Tensorial methods and renormalization in group field theories:
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| Main Author: | |
|---|---|
| Format: | Thesis Book |
| Language: | English |
| Published: |
Cham [u.a.]
Springer
2014
|
| Series: | Springer theses
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | XV, 226 S. Ill., graph. Darst. |
| ISBN: | 9783319058665 |
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MARC
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| 100 | 1 | |a Carrozza, Sylvain |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Tensorial methods and renormalization in group field theories |c Sylvain Carrozza |
| 264 | 1 | |a Cham [u.a.] |b Springer |c 2014 | |
| 300 | |a XV, 226 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer theses | |
| 502 | |a Zugl.: Orsay, Université Paris-Sud, Diss. | ||
| 650 | 4 | |a Quantentheorie | |
| 650 | 4 | |a Group theory | |
| 650 | 4 | |a Quantum theory | |
| 650 | 4 | |a Physics | |
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Record in the Search Index
| _version_ | 1804152538238287872 |
|---|---|
| adam_text | Contents
Introduction
and Motivation
............................
í
1.1
Why a Quantum Theory of Gravity Cannot
be Dispensed With
................................. 1
1
.2
Quantum Gravity and Quantization
..................... 5
1.3
On Scales and Renormalization With or Without
Background
..................................... 8
1.4
Purpose and Plan of the Thesis
........................ 10
References
.......................................... 12
Two Paths to Group Field Theories
....................... 17
2.1
Group Field Theories and Quantum General Relativity
....... 17
2.1.1
Loop Quantum Gravity
........................ 17
2.1.2
Spin Foams
................................ 22
2.1.3
Summing Over Spin Foams
..................... 27
2.1.4
Towards Well-Defined Quantum Field Theories
of Spin Networks
............................ 29
2.2
Group Field Theories and Random Discrete Geometries
....... 34
2.2.1
Matrix Models and Random Surfaces
.............. 34
2.2.2
Higher Dimensional Generalizations
............... 39
2.2.3
Bringing Discrete Geometry In
................... 41
2.3
A Research Direction
............................... 43
References
.......................................... 45
Colors and Tensor
Invariance
........................... 49
3.1
Colored Group Field Theories
......................... 50
3.1.1
Combinatorial and Topological Motivations
.......... 50
3.1.2
Motivation from Discrete Diffeomorphisms
.......... 52
3.2
Colored Tensor Models
............................. 53
3.2.
í
Models and Amplitudes
........................ 53
3.2.2
Degree and Existence of the Large
N
Expansion
...... 55
3.2.3
The World of Melons
......................... 55
XIII
XLV
Contents
3.3
Tensor
Invariance
................................. 57
3.3.1
From Colored Simplices to Tensor Invariant
Interactions
................................ 57
3.3.2
Generalization to GFTs
........................ 58
References
.......................................... 59
4
Large
N
Expansion in Topological Group Field Theories
....... 61
4.1
Colored Boulatov Model
............................ 61
4.1.1
Vertex Variables
............................. 61
4.1.2
Regularization and General Scaling Bounds
.......... 74
4.1.3
Topological Singularities
....................... 78
4.1.4
Domination of Melons
......................... 82
4.2
Colored Ooguri Model
.............................. 88
4.2.1
Edge Variables
.............................. 88
4.2.2
Regularization and General Scaling Bounds
.......... 96
4.2.3
Topological Singularities
....................... 102
4.2.4
Domination of Melons
......................... 105
References
.......................................... 109
5
Renormalization of
Tensorial
Group Field Theories:
Generalities
.........................................
Ill
5.1
Preliminaries: Renormalization of Local Field Theories
.......
Ill
5.1.1
Locality, Scales and Divergences
.................
