UID:
almahu_9947367770002882
Format:
1 online resource (447 p.)
ISBN:
1-281-79791-X
,
9786611797911
,
0-08-088021-5
Series Statement:
North-Holland mathematical library ; vol. 36
Content:
This monograph contains a functional analytic introduction to Dirac's formalism. The first part presents some new mathematical notions in the setting of triples of Hilbert spaces, mentioning the concept of Dirac basis. The second part introduces a conceptually new theory of generalized functions, integrating the notions of the first part.The last part of the book is devoted to a mathematical interpretation of the main features of Dirac's formalism. It involves a pairing between distributional bras and kets, continuum expansions and continuum matrices.
Note:
Description based upon print version of record.
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Front Cover; A Mathematical Introduction to Dirac's Formalism; Copyright Page; Contents; Section A: Some basic concepts and developments in Sobolev triples; CHAPTER I. CARLEMAN OPERATORS; I.1. Some measure theoretical prerequisites; I.2. Carleman operators; I.3. Operators of Carleman type; I.4. Strong Carleman operators; Some bibliographical notes and comments; CHAPTER II. A MEASURE THEORETICAL SOBOLEV LEMMA; II.1. An elementary illustration; II.2. Sobolev triples of Hilbert spaces; II.3. Federer measure spaces; II.4. A measure theoretical generalization of the Sobolev embedding theorem
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II.5. Some applicationsSome bibliographical notes and comments; CHAPTER III. DIRAC BASES; III.1. The concept of Dirac basis; III.2. Canonical Dirac bases; III.3. Dirac-Riesz bases; III.4. Canonical Dirac-Riesz bases; Some bibliographical notes and comments; CHAPTER IV. THE GENERALIZED EIGENVALUE PROBLEM FOR SELF-ADJOINT OPERATORS; IV.l. Commutative multiplicity theory; IV.2. An application of the measure theoretical Sobolev lemma; IV.3. A solution; IV.4. Some illustrations; Some bibliographical notes and comments; CHAPTER V. DIRECT RESOLUTIONS IN SOBOLEV TRIPLES
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V.l. Preliminaries and summaryV.2. Direct resolutions of the identity; V.3. A general construction of a direct resolution; V.4. Equivalence classes of direct resolutions, canonical direct resolutions; V.5. Generalized eigenprojections related to commutative von Neumann algebras; Some bibliographical notes and comments; Section B: A theory of generalized functions; CHAPTER I. ANALYTICITY SPACES, TRAJECTORY SPACES AND THEIR DUALITY; I.1. The analyticity space SX,A; I.2. The trajectory space T X,A; I.3. Pairing and duality of SX, A and TX,A; I.4. Sequence space representation
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Some bibliographical notes and commentsCHAPTER II. LINEAR MAPPINGS, TENSOR PRODUCTS AND KERNEL THEOREMS; II.1. Continuous linear mappings between analyticity and trajectory spaces; II.2. Topological tensor products of analyticity and trajectory spaces; II.3. Kernel theorems; II.4. Matrix representations; Some bibliographical notes and comments; CHAPTER III. ILLUSTRATIONS OF ANALYTICITY SPACES AND TRAJECTORY SPACES; III.1. Analyticity spaces based on the Laplacian operator; III.2. The spaces Sßa of Gelfand and Shilov; III.3. Analyticity spaces based on classical polynomials
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III.4. Analyticity spaces related to unitary representations of Lie groupsSome bibliographical notes and comments; Chapter IV. THE CONCEPT OF DIRAC BASIS LIFTED TO TRAJECTORY SPACES; IV.l. A measure theoretical Sobolev lemma for analyticity spaces; IV.2. Dirac bases in trajectory spaces; IV.3. Canonical Dirac bases in trajectory spaces; IV.4. The generalized eigenvalue problem for self-adjoint operators solved in the setting of trajectory spaces; Some bibliographical notes and comments; Section C: A mathematical interpretation of Dirac's formalism
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CHAPTER I. DIRAC'S FORMALISM ACCORDING TO DIRAC AND ITS RELATIONS WITH LINEAR ALGEBRA
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English
Additional Edition:
ISBN 0-444-70127-3
Language:
English
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