their interaction in this light, emphasizing the leading roles played by Hilbert in preparing the ground and by von Neumann in bringing them together during the crucial year of 1927, when he gave the modern, abstract definition of a Hilbert space and applied this concept to quantum mechanics (consolidated in his famous monograph from 1932). Subsequently, I give a very brief overview of three areas of functional analysis that have had fruitful interactions with quantum theory since 1932, namely unbounded operators, operator algebras, and distributions. The paper closes with some musings about the role of functional analysis in actual physics.