Feynman motives:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey [u.a.]
World Scientific
2010
|
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Physical Description: | XIII, 220 S. graph. Darst. |
| ISBN: | 9789814271202 9814271209 9789814304481 9814304484 |
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Record in the Search Index
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|---|---|
| adam_text | FEYNMAN
MOTIVES
This book presents recent and ongoing research work aimed at understanding the
mysterious relation between the computations of Feynman integrals in perturbative
quantum field theory and the theory of motives of algebraic varieties and their periods.
One of the main questions in the field is understanding when the residues of Feynman
integrals in perturbative quantum field theory evaluate to periods of mixed
Tate
motives.
The question originates
from
the occurrence of multiple
zeta
values in Feynman integrals
calculations observed by Broadhurst and Kreimer.
Two different approaches to the subject are described. The first, a bottom-up approach,
constructs explicit algebraic varieties and periods from Feynman graphs and parametric
Feynman integrals. This approach, which grew out of work of Bloch-Esnault-Kreimer
and was more recently developed in joint work of Paolo
Al uff
і
and the author, leads to
algebro-geometric and motivic versions of the Feynman rules of quantum field theory
and concentrates on explicit constructions of motives and classes in the Grothendieck
ring of varieties associated to Feynman integrals. While the varieties obtained in this
way can be arbitrarily complicated as
motivs»,
the part of the cohomology that is
involved in the Feynman integral computation might still be of the special mixed
Tate
kind. A second, top-down approach to the problem, developed in the work ot Alain
donnes
and the author, consists of comparing a Tannakian category constructed out of
the data of
renormal
ization of perturbative scalar field theories, obtained in the form
of a Riemann-liilbert correspondence, with Tannakian categories of mixecITate motives.
The book draws connections between these two approaches and gives an overview of
other ongoing directions of research in the field, outlining the many connections of
perturbative quantum field theory and renormalization to motives, singularity theory,
I lodge structures, arithmetic geometry, supermanifolds, algebraic and non-commutative
geometry .
The text is aimed at researchers in mathematical physics, high energy physics, number
theory and algebraic geometry. Partly based on lecture notes for a graduate course
given by the author at Caftech in the fall of
2008,
it can also· be. used fay graduate
students interested in working in this area.
Contents
Preface vii
Acknowledgments
ix
1.
Perturbative
quantum field theory and Feynman diagrams
1
1.1
A calculus exercise in Feynman integrals
.......... 1
1.2
From Lagrangian to effective action
............. 6
1.3
Feynman rules
........................ 9
1.4
Simplifying graphs: vacuum bubbles, connected graphs
. . 12
1.5
One-particle-irreducible graphs
............... 14
1.6
The problem of renormalization
............... 18
1.7
Gamma functions,
Schwinger
and Feynman parameters
. 20
1.8
Dimensional Regularization and Minimal Subtraction
. . 21
2.
Motives and periods
25
2.1
The idea of motives
..................... 25
2.2
Pure motives
......................... 28
2.3
Mixed motives and triangulated categories
......... 34
2.4
Motivic sheaves
........................ 37
2.5
The Grothendieck ring of motives
.............. 38
2.6
Tate
motives
......................... 39
2.7
The algebra of periods
.................... 44
2.8
Mixed
Tate
motives and the logarithmic extensions
.... 45
2.9
Categories and Galois groups
................ 49
2.10
Motivic Galois groups
.................... 50
xii
Feynman
Motives
3.
Feynman
integrals and algebraic varieties
53
3.1
The parametric Feynman integrals
............. 54
3.2
The graph hypersurfaces
................... 60
3.3
Landau varieties
....................... 65
3.4
Integrals in
affine
and
projective
spaces
.......... 67
3.5
Non-isolated singularities
.................. 71
3.6
Cremona transformation and dual graphs
......... 72
3.7
Classes in the Grothendieck ring
.............. 76
3.8
Motivic Feynman rules
.................... 78
3.9
Characteristic classes and Feynman rules
......... 81
3.10
Deletion-contraction relation
................ 84
3.11
Feynman integrals and periods
............... 93
3.12
The mixed
Tate
mystery
................... 94
3.13
From graph hypersurfaces to determinant hypersurfaces
. 97
3.14
Handling divergences
..................... 112
3.15
Motivic
zeta
functions and motivic Feynman rules
.... 115
4.
