Noncommutative geometry, quantum fields and motives:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2008
|
| Series: | American Mathematical Society Colloquium publications
volume 55 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverzeichnis: Seiten 749 - 761 |
| Physical Description: | XXII, 785 Seiten Illustrationen, Diagramme |
| ISBN: | 9780821842102 |
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| 100 | 1 | |a Connes, Alain |d 1947- |e Verfasser |0 (DE-588)112614760 |4 aut | |
| 245 | 1 | 0 | |a Noncommutative geometry, quantum fields and motives |c Alain Connes ; Matilde Marcolli |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c 2008 | |
| 264 | 4 | |c © 2008 | |
| 300 | |a XXII, 785 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a American Mathematical Society Colloquium publications |v volume 55 | |
| 500 | |a Literaturverzeichnis: Seiten 749 - 761 | ||
| 650 | 7 | |a Análise global |2 larpcal | |
| 650 | 7 | |a Corpos globais |2 larpcal | |
| 650 | 7 | |a Geometria algébrica |2 larpcal | |
| 650 | 7 | |a Mecânica quântica |2 larpcal | |
| 650 | 7 | |a Teoria analítica dos números |2 larpcal | |
| 650 | 7 | |a Teoria dos números |2 larpcal | |
| 650 | 7 | |a Álgebra |2 larpcal | |
| 650 | 4 | |a Noncommutative differential geometry | |
| 650 | 4 | |a Quantum field theory | |
| 650 | 0 | 7 | |a Nichtkommutative Differentialgeometrie |0 (DE-588)4311174-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Motiv |g Mathematik |0 (DE-588)4197596-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Quantenfeldtheorie |0 (DE-588)4047984-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Nichtkommutative Differentialgeometrie |0 (DE-588)4311174-9 |D s |
| 689 | 0 | 1 | |a Quantenfeldtheorie |0 (DE-588)4047984-5 |D s |
| 689 | 0 | 2 | |a Motiv |g Mathematik |0 (DE-588)4197596-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Marcolli, Matilde |d 1969- |e Verfasser |0 (DE-588)141750545 |4 aut | |
| 776 | 0 | 8 | |i Erscheint auch als |n Online-Ausgabe |z 978-1-4704-3201-0 |
| 830 | 0 | |a American Mathematical Society Colloquium publications |v volume 55 |w (DE-604)BV035417609 |9 55 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016685231&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016685231 | ||
Record in the Search Index
| _version_ | 1804137939250184192 |
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| adam_text | Contents
Preface
xiii
Chapter
1.
Quantum fields,
noncommutative
spaces, and motives
1
1.
Introduction
1
2.
Basics of perturbative QFT
7
2.1.
Lagrangian and Hamiltonian formalisms
8
2.2.
Lagrangian and the Feynman integral
10
2.3.
The Hamiltonian and canonical quantization
11
2.4.
The simplest example
13
2.5.
Green s functions
17
2.6.
Wick rotation and Euclidean Green s functions
18
3.
Feynman diagrams
22
3.1.
The simplest case
23
3.2.
The origins of renormalization
27
3.3.
Feynman graphs and rules
31
3.4.
Connected Green s functions
35
3.5.
The effective action and one-particle irreducible graphs
37
3.6.
Physically observable parameters
41
3.7.
The physics idea of renormalization
43
4.
Dimensional regularization
46
5.
The graph by graph method of Bogoliubov-Parasiuk-Hepp-
Zimmermann
52
5.1.
The simplest example of
subdivergence
54
5.2.
Superficial degree of divergence
58
5.3.
Subdivergences
and preparation
59
6.
The Connes-Kreimer theory of perturbative renormalization
66
6.1.
Commutative
Hopf
algebras and
affine
group schemes
67
6.2.
The
Hopf
algebra of Feynman graphs: discrete part
71
6.3.
The
Hopf
algebra of Feynman graphs: full structure
78
6.4.
BPHZ as a Birkhoff factorization
81
6.5.
Diffeographisms and diffeomorphisms
88
6.6.
The renormalization group
89
7.
Renormalization and the Riemann-Hilbert correspondence
95
7.1.
Counterterms and time-ordered exponentials
96
7.2.
Flat equisingular connections
103
7.3.
Equivariant principal bundles and the group G*
=
G x
Gm
114
CONTENTS
7.4. Tannakian
categories and
affine
group schemes
119
7.5.
Differential Galois theory and the local Riemann-Hilbert
correspondence
123
7.6.
Universal
Hopf
algebra and the Riemann-Hilbert
correspondence
128
8.
Motives in a nutshell
137
8.1.
Algebraic varieties and motives
137
8.2.
Pure motives
146
8.3.
Mixed motives
151
8.4.
Mixed Hodge structures
156
8.5.
