Unitary symmetry and combinatorics
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| Format: | Buch |
| Sprache: | English |
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New Jersey [u.a.]
World Scientific
2008
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| 005 | 20120419 | ||
| 007 | t| | ||
| 008 | 081209s2008 xx |||| 00||| eng d | ||
| 020 | |a 9812814728 |9 981-281472-8 | ||
| 020 | |a 9789812814722 |9 978-981-281472-2 | ||
| 035 | |a (OCoLC)273893634 | ||
| 035 | |a (DE-599)GBV583137725 | ||
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| 100 | 1 | |a Louck, James D. |d 1928- |e Verfasser |0 (DE-588)141071044 |4 aut | |
| 245 | 1 | 0 | |a Unitary symmetry and combinatorics |c James D. Louck |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2008 | |
| 300 | |a XXI, 619 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. 597 - 609 | ||
| 650 | 4 | |a Combinatorial analysis | |
| 650 | 4 | |a Eightfold way (Nuclear physics) | |
| 650 | 0 | 7 | |a Kombinatorik |0 (DE-588)4031824-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Kombinatorische Analysis |0 (DE-588)4164746-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Unitäre Symmetrie |0 (DE-588)4627368-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Kernphysik |0 (DE-588)4030340-8 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Kombinatorik |0 (DE-588)4031824-2 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Kombinatorische Analysis |0 (DE-588)4164746-4 |D s |
| 689 | 1 | 1 | |a Kernphysik |0 (DE-588)4030340-8 |D s |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-017010287 | |
Datensatz im Suchindex
| DE-19_call_number | 1705/SK 950 L886 1705/M 10 Lou=LS Theoretical Solid State Physics |
|---|---|
| DE-19_location | 95 |
| DE-BY-UBM_katkey | 4595670 |
| DE-BY-UBM_media_number | 99995295594 41618242510012 |
| _version_ | 1823054964678721536 |
| adam_text | Contents
Preface
vii
Notation xxi
1
Quantum
Angular
Momentum
1
1.1
Background and Viewpoint
................. 1
1.1.1
Euclidean and Cartesian 3-space
.......... 1
1.1.2
Newtonian physics
.................. 9
1.1.3
Nonrelativistic quantum physics
.......... 10
1.1.4
Unitary frame rotations
............... 17
1.2
Abstract Angular Momentum
................ 28
1.2.1
Brief background and history
............ 28
1.2.2
One angular momentum
............... 30
1.2.3
Two angular momenta
................ 35
1.3
5O(3,R)
andSÍ/(2) Soud
Harmonics
........... 43
1.3.1
Inner products
.................... 49
1.4
Combinatorial Features
................... 51
1.4.1
Combinatorial definition of Wigner-Clebsch-Gordan
coefficients
...................... 51
1.4.2
Magic square realization
............... 59
1.5 Kronecker
Product of Solid Harmonics
........... 61
1.6
SU(n) Solid Harmonics
................... 64
xiii
xiv CONTENTS
1.6.1 Definition
and properties of SU(n)
solid
harmonics
64
1.6.2 MacMahon and Schwinger
master theorems
.... 66
1.6.3
Combinatorial proof of the multiplication property
67
1.6.4
Maclaurin monomials
................ 71
1.6.5
Summary of relations
................ 74
1.7
Generalization to U{2)
.................... 75
1.7.1
Definition of
Č7(2)
solid harmonics
......... 75
1.7.2
Basic multiplication properties
........... 78
1.7.3
Indeterminate and derivative actions on the U{2)
solid harmonics
.................... 81
2
Composite Systems
83
2.1
General Setting
........................ 83
2.1.1
Angular momentum state vectors of a composite
system
......................... 83
2.1.2
Group actions in a composite system
........ 89
2.1.3
Standard form of the
Kronecker
direct sum
.... 90
2.1.4
Reduction of
Kronecker
products
.......... 93
2.1.5
Recoupling matrices
................. 94
2.2
Binary Coupling Theory
................... 97
2.2.1
Binary trees
......................100
2.2.2
Standard labeling of binary trees
..........107
2.2.3
Generalized WCG coefficients defined in terms of
binary trees
......................109
2.2.4
Binary coupled state vectors
............112
2.2.5
Binary reduction of
Kronecker
products
......114
2.2.6
Binary recoupling matrices
.............115
2.2.7
Triangle patterns and triangle coefficients
.....119
2.2.8
Racah coefficients
..................131
2.2.9
Recoupling matrices for
η
= 3............137
2.2.10
Recoupling matrices for
η
= 4............140
CONTENTS xv
2.2.11
Structure
of general triangle coefficients and
recoupling matrices
..................152
2.3
Classification of Recoupling Matrices
............156
3
Graphs and Adjacency Diagrams
163
3.1
Binary Trees and Trivalent Trees
.............. 163
3.2
Nonisomorphic Trivalent Trees
............... 172
3.2.1
Shape labels of binary trees
............. 184
3.3
Cubic Graphs and Trivalent Trees
............. 189
3.4
Cubic Graphs
......................... 198
3.4.1
Factoring properties of cubic graphs
........ 200
3.4.2
Join properties of cubic graphs
........... 213
3.4.3
Cubic graph matrices
................ 221
3.4.4
Labeled cubic graphs
................. 226
3.4.5
Summary and unsolved problems
.......... 228
4
Generating Functions
229
4.1
Pfaffians and Double Pfaffians
................ 230
4.2
Skew-Symmetric Matrix
................... 232
4.3
Triangle Monomials
..................... 238
4.4
Coupled Wave Functions
................... 239
4.5
Recoupling Coefficients
.................... 241
4.6
Special Cases
.........................
