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Group representation for quantum theory [electronic resource]

Group representation for quantum theory [electronic resource]

Material type
E-Book(소장)
Personal Author
Hayashi, Masahito.
Title Statement
Group representation for quantum theory [electronic resource] / Masahito Hayashi.
Publication, Distribution, etc
Cham :   Springer,   c2017.  
Physical Medium
1 online resource (xxviii, 338 p.) : ill.
ISBN
9783319449043 9783319449067 (eBook)
요약
This book explains the group representation theory for quantum theory in the language of quantum theory. As is well known, group representation theory is very strong tool for quantum theory, in particular, angular momentum, hydrogen-type Hamiltonian, spin-orbit interaction, quark model, quantum optics, and quantum information processing including quantum error correction. To describe a big picture of application of representation theory to quantum theory, the book needs to contain the following six topics, permutation group, SU(2) and SU(d), Heisenberg representation, squeezing operation, Discrete Heisenberg representation, and the relation with Fourier transform from a unified viewpoint by including projective representation. Unfortunately, although there are so many good mathematical books for a part of six topics, no book contains all of these topics because they are too segmentalized. Further, some of them are written in an abstract way in mathematical style and, often, the materials are too segmented. At least, the notation is not familiar to people working with quantum theory. Others are good elementary books, but do not deal with topics related to quantum theory. In particular, such elementary books do not cover projective representation, which is more important in quantum theory. On the other hand, there are several books for physicists. However, these books are too simple and lack the detailed discussion. Hence, they are not useful for advanced study even in physics. To resolve this issue, this book starts with the basic mathematics for quantum theory. Then, it introduces the basics of group representation and discusses the case of the finite groups, the symmetric group, e.g. Next, this book discusses Lie group and Lie algebra. This part starts with the basics knowledge, and proceeds to the special groups, e.g., SU(2), SU(1,1), and SU(d). After the special groups, it explains concrete applications to physical systems, e.g., angular momentum, hydrogen-type Hamiltonian, spin-orbit interaction, and quark model. Then, it proceeds to the general theory for Lie group and Lie algebra. Using this knowledge, this book explains the Bosonic system, which has the symmetries of Heisenberg group and the squeezing symmetry by SL(2,R) and Sp(2n,R). Finally, as the discrete version, this book treats the discrete Heisenberg representation which is related to quantum error correction. To enhance readers' undersnding, this book contains 54 figures, 23 tables, and 111 exercises with solutions.
General Note
Title from e-Book title page.  
Content Notes
Foundation of Quantum Theory -- Group Representation -- Representations of Lie Group and Lie Algebra (Basics) -- Representations of Lie Group and Lie Algebra (Special Case) -- Representations of Lie Group and Lie Algebra (General Case) -- Bosonic System -- Discretization of Bosonic System.
Bibliography, Etc. Note
Includes bibliographical references (p. 321-332) and index.
이용가능한 다른형태자료
Issued also as a book.  
Subject Added Entry-Topical Term
Representations of groups. Quantum theory.
Short cut
URL
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020 ▼a 9783319449043
020 ▼a 9783319449067 (eBook)
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050 4 ▼a QC173.96-174.52
082 0 4 ▼a 512.22 ▼2 23
084 ▼a 512.22 ▼2 DDCK
090 ▼a 512.22
100 1 ▼a Hayashi, Masahito.
245 1 0 ▼a Group representation for quantum theory ▼h [electronic resource] / ▼c Masahito Hayashi.
260 ▼a Cham : ▼b Springer, ▼c c2017.
300 ▼a 1 online resource (xxviii, 338 p.) : ▼b ill.
500 ▼a Title from e-Book title page.
504 ▼a Includes bibliographical references (p. 321-332) and index.
505 0 ▼a Foundation of Quantum Theory -- Group Representation -- Representations of Lie Group and Lie Algebra (Basics) -- Representations of Lie Group and Lie Algebra (Special Case) -- Representations of Lie Group and Lie Algebra (General Case) -- Bosonic System -- Discretization of Bosonic System.
520 ▼a This book explains the group representation theory for quantum theory in the language of quantum theory. As is well known, group representation theory is very strong tool for quantum theory, in particular, angular momentum, hydrogen-type Hamiltonian, spin-orbit interaction, quark model, quantum optics, and quantum information processing including quantum error correction. To describe a big picture of application of representation theory to quantum theory, the book needs to contain the following six topics, permutation group, SU(2) and SU(d), Heisenberg representation, squeezing operation, Discrete Heisenberg representation, and the relation with Fourier transform from a unified viewpoint by including projective representation. Unfortunately, although there are so many good mathematical books for a part of six topics, no book contains all of these topics because they are too segmentalized. Further, some of them are written in an abstract way in mathematical style and, often, the materials are too segmented. At least, the notation is not familiar to people working with quantum theory. Others are good elementary books, but do not deal with topics related to quantum theory. In particular, such elementary books do not cover projective representation, which is more important in quantum theory. On the other hand, there are several books for physicists. However, these books are too simple and lack the detailed discussion. Hence, they are not useful for advanced study even in physics. To resolve this issue, this book starts with the basic mathematics for quantum theory. Then, it introduces the basics of group representation and discusses the case of the finite groups, the symmetric group, e.g. Next, this book discusses Lie group and Lie algebra. This part starts with the basics knowledge, and proceeds to the special groups, e.g., SU(2), SU(1,1), and SU(d). After the special groups, it explains concrete applications to physical systems, e.g., angular momentum, hydrogen-type Hamiltonian, spin-orbit interaction, and quark model. Then, it proceeds to the general theory for Lie group and Lie algebra. Using this knowledge, this book explains the Bosonic system, which has the symmetries of Heisenberg group and the squeezing symmetry by SL(2,R) and Sp(2n,R). Finally, as the discrete version, this book treats the discrete Heisenberg representation which is related to quantum error correction. To enhance readers' undersnding, this book contains 54 figures, 23 tables, and 111 exercises with solutions.
530 ▼a Issued also as a book.
538 ▼a Mode of access: World Wide Web.
650 0 ▼a Representations of groups.
650 0 ▼a Quantum theory.
856 4 0 ▼u https://oca.korea.ac.kr/link.n2s?url=https://doi.org/10.1007/978-3-319-44906-7
945 ▼a KLPA
991 ▼a E-Book(소장)

