Indefinite inner product spaces
By definition, an indefinite inner product space is a real or complex vector space together with a symmetric (in the complex case: hermi tian) bilinear form prescribed on it so that the corresponding quadratic form assumes both positive and negative values. The most important special case arises when a Hilbert space is considered as an orthogonal direct sum of two subspaces, one equipped with the original inner prod uct, and the other with -1 times the original inner product. The subject first …
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- Full title :Indefinite inner product spaces.
Summary
By definition, an indefinite inner product space is a real or complex vector space together with a symmetric (in the complex case: hermi tian) bilinear form prescribed on it so that the corresponding quadratic form assumes both positive and negative values. The most important special case arises when a Hilbert space is considered as an orthogonal direct sum of two subspaces, one equipped with the original inner prod uct, and the other with -1 times the original inner product. The subject first appeared thirty years ago in a paper of Dirac [1] on quantum field theory (d. also Pauli [lJ). Soon afterwards, Pontrja gin [1] gave the first mathematical treatment of an indefinite inner prod uct space. Pontrjagin was unaware of the investigations of Dirac and Pauli; on the other hand, he was inspired by a work of Sobolev [lJ, unpublished up to 1960, concerning a problem of mechanics. The attempts of Dirac and Pauli to apply the concept and elemen tary properties of indefinite inner product spaces to field theory have been renewed by several authors. At present it is not easy to judge which of their results will contribute to the final form of this part of physics. The following list of references should serve as a guide to the extensive literature: Bleuler [1], Gupta [lJ, Kallen and Pauli [lJ, Heisen berg [lJ-[4J, Bogoljubov, Medvedev and Polivanov [lJ, K.L. Nagy [lJ-[3], Berezin [lJ, Arons, Han and Sudarshan [1], Lee and Wick [1J.
Read More ↓Edition Statements
- Publication Date:1974.
Publication Statements
- Publisher(s):Springer
- Place of Publication:Berlin
- Language:eng
- Publication Date:1974.
The Physical Object
- Description:1 online resource (ix, 223 pages)
Edition Identifiers
- ISBN:9783642655678, 364265567X
- DOI:10.1007/978-3-642-65567-8
- oclc:ocn680712552
- LC Classification:QA322.5
- Dewey Decimal:512/.523
Table of Contents
I. Inner Product Spaces without Topology -- 1. Vector Spaces -- 2. Inner Products -- 3. Orthogonality -- 4. Isotropic Vectors -- 5. Maximal Non-degenerate Subspaces -- 6. Maximal Semi-definite Subspaces -- 7. Maximal Neutral Subspaces -- S. Projections of Vectors on Subspaces -- 9. Ortho-complemented Subspaces -- 10. Dual Pairs of Subspaces -- 11. Fundamental Decompositions -- Notes to Chapter I -- II. Linear Operators in Inner Product Spaces without Topology -- 1. Linear Operators in Vector Spaces -- 2. Isometric Operators -- 3. Symmetric Operators -- 4. Cayley Transformations -- 5. Principal Vectors of Cayley Transforms -- 6. Pairs of Inner Products: Semi-boundedness -- 7. Pairs of Inner Products: Sign -- 8. Plus-operators -- 9. Pesonen Operators -- 10. Fundamental Projectors -- 11. Fundamental Symmetries. Angular Operators -- Notes to Chapter II -- III. Partial Majorants and Admissible Topologies on Inner Product Spaces -- 1. Locally Convex Topologies on Vector Spaces -- 2. Partial Majorants. The Weak Topology -- 3. Metrizable Partial Majorants -- 4. The Polar of a Normed Partial Majorant -- 5. Admissible Topologies -- 6. Orthogonal Companions and Admissible Topologies -- 7. Pr…
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