245
4.6.1
Cubic graph geometry of the
6 -
j
and
9 -
j
coefficients
......................255
4.7
Concluding Remarks
.....................259
5
The D* -Polynomials: Form 261
5.1
Overview
...........................
261
5.2
Defining Relations
......................
268
5.2.1
Another proof of the multiplication rule
......277
xvi CONTENTS
5.3
Restriction
to Fewer Variables
................278
5.4
Vector Space Aspects
.....................280
5.5
Fundamental Structural Relations
.............283
5.5.1
Structural relations
..................283
5.5.2
Principal objectives and tasks
............284
6
Operator Actions in Hilbert Space
285
6.1
Introductory Remarks
.................... 285
6.2
Action of Fundamental Shift Operators
.......... 286
6.3
Digraph Interpretation
.................... 295
6.4
Algebra of Shift Operators
.................. 301
6.5
Hilbert Space and ^-Polynomials
............ 304
6.6
Shift Operator Polynomials
................. 308
6.7 Kronecker
Product Reduction
................ 312
6.8
More on Explicit Operator Actions
............. 318
7
The Dx -Polynomials: Structure
325
7.1
The C A) Matrices
.....................325
7.2
Reduction of
ΣΡ(Ζ)
<g> DX(Z)
................329
7.3
Binary Tree Structure:
С 4
-Coefficients
..........335
7.3.1
The Dx -polynomials for partitions
with two nonzero parts
...............342
7.3.2
Recurrence relation for the Dx—polynomials
. . . 351
8
The General Linear and Unitary Groups
355
8.1
Background and Review
...................355
8.2
GL(n, C) and its Unitary Subgroup U(n)
.........358
8.3
Commuting Hermitian
Observables
.............368
8.3.1
The Gelfand invariants
................368
8.4
Differential Operator Actions
................370
CONTENTS xvii
8.5
Eigenvalues of the
Gelfand
Invariants
............371
8.5.1
A new class of symmetric functions
........373
8.5.2
The general eigenvalues of the Gelfand invariants
375
9
Tensor Operator Theory
379
9.1
Introduction
..........................379
9.1.1
A basis of irreducible tensor operators
.......382
9.2
Unit Tensor Operators
....................386
9.2.1
Summary of properties of unit tensor operators
. . 389
9.2.2
Explicit unit tensor operators
............392
9.3
Canonical Tensor Operators
.................394
9.3.1
Application of the factorization lemma to canoni¬
cal tensor operators
.................399
9.3.2
Subgroup conditions and reduced matrix elements
403
9.4
Properties of Reduced Matrix Elements
..........408
9.4.1
Unit
projective
operators
..............410
9.4.2
The pattern calculus
.................415
9.4.3
Shift
invariance
of
Kostka
and
Littlewood-Richardson numbers
...........421
9.5
The Unitary Group t/(3)
...................422
9.5.1
The U(S) canonical tensor operators
........422
9.6
The
Ï7(3)
Characteristic Null Spaces
............425
9.7
The U(3)
:
U(2) Unit
Projective
Operators
........432
9.7.1
Coupling rules and Racah coefficients
.......433
9.7.2
Limit relations
....................440
10
Compendium A. Basic Algebraic Objects
447
10.1
Groups
.............................
U7
10.1.1
Group actions
.....................
^9
10.2
Rings
.............................
451
10.2.1
Rings of polynomials
.................