Ill
5.1.2
Perturbati ve
Renormalization Through
a Multiscale Decomposition
..................... 113
5.2
Locality and Propagation in GFT
...................... 117
5.2.1
Simplicial and
Tensorial
Interactions
............... 118
5.2.2
Constraints and Propagation
..................... 119
5.3
A Class of Models with Closure Constraint
............... 121
5.3.1
Definition
.................................. 121
5.3.2
Graph-Theoretic and Combinatorial Tools
........... 124
5.4
Multiscale Expansion and Power-Counting
................ 131
5.4.1
Multiscale Decomposition
...................... 131
5.4.2
Propagator Bounds
........................... 132
5.4.3
Abelian Power-Counting
....................... 134
5.5
Classification of Just-Renormalizable Models
.............. 137
5.5.1
Analysis of the Abelian Divergence Degree
.......... 137
5.5.2
Just-Renormalizable Models
..................... 142
5.5.3
Properties of
Melonie
Subgraphs
.................. 143
References
.......................................... 149
6
Super-Renormalizable U(l) Models in Four Dimensions
........ 151
6.1
Divergent Subgraphs and Wick Ordering
................. 151
6.1
Л
A Bound on the Divergence Degree
............... 152
Contents xv
6.1.2
Classification
of Divergences
.................... 153
6.1.3
Localization Operators
......................... 155
6.1.4
Melordering
................................ 156
6.1.5
Vacuum Submelonic Counter-Terms
............... 159
6.2
Finiteness of the Renormalized Series
................... 160
6.2.1
Classification of Forests
........................ 161
6.2.2
Power-Counting of Renormalized Amplitudes
........ 163
6.2.3
Sum Over Scale Attributions
.................... 165
6.3
Example: Wick-Ordering of
a
φ6
Interaction
.............. 167
References
.......................................... 169
7
Just-Renormalizable SU(2) Model in Three Dimensions
........ 171
7.1
The Model and Its Divergences
........................ 171
7.1.1
Regularization and Counter-Terms
................ 171
7.1.2
List of Divergent Subgraphs
..................... 174
7.2
Non-Abelian Multiscale Expansion
..................... 175
7.2.1
Power-Counting Theorem
...................... 175
7.2.2
Contraction of High
Melonie
Subgraphs
............ 179
7.3
Perturbative Renormalizability
........................ 186
7.3.1
Effective and Renormaiized Expansions
............ 188
7.3.2
Classification of Forests
........................ 194
7.3.3
Convergent Power-Counting for Renormalized
Amplitudes
................................. 197
7.3.4
Sum Over Scale Attributions
.................... 199
7.4
Renormalization Group Flow
......................... 201
7.4.1
Approximation Scheme
........................ 202
7.4.2
Truncated Equations for the Counter-Terms
.......... 204
7.4.3
Physical Coupling Constants: Towards
Asymptotic Freedom
.......................... 210
7.4.4
Mass and Consistency of the Assumptions
........... 211
References
.......................................... 212
8
Conclusions and Perspectives
............................ 213
8.1
The
Ì/N
Expansion in Colored GFTs
................... 213
8.1.1
Achievements
............................... 213
8.1.2
Discussion and Outlook
........................ 215
8.2
Renormalization of TGFTs
........................... 216
8.2.1
Achievements
............................... 216
8.2.2
Discussion and Outlook
........................ 218
References
.......................................... 219
Appendix A: Technical Appendix
............................ 221
The main focus of this diesis is the mathematical structure of Group
Field Theories (GFTs) from the point of view of renormalization theory.
Such quantum field theories are found in approaches to quantum gravity
related, on the one hand, to Loop Quantum Gravity (LQG) and, on the
other, to matrix- and tensor models. Background material on these top¬
ics, including conceptual and technical aspects, are introduced in the first
chapters, The work then goes on to explain how the standard tools of
Quantum Field Theory can be generalized to GFTs, and exploited to study
Ее
large cut-off behaviour and «normalization group transformations
of the latter. Among the new results derived
m
this context are a proof of
renormalizability of a three-dimensional GFT with gauge group SUM,
which opens the way to applications of the formalism to quantum gravity.
|
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| author | Carrozza, Sylvain |
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| ctrlnum | (OCoLC)892961776 (DE-599)BVBBV042084826 |
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| isbn | 9783319058665 |
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| spelling | Carrozza, Sylvain Verfasser aut Tensorial methods and renormalization in group field theories Sylvain Carrozza Cham [u.a.] Springer 2014 XV, 226 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer theses Zugl.: Orsay, Université Paris-Sud, Diss. Quantentheorie Group theory Quantum theory Physics Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Quantentheorie (DE-588)4047992-4 gnd rswk-swf (DE-588)4113937-9 Hochschulschrift gnd-content Gruppentheorie (DE-588)4072157-7 s Quantentheorie (DE-588)4047992-4 s 1\p DE-604 Erscheint auch als Online-Ausgabe 978-3-319-05867-2 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027525826&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027525826&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Carrozza, Sylvain Tensorial methods and renormalization in group field theories Quantentheorie Group theory Quantum theory Physics Gruppentheorie (DE-588)4072157-7 gnd Quantentheorie (DE-588)4047992-4 gnd |
| subject_GND | (DE-588)4072157-7 (DE-588)4047992-4 (DE-588)4113937-9 |
| title | Tensorial methods and renormalization in group field theories |
| title_auth | Tensorial methods and renormalization in group field theories |
| title_exact_search | Tensorial methods and renormalization in group field theories |
| title_full | Tensorial methods and renormalization in group field theories Sylvain Carrozza |
| title_fullStr | Tensorial methods and renormalization in group field theories Sylvain Carrozza |
| title_full_unstemmed | Tensorial methods and renormalization in group field theories Sylvain Carrozza |
| title_short | Tensorial methods and renormalization in group field theories |
| title_sort | tensorial methods and renormalization in group field theories |
| topic | Quantentheorie Group theory Quantum theory Physics Gruppentheorie (DE-588)4072157-7 gnd Quantentheorie (DE-588)4047992-4 gnd |
| topic_facet | Quantentheorie Group theory Quantum theory Physics Gruppentheorie Hochschulschrift |
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