Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory:
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
1977
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| Series: | Lecture notes in physics
63 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 280 S. graph. Darst. |
| ISBN: | 354008150X 038708150X |
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| 245 | 1 | 0 | |a Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory |c V. K. Dobrev ... |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1977 | |
| 300 | |a X, 280 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in physics |v 63 | |
| 650 | 4 | |a Analyse harmonique | |
| 650 | 4 | |a Champs, Théorie quantique des | |
| 650 | 7 | |a Harmonische analyse |2 gtt | |
| 650 | 7 | |a Kwantumveldentheorie |2 gtt | |
| 650 | 4 | |a Lorentz, Transformations de | |
| 650 | 7 | |a Lorentz-groepen |2 gtt | |
| 650 | 4 | |a Harmonic analysis | |
| 650 | 4 | |a Lorentz transformations | |
| 650 | 4 | |a Quantum field theory | |
| 650 | 0 | 7 | |a Harmonische Analyse |0 (DE-588)4023453-8 |2 gnd |9 rswk-swf |
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Record in the Search Index
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| adam_text | CONTENTS
Introduction 1
Part One Type I Representations and Intertwining Operators for 0(2h+l,l) 10
I. Elementary representations of the pseudo orthogonal group
I¦ Group structure. Preliminaries
I.A The group 0(2h + I,]) and its Lie algebra 11
l.B Subgroups and decompositions 12
l.C The compactified Euclidean space as a homogeneous space of G ^g
l.D Matrix realization of various subgroups of G. Construction of
the Bruhat decomposition 18
l.E Relationship between the Bruhat and the Iwasawa decomposition
The Haar measure 25
2. Induced representations. Definition and various realizations
?.A Synopsis on the irreducible representations of the orthogonal
group ^8
2.B Covariant vector valued functions on G. Definition of
the induced representations 32
2.C The compact picture. K content of the elementary representations 34
2.D The noncompact picture: x space realization 37
3. Further properties of the elementary representations
3.A Equivalence, irreducibility, completeness 41
3.B Characters of elementary representations 47
3 cThe spherical trace function.The character of a sub
quotient of an elementary representation 53
3.D The principal series of unitary representations 54
3.E Infinitesimal generators and Casimir operators of the
elementary representations 57
VI
II. Intertwining distributions and their Fourier transform
4. Intertwining operators: x space realization
4.A Group theoretical definition of the intertwining operators 60
4.B The intertwining distributions in the noncompact picture 64
5. Momentum space expansion of the intertwining distribution and
positivity
5.A Fourier transform of G .(x;3 ,3».) 66
5.B Harmonic expansion of G (p) 69
A
5.C Normalization and positivity for nonexceptional representations.
Complementary series of unitary IR s 75
5.D Wightman positivity 81
III. Properties of elementary representations at exceptional integer
points
6. Nondecomposable representations and intertwining differential
operators
£.A Subrepresentations of exceptional elementary representations 85
6.B Intertwining differential operators. Partial equivalence among
the representations y,. ~ 89
iv
6.C Hermitian forms on invariant subspaces. Exceptional series
of unitary representations 94
.D Differential identities between hermitian forms for exceptional
representations 101
7. Discrete series of unitary representations
7.A Definition and general properties of the discrete series of
S0f(2n,l) 103
VII
7. B. Unitarily induced representations on G/K 107
7. C. Realization of the unitary representation U+ in the
5
space X s+ (NA) 11°
7. D. K invariants. Solution of the eigenvalue problem for the
Casimir operator. The discrete series LUo 115
7. E. Two point Green function. Equivalence of U^ with the
subrepresentation of Y, acting in D 122
8. The Plancherel theorem. Concluding; remarks
8. A. Harmonic analysis of the left regular representation of
B0t(2h + 1, l; for integer h 128
8. B. Harmonic analysis on S0t(2n,l;. The role of the discrete
series 131
8. C, Synopsis on unitary type 1 representations. Summary of
equivalence relations 135
Appendix A. Symmetric tensor representations of S0(n) and
their decomposition in IR s of S0(n 1) ^ *g
A.1 Harmonic extension of homogeneous polynomial functions
on the light cone 138
A.2 S0(n 1) expansion of homogeneous polynomials. The zonal
spherical functions 141
A. 3 Evaluation of the proportionality constant a^ between the
scalar products in jj and U 143
A.4 Derivation of factorized expression for the projection
operators f] s 144
A.5 Interior differentiation on the complex cone. Expression
for the convolution of two tensors in terms of homogeneous
polynomials 149
Appendix B. The special cases h=1 and h=^ . Relation to the 153
formalism of two by two matrices
B.1 Reduction of the representation % of 0/t (3, 1) into elemen
VIII
tary representations of SL(2,C) 153
B.2 Vanishing of the projection operators I I for s 1 155
B. 3 The structure of exceptional representations for h=1 157
B.4 Elementary representations of S0t(2,1). The analytic discrete
series 159
Appendix C. Positivity of the invariant scalar product in the 165
subspace ~Qev of C^
C.1 The problem. Asymptotic expansion of f(p,z ) for p 0 165
C.2 Existence of nontrivial positive semidefinite hermitian form
(f ,GJ* f) on C 167
Part Two. Conformal Partial Wave Analysis 173
IV. Clebsch Gordan expansion of the tensor product of two
unitary principal or supplementary series representations
9. The Kronecker product of two elementary representations
as an induced representation on G/MA 175
10. Construction of the Clebsch Gordan expansion
10.A.Clebsch Gordan kernels 18/|
10.B.Application of the Plancherel theorem to the Kronecker
product of two principal series representations 192
10.C.Odd space time dimension 2h 200
10.D.Analytic continuation in c^ and c 203
IX
11. Special cases and further properties of the
expansion formula
11.A. The Clebsch Gordan kernel for two class I representations.
