Quantum theory and functional analysis
Abstract
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.
Cite
@article{arxiv.1911.06630,
title = {Quantum theory and functional analysis},
author = {Klaas Landsman},
journal= {arXiv preprint arXiv:1911.06630},
year = {2019}
}
Comments
17 pages, to appear in the Oxford Handbook of the History of Interpretations and Foundations of Quantum Mechanics, ed. O. Freire (Oxford University Press, 2021)