Tate
motives, periods, and quantum fields
159
9.
The Standard Model of elementary particles
160
9.1.
Particles and interactions
162
9.2.
Symmetries
163
9.3.
Quark mixing: the
CKM
matrix
166
9.4.
The Standard Model Lagrangian
166
9.5.
Quantum level: anomalies, ghosts, gauge fixing
170
9.6.
Massive neutrinos
174
9.7.
The Standard Model minimally coupled to gravity
179
9.8.
Higher derivative terms in gravity
183
9.9.
Symmetries as diffeomorphisms
184
10.
The framework of (metric)
noncommutative
geometry
186
10.1.
Spectral geometry
187
10.2.
Spectral triples
190
10.3.
The real part of a real spectral triple
192
10.4. Hochschild
and cyclic cohomology
193
10.5.
The local index cocycle
198
10.6.
Positivity
in
Hochschild
cohomology and Yang-Mills action
201
10.7.
Cyclic cohomology and Chern-Simons action
202
10.8.
Inner fluctuations of the metric
203
11.
The spectral action principle
206
11.1.
Terms in
Л2
in the spectral action and scalar curvature
210
11.2.
Seeley-DeWitt coefficients and Gilkey s theorem
216
11.3.
The generalized Lichnerowicz formula
217
11.4.
The Einstein-Yang-Mills system
218
11.5.
Scale independent terms in the spectral action
223
11.6.
Spectral action with dilaton
227
12.
Noncommutative
geometry and the Standard Model
230
13.
The finite
noncommutative
geometry
234
13.1.
The
subalgebra
and the order one condition
238
13.2.
The bimodule
H f
and
fermions
240
13.3.
Unimodularity and hypercharges
243
13.4.
The classification of Dirac operators
246
13.5.
Moduli space of Dirac operators and Yukawa parameters
252
13.6.
The intersection pairing of the finite geometry
255
CONTENTS
vii
14.
The product geometry
257
14.1.
The real part of the product geometry
258
15.
Bosons as inner fluctuations
259
15.1.
The local gauge transformations
259
15.2.
Discrete part of the inner fluctuations and the Higgs field
260
15.3.
Powers of Z^0 1)
262
15.4.
Continuous part of the inner fluctuations and gauge
bosons
265
15.5.
Independence of the boson fields
269
15.6.
The Dirac operator and its square
269
16.
The spectral action and the Standard Model Lagrangian
271
16.1.
The asymptotic expansion of the spectral action on
M x F
271
16.2.
Fermionic action and Pfaffian
275
16.3.
Fermion doubling, Pfaffian and
Majorana
fermions
277
17.
The Standard Model Lagrangian from the spectral action
280
17.1.
Change of variables in the asymptotic formula and unification
281
17.2.
Coupling constants at unification
282
17.3.
The coupling of
fermions
284
17.4.
The mass relation at unification
292
17.5.
The see-saw mechanism
293
17.6.
The mass relation and the top quark mass
295
17.7.
The self-interaction of the gauge bosons
298
17.8.
The minimal coupling of the Higgs field
300
17.9.
The Higgs field self-interaction
302
17.10.
The Higgs scattering parameter and the Higgs mass
304
17.11.
The gravitational terms
306
17.12.
The parameters of the Standard Model
308
18.
Functional integral
309
18.1.
Real orientation and volume form
311
18.2.
The reconstruction of spin manifolds
313
18.3.
Irreducible finite geometries of
iťO-dimension
6 314
18.4.
The functional integral and open questions
316
19.
Dimensional regularization and
noncommutative
geometry
318
19.1.
Chir
al
anomalies
318
19.2.
The spaces Xz
322
19.3.
Chiral gauge transformations
325
19.4.
Finiteness of anomalous graphs and relation with
residues
326
19.5.
The simplest anomalous graphs
329
19.6.
Anomalous graphs in dimension
2
and the local index cocycle
335
Chapter
2.
The Riemann
zeta
function and
noncommutative
geometry
341
1.
Introduction
341
2.
Counting primes and the
zeta
function
345
3.
Classical and quantum mechanics of
zeta
351
viii CONTENTS
3.1.
Spectral
lines and the Riemann flow
352
3.2.
Symplectic volume and the scaling Hamiltonian
354
3.3.
Quantum system and prolate functions
356
4.
Principal values from the local trace formula
362
4.1.
Normalization of
Haar
measure on a modulated group
364
4.2.
Principal values
366
5.
Quantum states of the scaling flow
370
5.1.
Quantized calculus
372
5.2.
Proof of Theorem
2.18 375
6.
The map
<£ 377
6.1.
Hermite-Weber approximation and Riemann s
ξ
function
378
7.
The
adèle
class space: finitely many degrees of freedom
381
7.1.