Symmetry and normalization 206
11.B. Identities for the Clebsch Gordan kernels at exceptional
integer points 213
ll.C. Tensor product representation and Clebsch Gordan
expansion for distributions 217
V. Dynamical derivation of vacuum operator product expansion in
Euclidean conformal quantum field theory
12. Renormalizable models of self interacting scalar fields.
Dynamical equations for Euclidean Green functions
12.A. A 6 dimensional model. Euclidean Green functions. Generating
functionals 219
12.B. Graphical notation, li and 2i kernels 220
12.C. Dynamical equations. Stress energy tensor. Ward identities 221
12.D. A more realistic model 224
13. Invariance and invariant solutions of the dynamical equations.
Conformal partial wave expansion for the Euclidean Green functions
13.A. Euclidean conformal invariance of the equations 225
13.B. Conformal invariant 2 and 3 point functions 227
13.C. Skeleton diagram expansion 230
13.D. Conformal partial wave expansion 232
13.E. Further expansions 234
Ik. Implications of the dynamical equations.
Pole structure of conformal partial waves
I t .A. Poles in the conformal partial waves implied by the vertex
bootstrap equations 238
I t.B. Pole structure of the n point partial waves. Expression for the
residues 240
l^.C. Basic conformal covariant tensor fields. Analyticity assumption 241
15. Derivation of an operator product expansion for vacuum expec¬
tation values
X
15 A Another form of the conformal expansion involving a
Minkowski momentum space integral. The Q kernels 244
15 B The vacuum operator product expansion 249
15.C Wightman positivity for the A—point function 253
16. The problem of crossing symmetry. Concluding remarks
16.A Crossing symmetry and duality 254
16.B A crossing symmetry representation for the A—point
function 256
16.u Summary and discussion 257
Appendix P. Proof of lemma 1o.5. 259
Appendix K. A summation formula involving ratios of T functions 260
Appendix F. Partial Fourier transform of VCx^XgX^) and related
formulas 262
F.1 Fourier transform in x, 262
F.2 Derivation of Eq.(13.36) for the conformal partial wave 263
Appendix G. Identities between Q and Y functions for
partially equivalent representations 267
References 268
Figures 1, 2, 3 278 280
|
| adam_txt |
CONTENTS
Introduction 1
Part One Type I Representations and Intertwining Operators for 0(2h+l,l) 10
I. Elementary representations of the pseudo orthogonal group
I¦ Group structure. Preliminaries
I.A The group 0(2h + I,]) and its Lie algebra 11
l.B Subgroups and decompositions 12
l.C The compactified Euclidean space as a homogeneous space of G ^g
l.D Matrix realization of various subgroups of G. Construction of
the Bruhat decomposition 18
l.E Relationship between the Bruhat and the Iwasawa decomposition
The Haar measure 25
2. Induced representations. Definition and various realizations
?.A Synopsis on the irreducible representations of the orthogonal
group ^8
2.B Covariant vector valued functions on G. Definition of
the induced representations 32
2.C The compact picture. K content of the elementary representations 34
2.D The noncompact picture: x space realization 37
3. Further properties of the elementary representations
3.A Equivalence, irreducibility, completeness 41
3.B Characters of elementary representations 47
3 cThe spherical trace function.The character of a sub
quotient of an elementary representation 53
3.D The principal series of unitary representations 54
3.E Infinitesimal generators and Casimir operators of the
elementary representations 57
VI
II. Intertwining distributions and their Fourier transform
4. Intertwining operators: x space realization
4.A Group theoretical definition of the intertwining operators 60
4.B The intertwining distributions in the noncompact picture 64
5. Momentum space expansion of the intertwining distribution and
positivity
5.A Fourier transform of G .(x;3 ,3».) 66
5.B Harmonic expansion of G (p) 69
A
5.C Normalization and positivity for nonexceptional representations.
