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OPERATORS AND REPRESENTATION THEORY : canonical models for algebras of operators arising in... quantum mechanics.
Math/Physics/Astronomy Library QA326 .J67 2017
Available
- Format:
- Book
- Author/Creator:
- Jørgensen, Palle E. T., 1947-
- Language:
- English
- Subjects (All):
- Operator algebras.
- Representations of algebras.
- Physical Description:
- xxv, 276 pages ; 24 cm
- Place of Publication:
- [Place of publication not identified] : DOVER PUBNS, 2017.
- Summary:
- Suitable for advanced undergraduates and graduate students in mathematics and physics, this three-part treatment of operators and representation theory begins with background material on definitions and terminology as well as on operators in Hilbert space. The introductory section concludes with a look at the imprimitivity theorem, which grounds in more mathematical language the work of Wigner on representations of the Poincaré and Galilei groups. The second part of the monograph addresses the algebras of operators in Hilbert space, broadening the mathematics used in earlier versions of quantum theory. There are many examples in which the Hamiltonian, the operator that translates a quantum system in time, can be written as a polynomial in elements of an underlying Lie algebra. This section deals with properties of such operators. Part 3 explores covariant representation and connections, with a particular focus on infinite-dimensional Lie algebras. Connections to mathematical physics are stressed throughout the text, which concludes with three helpful appendixes, including a Guide to the Literature. Book jacket.
- Contents:
- Part 1 Background Material
- Chapter 1 Introduction and Overview 2
- Chapter 2 Definitions and Terminology 6
- §2.1 Notation 8
- §2.2 Brackets 8
- §2.3 Representations 8
- §2.4 Units 9
- §2.5 Projective Representations 9
- §2.6 Central Extensions 10
- §2.7 Modules 10
- §2.8 Norm and Unboundedness 10
- §2.9 C*-norms 11
- §2.10 Groups, Algebras, and Operators 12
- Chapter 3 Operators in Hilbert Space 21
- §3.1 Domain and Graph 21
- §3.2 The Adjoint Operator 22
- §3.3 Selfadjoint Operators 23
- §3.4 The Operator T*T 23
- §3.5 The Polar Decomposition 24
- §3.6 Normal Operators 25
- §3.7 Functional Calculus 26
- §3.8 Real and Imaginary Parts 27
- §3.9 Positive Operators 28
- Chapter 4 The Imprimitivity Theorem 30
- §4.1 Integration and Unitary Equivalence 31
- §4.2 Applications 37
- Part 2 Algebras of Operators on Hilbert Space
- Chapter 5 Domains of Representations 44
- §5.1 Introduction 45
- §5.2 Selfadjoint Representations 48
- §5.3 The Derived Representation 51
- §5.4 Integrability of Selfadjoint Representations 57
- §5.5 Scalar Casimir Operator: A Special Case 61
- Chapter 6 Operators in the Enveloping Algebra 70
- §6.1 Central Elements 72
- §6.2 Second Order Elements 75
- §6.3 The Element x + iy 83
- §6.4 Higher Order Elements 96
- Chapter 7 Spectral Theory 111
- §7.0 Survey of Results 112
- §7.1 Discrete Spectrum 117
- §7.2 Trace Formula: High Energy Behavior 123
- §7.3 Continuous Spectrum 128
- §7.4 Lebesgue Spectrum 135
- Part 3 Covariant Representations and Connections
- Chapter 8 Infinite-Dimensional Lie Algebras 143
- §8.1 Rotation Algebras 144
- §8.2 Completions of 21<sub>c</sub> (L) 157
- §8.3 Extensions of Γ by u (A(R)) 169
- §8.4 Extensions and u(n, 1) 182
- §8.5 Vector Fields on the Circle 197
- §8.6 Noncommutative Tori 207.
- Notes:
- Includes bibliographical references and index.
- ISBN:
- 9780486815725
- 0486815722
- OCLC:
- 962822876
- Publisher Number:
- 99974740936
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