452
xviii CONTENTS
10.2.2
Vector
spaces of polynomials
............455
10.3
Abstract Hubert Spaces
...................456
10.3.1
Inner product spaces
................. 456
10.3.2
Linear operators
................... 458
10.3.3
Inner products and linear operators
........ 460
10.3.4
Orthonormalization methods
............ 461
10.3.5
Matrix representations of linear operators
..... 464
10.4
Properties of Matrices
.................... 465
10.4.1
Properties of normal matrices
............ 467
10.4.2
Inner product on the space of complex matrices
. . 468
10.4.3
Exponentiated matrices
............... 469
10.4.4
Lie bracket polynomials
............... 476
10.5
Tensor Product Spaces
.................... 486
10.6
Vector Spaces of Polynomials
................ 488
10.7
Group Representations
.................... 494
10.7.1
Irreducible representations of groups
........499
10.7.2
Schur s lemmas
....................501
11
Compendium B: Combinatorial Objects
505
11.1
Partitions and Tableaux
...................506
11.1.1
Restricted compositions
...............512
11.1.2
Linear ordering of sequences and partitions
.... 513
11.2
Young Frames and Tableaux
.................514
11.2.1
Skew tableaux
....................520
11.3
Gelfand-Tsetlin Patterns
...................523
11.3.1
Combinatorial origin of Gelfand-Tsetlin patterns
. 523
11.3.2
Group theoretical origin of Gelfand-Tsetlin
patterns
........................528
11.3.3
Skew Gelfand-Tsetlin patterns
...........528
11.3.4
Notations
.......................532
CONTENTS xix
11.3.5
Triangular
and skew Gelfand-Tsetlin patterns
... 533
11.3.6
Words and lattice permutations
...........535
11.3.7
Lattice permutations and Littlewood-Richardson
numbers
.......................538
11.3.8
Kostka
and Littlewood-Richardson numbers
.... 550
11.4
Generating Functions and Relations
............556
11.4.1
Notations
....................... 557
11.4.2
Counting relations
.................. 558
11.4.3
Generating relations of functions
.......... 559
11.4.4
Operator generated functions
............ 561
11.5 Multivariable
Special Functions
.............. 563
11.5.1
Solid harmonics
....................564
11.5.2
Double harmonic functions in R3
..........565
11.5.3
Hypergeometric functions
..............565
11.5.4
MacMahon s master theorem
............568
11.5.5
Power of a determinant
...............568
11.6
Symmetric Functions
.....................570
11.6.1
Introduction
..................... 570
11.6.2
Four basic symmetric functions
........... 571
11.6.3
Vandermonde determinants
............. 572
11.6.4 Schur
functions
.................... 573
11.6.5
Dual bases for symmetric functions
......... 578
11.7
Sylvester s Identity
...................... 580
11.8
Derivation of
Weyľs
Dimension Formula
.......... 582
11.8.1
Vandermonde determinants, Bernoulli polynomi¬
als, and Weyl s formula
...............582
11.9
Other Topics
.........................585
11.9.1
Alternating sign matrices
..............585
11.9.2
Binary trees, graphs, and digraphs
.........595
11.9.3
Umbral
calculus and double tableau calculus
. . . 595
xx CONTENTS
11.9.4 Special
functions...................
596
11.9.5
Other generalizations
................596
Bibliography
597
Index
611
|
| any_adam_object | 1 |
| author | Louck, James D. 1928- |
| author_GND | (DE-588)141071044 |
| author_facet | Louck, James D. 1928- |
| author_role | aut |
| author_sort | Louck, James D. 1928- |
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| callnumber-label | QA167 |
| callnumber-raw | QA167 |
| callnumber-search | QA167 |
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| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 950 UK 3000 |
| ctrlnum | (OCoLC)273893634 (DE-599)GBV583137725 |
| dewey-full | 511.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.6 |
| dewey-search | 511.6 |
| dewey-sort | 3511.6 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV035203869 |
| illustrated | Not Illustrated |
| indexdate | 2025-02-03T17:28:01Z |
| institution | BVB |
| isbn | 9812814728 9789812814722 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017010287 |
| oclc_num | 273893634 |
| open_access_boolean | |
| owner | DE-29T DE-703 DE-11 DE-19 DE-BY-UBM |
| owner_facet | DE-29T DE-703 DE-11 DE-19 DE-BY-UBM |
| physical | XXI, 619 S. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | World Scientific |
| record_format | marc |
| spellingShingle | Louck, James D. 1928- Unitary symmetry and combinatorics Combinatorial analysis Eightfold way (Nuclear physics) Kombinatorik (DE-588)4031824-2 gnd Kombinatorische Analysis (DE-588)4164746-4 gnd Unitäre Symmetrie (DE-588)4627368-2 gnd Kernphysik (DE-588)4030340-8 gnd |
| subject_GND | (DE-588)4031824-2 (DE-588)4164746-4 (DE-588)4627368-2 (DE-588)4030340-8 |
| title | Unitary symmetry and combinatorics |
| title_auth | Unitary symmetry and combinatorics |
| title_exact_search | Unitary symmetry and combinatorics |
| title_full | Unitary symmetry and combinatorics James D. Louck |
| title_fullStr | Unitary symmetry and combinatorics James D. Louck |
| title_full_unstemmed | Unitary symmetry and combinatorics James D. Louck |
| title_short | Unitary symmetry and combinatorics |
| title_sort | unitary symmetry and combinatorics |
| topic | Combinatorial analysis Eightfold way (Nuclear physics) Kombinatorik (DE-588)4031824-2 gnd Kombinatorische Analysis (DE-588)4164746-4 gnd Unitäre Symmetrie (DE-588)4627368-2 gnd Kernphysik (DE-588)4030340-8 gnd |
| topic_facet | Combinatorial analysis Eightfold way (Nuclear physics) Kombinatorik Kombinatorische Analysis Unitäre Symmetrie Kernphysik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017010287&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT louckjamesd unitarysymmetryandcombinatorics |