Complementary series of unitary IR's 75
5.D Wightman positivity 81
III. Properties of elementary representations at exceptional integer
points
6. Nondecomposable representations and intertwining differential
operators
£.A Subrepresentations of exceptional elementary representations 85
6.B Intertwining differential operators. Partial equivalence among
the representations y,. ~ 89
iv
6.C Hermitian forms on invariant subspaces. Exceptional series
of unitary representations 94
.D Differential identities between hermitian forms for exceptional
representations 101
7. Discrete series of unitary representations
7.A Definition and general properties of the discrete series of
S0f(2n,l) 103
VII
7. B. Unitarily induced representations on G/K 107
7. C. Realization of the unitary representation U+ in the
5
space X s+ (NA) 11°
7. D. K invariants. Solution of the eigenvalue problem for the
Casimir operator. The discrete series LUo 115
7. E. Two point Green function. Equivalence of U^ with the
subrepresentation of Y, acting in D 122
8. The Plancherel theorem. Concluding; remarks
8. A. Harmonic analysis of the left regular representation of
B0t(2h + 1, l; for integer h 128
8. B. Harmonic analysis on S0t(2n,l;. The role of the discrete
series 131
8. C, Synopsis on unitary type 1 representations. Summary of
equivalence relations 135
Appendix A. Symmetric tensor representations of S0(n) and
their decomposition in IR's of S0(n 1) ^ *g
A.1 Harmonic extension of homogeneous polynomial functions
on the light cone 138
A.2 S0(n 1) expansion of homogeneous polynomials. The zonal
spherical functions 141
A. 3 Evaluation of the proportionality constant a^ between the
scalar products in jj and U 143
A.4 Derivation of factorized expression for the projection
operators f] s 144
A.5 Interior differentiation on the complex cone. Expression
for the convolution of two tensors in terms of homogeneous
polynomials 149
Appendix B. The special cases h=1 and h=^ . Relation to the 153
formalism of two by two matrices
B.1 Reduction of the representation % of 0/t'(3, 1) into elemen
VIII
tary representations of SL(2,C) 153
B.2 Vanishing of the projection operators I I for s 1 155
B. 3 The structure of exceptional representations for h=1 157
B.4 Elementary representations of S0t(2,1). The analytic discrete
series 159
Appendix C. Positivity of the invariant scalar product in the 165
subspace ~Qev of C^
C.1 The problem. Asymptotic expansion of f(p,z ) for p 0 165
C.2 Existence of nontrivial positive semidefinite hermitian form
(f ,GJ* f) on C 167
Part Two. Conformal Partial Wave Analysis 173
IV. Clebsch Gordan expansion of the tensor product of two
unitary principal or supplementary series representations
9. The Kronecker product of two elementary representations
as an induced representation on G/MA 175
10. Construction of the Clebsch Gordan expansion
10.A.Clebsch Gordan kernels 18/|
10.B.Application of the Plancherel theorem to the Kronecker
product of two principal series representations 192
10.C.Odd space time dimension 2h 200
10.D.Analytic continuation in c^ and c 203
IX
11. Special cases and further properties of the
expansion formula
11.A. The Clebsch Gordan kernel for two class I representations.
Symmetry and normalization 206
11.B. Identities for the Clebsch Gordan kernels at exceptional
integer points 213
ll.C. Tensor product representation and Clebsch Gordan
expansion for distributions 217
V. Dynamical derivation of vacuum operator product expansion in
Euclidean conformal quantum field theory
12. Renormalizable models of self interacting scalar fields.
Dynamical equations for Euclidean Green functions
12.A. A 6 dimensional model. Euclidean Green functions. Generating
functionals 219
12.B. Graphical notation, li and 2i kernels 220
12.C. Dynamical equations. Stress energy tensor. Ward identities 221
12.D. A more realistic model 224
13. Invariance and invariant solutions of the dynamical equations.
Conformal partial wave expansion for the Euclidean Green functions
13.A. Euclidean conformal invariance of the equations 225
13.B. Conformal invariant 2 and 3 point functions 227
13.C. Skeleton diagram expansion 230
13.D. Conformal partial wave expansion 232
13.E. Further expansions 234
Ik. Implications of the dynamical equations.