Feynman integrals and Gelfand-Leray forms
119
4.1
Oscillatory integrals
..................... 119
4.2
Leray regularization of Feynman integrals
......... 121
5.
Connes-Kreimer theory in a nutshell
127
5.1
The Bogolyubov recursion
.................. 128
5.1.1
Step
1:
Preparation
................. 128
5.1.2
Step
2:
Counterterms
................ 128
5.1.3
Step
3:
Renormalized values
............ 129
5.2 Hopf
algebras and
affine
group schemes
.......... 130
5.3
The Connes-Kreimer
Hopf
algebra
............. 133
5.4
Birkhoff factorization
..................... 135
5.5
Factorization and Rota-Baxter algebras
.......... 137
5.6
Motivic Feynman rules and Rota-Baxter structure
.... 139
6.
The Riemann-ffilbert correspondence
143
6.1
From divergences to iterated integrals
........... 143
6.2
From iterated integrals to differential systems
....... 145
6.3
Flat equisingular connections and vector bundles
..... 146
6.4
The cosmic Galois group
................. 147
Contents xiii
7.
The geometry of DimReg
151
7.1
The motivic geometry of DimReg
.............. 151
7.2
The
noncommutative
geometry of DimReg
......... 155
8.
Renormalization, singularities, and Hodge structures
167
8.1
Projective
Radon transform
................. 167
8.2
The polar filtration and the Milnor fiber
.......... 170
8.3
DimReg and mixed Hodge structures
............ 173
8.4
Regular and irregular singular connections
......... 176
9.
Beyond scalar theories
185
9.1
Supermanifolds
........................ 185
9.2
Parametric Feynman integrals and supermanifolds
.... 190
9.3
Graph supermanifolds
.................... 199
9.4
Noncommutative
field theories
............... 201
Bibliography
207
Index
215
|
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| dewey-ones | 530 - Physics |
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| discipline | Physik Mathematik |
| format | Book |
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| isbn | 9789814271202 9814271209 9789814304481 9814304484 |
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| spelling | Marcolli, Matilde 1969- Verfasser (DE-588)141750545 aut Feynman motives Mathilde Marcolli New Jersey [u.a.] World Scientific 2010 XIII, 220 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Feynman integrals Motives (Mathematics) Quantum field theory Pfadintegral (DE-588)4173973-5 gnd rswk-swf Algebraische Varietät (DE-588)4581715-7 gnd rswk-swf Quantenfeldtheorie (DE-588)4047984-5 gnd rswk-swf Motiv Mathematik (DE-588)4197596-0 gnd rswk-swf Pfadintegral (DE-588)4173973-5 s Quantenfeldtheorie (DE-588)4047984-5 s Motiv Mathematik (DE-588)4197596-0 s Algebraische Varietät (DE-588)4581715-7 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018615153&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018615153&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Marcolli, Matilde 1969- Feynman motives Feynman integrals Motives (Mathematics) Quantum field theory Pfadintegral (DE-588)4173973-5 gnd Algebraische Varietät (DE-588)4581715-7 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd Motiv Mathematik (DE-588)4197596-0 gnd |
| subject_GND | (DE-588)4173973-5 (DE-588)4581715-7 (DE-588)4047984-5 (DE-588)4197596-0 |
| title | Feynman motives |
| title_auth | Feynman motives |
| title_exact_search | Feynman motives |
| title_full | Feynman motives Mathilde Marcolli |
| title_fullStr | Feynman motives Mathilde Marcolli |
| title_full_unstemmed | Feynman motives Mathilde Marcolli |
| title_short | Feynman motives |
| title_sort | feynman motives |
| topic | Feynman integrals Motives (Mathematics) Quantum field theory Pfadintegral (DE-588)4173973-5 gnd Algebraische Varietät (DE-588)4581715-7 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd Motiv Mathematik (DE-588)4197596-0 gnd |
| topic_facet | Feynman integrals Motives (Mathematics) Quantum field theory Pfadintegral Algebraische Varietät Quantenfeldtheorie Motiv Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018615153&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018615153&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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