Geometry of the semi-local
adèle
class space
383
7.2.
The Hubert space L2(XS) and the trace formula
388
8.
Weil s formulation of the explicit formulas
396
8.1.
L-functions
396
8.2.
Weil s explicit formula
398
8.3.
Fourier transform on
Ск
399
8.4.
Computation of the principal values
400
8.5.
Reformulation of the explicit formula
406
9.
Spectral realization of critical zeros of L-functions
407
9.1.
L-functions and homogeneous distributions on
Ак
409
9.2.
Approximate units in the Sobolev spaces
L¿(Ck)
414
9.3.
Proof of Theorem
2.47 416
10.
A Lefschetz formula for Archimedean local factors
421
10.1.
Archimedean local L-factors
422
10.2.
Asymptotic form of the number of zeros of L-functions
423
10.3.
Weil form of logarithmic derivatives of local factors
424
10.4.
Lefschetz formula for complex places
427
10.5.
Lefschetz formula for real places
428
10.6.
The question of the spectral realization
431
10.7.
Local factors for curves
434
10.8.
Analogy with dimensional regularization
435
Chapter
3.
Quantum statistical mechanics and Galois symmetries
437
1.
Overview: three systems
437
2.
Quantum statistical mechanics
442
2.1.
Observables
and time evolution
444
2.2.
The KMS condition
445
2.3.
Symmetries
449
2.4.
Warming up and cooling down
451
2.5.
Pushforward of KMS states
451
3.
Q-lattices and commensurability
452
4.
1-dimensional Q-lattices
454
4.1.
The Bost-Connes system
458
CONTENTS ix
4.2. Hecke
algebras
459
4.3.
Symmetries of the
ВС
system
461
4.4.
The arithmetic
subalgebra
462
4.5.
Class field theory and the
Kronecker-
Weber theorem
470
4.6.
KMS states and class field theory
474
4.7.
The class field theory problem: algebras and fields
476
4.8.
The Shimura variety of Gm
479
4.9.
QSM and QFT of 1-dimensional Q-lattices
481
5.
2-dimensional Q-lattices
483
5.1.
Elliptic curves and
Tate
modules
486
5.2.
Algebras and groupoids
488
5.3.
Time evolution and regular representation
493
5.4.
Symmetries
495
6.
The modular field
501
6.1.
The modular field of level
N =1 502
6.2.
Modular field of level
N 504
6.3.
Modular functions and modular forms
511
6.4.
Explicit computations for
N = 2
and
N = 4 513
6.5.
The modular field
F
and Q-lattices
514
7.
Arithmetic of the GL2 system
518
7.1.
The arithmetic
subalgebra:
explicit elements
518
7.2.
The arithmetic
subalgebra:
definition
521
7.3.
Division relations in the arithmetic algebra
527
7.4.
KMS states
532
7.5.
Action of symmetries on KMS states
541
7.6.
Low-temperature KMS states and Galois action
542
7.7.
The high temperature range
543
7.8.
The Shimura variety of GL2
545
7.9.
The
noncommutative
boundary of modular curves
546
7.10.
Compatibility between the systems
549
8.
KMS states and complex multiplication
551
8.1.
1-dimensional K-lattices
551
8.2.
K-lattices and Q-lattices
553
8.3.
Adelic description of K-lattices
554
8.4.
Algebra and time evolution
556
8.5.
K-lattices and ideals
558
8.6.
Arithmetic
subalgebra
559
8.7.
Symmetries
560
8.8.
Low-temperature KMS states and Galois action
563
8.9.
High temperature KMS states
569
8.10.
Comparison with other systems
572
9.
Quantum statistical mechanics of Shimura varieties
574
Chapter
4.
Endomotives, thermodynamics, and the Weil explicit
formula
577
CONTENTS
1. Morphisms
and categories of
noncommutative
spaces
582
1.1.
The
ÜT/f-category 582
1.2.
The cyclic category
585
1.3.
The non-unital case
588
1.4.
Cyclic (co)homology
589
2.
Endomotives
591
2.1.
Algebraic endomotives
594
2.2.
Analytic endomotives
598
2.3.
Galois action
600
2.4.
Uniform systems and measured endomotives
603
2.5.
Compatibility of endomotives categories
604
2.6.
Self-maps of algebraic varieties
606
2.7.
The Bost-Connes endomotive
607
3.
Motives and noncommutative spaces: higher dimensional
perspectives
610
3.1.
Geometric correspondences
610
3.2.
Algebraic cycles and
.ří-theory
612
4.
A thermodynamic Frobenius in characteristic zero
615
4.1.
Tomita s theory and the modular automorphism group
616
4.2.
Regular extremal KMS states (cooling)
618
4.3.
The dual system
622
4.4.