Holdings Information

No. Location Call Number Accession No. Availability Due Date Make a Reservation Service
No. 1 Location Main Library/e-Book Collection/ Call Number CR 512.22 Accession No. E14014311 Availability Loan can not(reference room) Due Date Make a Reservation Service M

Contents information

Table of Contents

Intro -- Preface -- Preface to the Japanese Version -- Contents -- About the Author -- Symbols -- 1 Mathematical Foundation for Quantum System -- 1.1 System, State, and Measurement -- 1.2 Composite System -- 1.2.1 Tensor Product System -- 1.2.2 Entangled State -- 1.3 Many-Body System -- 1.4 Hamiltonian -- 1.4.1 Dynamics and Hamiltonian -- 1.4.2 Simultaneous Diagonalization -- 1.4.3 Relation to Representation -- 1.5 Relation to Symmetry -- 1.6 Remark for Unbounded Case* -- 2 Group Representation Theory -- 2.1 Group and Homogeneous Space -- 2.1.1 Group -- 2.1.2 Homogeneous Space -- 2.2 Extension of Group -- 2.2.1 General Case -- 2.2.2 Central Extension of a Commutative Group -- 2.2.3 Examples for Central Extensions -- 2.3 Representation and Projective Representation -- 2.3.1 Definitions of Representation and Projective Representation -- 2.3.2 Schur''s Lemma -- 2.3.3 Representations of Direct Product Group -- 2.4 Projective Representation and Extension of Group -- 2.4.1 Factor System of Projective Representation -- 2.4.2 Irreducibility and Projective Representation -- 2.4.3 Extension by U(1) -- 2.5 Semi Direct Product and Its Representation -- 2.5.1 From HK to K and H -- 2.5.2 From K and H to HK* -- 2.6 Real Representation and Complex Conjugate Representation -- 2.6.1 Real Linear Space and Its Complexification -- 2.6.2 Real Representation -- 2.6.3 Complex Conjugate Representation -- 2.7 Representation on Composite System -- 2.8 Fourier Transform for Finite Group -- 2.8.1 Discrete Fourier Transform -- 2.8.2 Character and Orthogonality -- 2.8.3 Fourier Transform -- 2.9 Representation of Permutation Group and Young Diagram -- 2.9.1 Young Diagram and Young Tableau -- 2.9.2 Permutation Group and Young Diagram -- 2.9.3 Plancherel Measure -- 3 Foundation of Representation Theory of Lie Group and Lie Algebra -- 3.1 Lie Group -- 3.1.1 Basic Examples -- 3.1.2 Symmetry in Analytical Mechanics -- 3.1.3 Complex Lie Group -- 3.1.4 Other Examples of Real Lie Groups -- 3.2 Lie Algebra -- 3.3 Relation Between Lie Group and Lie Algebra I -- 3.3.1 Infinitesimal Transformation and Lie Algebra -- 3.3.2 Examples -- 3.3.3 Central Extension of Real Lie Algebra -- 3.4 Representation of Lie Algebra -- 3.4.1 Representation of Real Lie Algebra -- 3.4.2 Real Representation -- 3.4.3 Representation of Complex Lie Algebra -- 3.4.4 Adjoint Representation -- 3.4.5 Projective Representation -- 3.4.6 Semi Direct Product Lie Algebra and Representation -- 3.5 Killing Form and Compactness -- 3.5.1 Killing Form -- 3.5.2 Compactness of Real Lie Algebra mathfrakg -- 3.5.3 Casimir Operator -- 3.6 Relation Between Lie Group and Lie Algebra II -- 3.6.1 Universal Covering Group -- 3.6.2 Relation to Representation -- 3.6.3 Projective Representation -- 3.6.4 Representation for Complex Lie Groups -- 3.7 Invariant Measures on Group and Homogeneous Space -- 3.8 Fourier Transform on Lie Group -- 3.8.1 Commutative Case -- 3.8.2 Non-commutative Case -- 4 Representations of Typical Lie Groups and Typi.

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