文档介绍：Contemporary Mathematics
A Selective History of the Stone-von Neumann Theorem
Jonathan Rosenberg
Abstract. The names of Stone and von Neumann are intertwined in what
is now known as the Stone-von Neumann Theorem. We discuss the origins of
this theorem, the contributions to it of Stone and von Neumann, the ways the
theorem has been reformulated, and some of the varied mathematics that has
grown out of it. At the end we discuss certain generalizations or analogues of
the Stone-von Neumann Theorem which are still subjects of current research,
such as a new C∗-algebra attached to the mutation relations of
quantum field theory, and supersymmetric versions of the theorem.
Introduction
As this volume is dedicated in honor of the hundredth birthdays of Stone and
von Neumann, it seems totally appropriate to consider the history of the one theo-
rem that bears both of their names. This history turns out to involve a surprisingly
large swath of 20th century mathematics: from quantization to group represen-
tations, operator algebras, and even number theory. For other points of view on
topics related to the Stone-von Neumann Theorem, I would mend [24] and
[59]. I want to thank Bob Doran and Dick Kadison anizing a fascinating
session in Baltimore and for originating the idea of this volume.
Some of the original sources analyzed here, especially in sections 1 and 2, are
in German, and some of the sources in section 4 are in French. To make this paper
understandable to as wide an audience as possible, I have translated some of the
relevant passages in footnotes. Any mistakes in the translations are clearly my
responsibility.
1. The early history of quantum mechanics,
and origins of the problem
I am petent to explain in detail the early history of quantum mechan-
ics, though there is no doubt that this is a fascinating chapter in the history of
science. So this section will be very brief, and will concentrate just on the problem
that mot