Pole structure of conformal partial waves
I't .A. Poles in the conformal partial waves implied by the vertex
bootstrap equations 238
I't.B. Pole structure of the n point partial waves. Expression for the
residues 240
l^.C. Basic conformal covariant tensor fields. Analyticity assumption 241
15. Derivation of an operator product expansion for vacuum expec¬
tation values
X
15 A Another form of the conformal expansion involving a
Minkowski momentum space integral. The Q kernels 244
15 B The vacuum operator product expansion 249
15.C Wightman positivity for the A—point function 253
16. The problem of crossing symmetry. Concluding remarks
16.A Crossing symmetry and duality 254
16.B A crossing symmetry representation for the A—point
function 256
16.u Summary and discussion 257
Appendix P. Proof of lemma 1o.5. 259
Appendix K. A summation formula involving ratios of T functions 260
Appendix F. Partial Fourier transform of VCx^XgX^) and related
formulas 262
F.1 Fourier transform in x, 262
F.2 Derivation of Eq.(13.36) for the conformal partial wave 263
Appendix G. Identities between Q and Y functions for
partially equivalent representations 267
References 268
Figures 1, 2, 3 278 280 |
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| id | DE-604.BV022146990 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:17:57Z |
| indexdate | 2024-07-09T20:51:21Z |
| institution | BVB |
| isbn | 354008150X 038708150X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015361628 |
| oclc_num | 2873466 |
| open_access_boolean | |
| owner | DE-706 DE-11 |
| owner_facet | DE-706 DE-11 |
| physical | X, 280 S. graph. Darst. |
| publishDate | 1977 |
| publishDateSearch | 1977 |
| publishDateSort | 1977 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in physics |
| series2 | Lecture notes in physics |
| spelling | Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory V. K. Dobrev ... Berlin [u.a.] Springer 1977 X, 280 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in physics 63 Analyse harmonique Champs, Théorie quantique des Harmonische analyse gtt Kwantumveldentheorie gtt Lorentz, Transformations de Lorentz-groepen gtt Harmonic analysis Lorentz transformations Quantum field theory Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Quantenfeldtheorie (DE-588)4047984-5 gnd rswk-swf Lorentz-Gruppe (DE-588)4036335-1 gnd rswk-swf Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Physik (DE-588)4045956-1 gnd rswk-swf Physik (DE-588)4045956-1 s DE-604 Gruppentheorie (DE-588)4072157-7 s Quantenfeldtheorie (DE-588)4047984-5 s Harmonische Analyse (DE-588)4023453-8 s 1\p DE-604 Lorentz-Gruppe (DE-588)4036335-1 s 2\p DE-604 Dobrev, Vladimir K. Sonstige (DE-588)1035099098 oth Lecture notes in physics 63 (DE-604)BV000003166 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015361628&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory Lecture notes in physics Analyse harmonique Champs, Théorie quantique des Harmonische analyse gtt Kwantumveldentheorie gtt Lorentz, Transformations de Lorentz-groepen gtt Harmonic analysis Lorentz transformations Quantum field theory Harmonische Analyse (DE-588)4023453-8 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd Lorentz-Gruppe (DE-588)4036335-1 gnd Gruppentheorie (DE-588)4072157-7 gnd Physik (DE-588)4045956-1 gnd |
| subject_GND | (DE-588)4023453-8 (DE-588)4047984-5 (DE-588)4036335-1 (DE-588)4072157-7 (DE-588)4045956-1 |
| title | Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory |
| title_auth | Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory |
| title_exact_search | Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory |
| title_exact_search_txtP | Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory |
| title_full | Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory V. K. Dobrev ... |
| title_fullStr | Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory V. K. Dobrev ... |
| title_full_unstemmed | Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory V. K. Dobrev ... |
| title_short | Harmonic analysis on the n-dimensional Lorentz group and its application to conformal quantum field theory |
| title_sort | harmonic analysis on the n dimensional lorentz group and its application to conformal quantum field theory |
| topic | Analyse harmonique Champs, Théorie quantique des Harmonische analyse gtt Kwantumveldentheorie gtt Lorentz, Transformations de Lorentz-groepen gtt Harmonic analysis Lorentz transformations Quantum field theory Harmonische Analyse (DE-588)4023453-8 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd Lorentz-Gruppe (DE-588)4036335-1 gnd Gruppentheorie (DE-588)4072157-7 gnd Physik (DE-588)4045956-1 gnd |
| topic_facet | Analyse harmonique Champs, Théorie quantique des Harmonische analyse Kwantumveldentheorie Lorentz, Transformations de Lorentz-groepen Harmonic analysis Lorentz transformations Quantum field theory Harmonische Analyse Quantenfeldtheorie Lorentz-Gruppe Gruppentheorie Physik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015361628&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000003166 |
| work_keys_str_mv | AT dobrevvladimirk harmonicanalysisonthendimensionallorentzgroupanditsapplicationtoconformalquantumfieldtheory |