Bose Algebras: The Complex and Real Wave Representations
The mathematics of Bose-Fock spaces is built on the notion of a commutative algebra and this algebraic structure makes the theory appealing both to mathematicians with no background in physics and to theorectical and mathematical physicists who will at on
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Tarben T. Nielsen
Bose Algebras: The Complex and Real Wave Representations
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Torben T. Nielsen Mathematical Institute, Arhus University and DIAX Telecommunications A/S Fselledvej 17, 7600 Struer, Denmark
Mathematics Subject Classification (1980): 81C99, 81D05, 47B47
ISBN 3-540-54041-5 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54041-4 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
Contents
O.
Introduction
1.
The Bose algebra
2.
Lifting operators to
3.
The coherent vectors in
4.
The Wick ordering and the Weyl relations
45
5.
Some special operators
53
6.
The complex wave representation
66
7.
The real wave representation
72
8.
Bose algebras of operators
79
9.
Wave representations of
4
fOK,
23
fK fK
* f(K+K)
33
89
10. Appendix 1: Halmos'
94
11. Appendix 2: Gaussian measures
96
12. References
130
13. Subject index
132
Introduction The aim of this paper is to present some results, which are consequences of an algebraization of the concept of Bose-Fock space. Though having their origin and principal area of application in theoretical physics, the usefulness of Bose-Fock spaces extends much further. They appear in digital signal processing [26], filtering theory [8] and in a new formalized description of the process of communication within the human society [22]. Bose-Fock spaces were brought to the attention of mathematicians by Irving E. Segal [15,16,17] and others (cf.[3]) in the sixties. In this paper both the Bose-Fock space and the accompanying creation and annihilation operator formalism have been combined to form a single well known mathematical object, namely an algebra. Given a Hilbert space *,, consider the free commutative algebra f * O generated by the linear space * and a fixed multiplicative unit 0, called the vacuum. The scalar product from * is extended over T0* in such a way that the adjoints to the operators of multiplication by elements in are derivations in the algebra. The algebra f provided O wi th such an extension of the scalar product is known as the Bose algebra with base *, (in physical literature the base space is called "the one-boson-space").
*
*
The main advantage of treating the Bose-Fock spaces in this manner is th
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