Field extensions and duality of factors (an analogy)
623
4.5.
Low temperature KMS states and scaling
626
4.6.
The kernel of the dual trace
631
4.7.
Holomorphic modules
634
4.8.
The cooling morphism (distillation)
636
4.9.
Distillation of the Bost-Connes endomotive
638
4.10.
Spectral realization
648
5.
A cohomological Lefschetz trace formula
650
5.1.
The
adèle
class space of a global field
651
5.2.
The cyclic module of the
adèle
class space
652
5.3.
The restriction map to the
idèle
class group
653
5.4.
The Morita equivalence and cokernel for
К
=
Q
654
5.5.
The cokernel of
p
for general global fields
656
5.6.
Trace pairing and vanishing
670
5.7.
Weil s explicit formula as a trace formula
671
5.8.
Weil
positivity
and the Riemann Hypothesis
672
6.
The Weil proof for function fields
674
6.1.
Function fields and their
zeta
functions
675
6.2.
Correspondences and divisors in
C x C
678
6.3.
Frobenius correspondences and effective divisors
680
6.4.
Positivity
in the Weil proof
682
7.
A noncommutative geometry perspective
685
7.1.
Distributional trace of a flow
686
7.2.
The periodic orbits of the action of CK on XK
690
CONTENTS xi
7.3. Probenius
(scaling) correspondence and the trace formula
691
7.4.
The Fubini theorem and trivial correspondences
693
7.5.
The curve inside the
adèle
class space
694
7.6.
Vortex configurations (an analogy)
712
7.7.
Building a dictionary
721
8.
The analogy between QG and RH
723
8.1.
KMS states and the electroweak phase transition
723
8.2.
Observables
in QG
727
8.3.
Invertibility at low temperature
729
8.4.
Spectral correspondences
730
8.5.
Spectral cobordisms
730
8.6.
Scaling action
730
8.7.
Moduli spaces for Q-lattices and spectral correspondences
731
Appendix
733
1.
Operator algebras
733
1.1.
C*-algebras
733
1.2. Von
Neumann algebras
734
1.3.
The passing of time
738
2.
Galois theory
741
Bibliography
749
Index
763
|
| adam_txt |
Contents
Preface
xiii
Chapter
1.
Quantum fields,
noncommutative
spaces, and motives
1
1.
Introduction
1
2.
Basics of perturbative QFT
7
2.1.
Lagrangian and Hamiltonian formalisms
8
2.2.
Lagrangian and the Feynman integral
10
2.3.
The Hamiltonian and canonical quantization
11
2.4.
The simplest example
13
2.5.
Green's functions
17
2.6.
Wick rotation and Euclidean Green's functions
18
3.
Feynman diagrams
22
3.1.
The simplest case
23
3.2.
The origins of renormalization
27
3.3.
Feynman graphs and rules
31
3.4.
Connected Green's functions
35
3.5.
The effective action and one-particle irreducible graphs
37
3.6.
Physically observable parameters
41
3.7.
The physics idea of renormalization
43
4.
Dimensional regularization
46
5.
The graph by graph method of Bogoliubov-Parasiuk-Hepp-
Zimmermann
52
5.1.
The simplest example of
subdivergence
54
5.2.
Superficial degree of divergence
58
5.3.
Subdivergences
and preparation
59
6.
The Connes-Kreimer theory of perturbative renormalization
66
6.1.
Commutative
Hopf
algebras and
affine
group schemes
67
6.2.
The
Hopf
algebra of Feynman graphs: discrete part
71
6.3.
The
Hopf
algebra of Feynman graphs: full structure
78
6.4.
BPHZ as a Birkhoff factorization
81
6.5.
Diffeographisms and diffeomorphisms
88
6.6.
The renormalization group
89
7.
Renormalization and the Riemann-Hilbert correspondence
95
7.1.
Counterterms and time-ordered exponentials
96
7.2.
Flat equisingular connections
103
7.3.
Equivariant principal bundles and the group G*
=
G x
Gm
114
CONTENTS
7.4. Tannakian
categories and
affine
group schemes
119
7.5.
Differential Galois theory and the local Riemann-Hilbert
correspondence
123
7.6.
Universal
Hopf
algebra and the Riemann-Hilbert
correspondence
128
8.
Motives in a nutshell
137
8.1.
Algebraic varieties and motives
137
8.2.
Pure motives
146
8.3.
Mixed motives
151
8.4.
Mixed Hodge structures
156
8.5.
Tate
motives, periods, and quantum fields
159
9.
The Standard Model of elementary particles
160
9.1.
Particles and interactions
162
9.2.
Symmetries
163
9.3.
Quark mixing: the
CKM
matrix
166
9.4.
The Standard Model Lagrangian
166
9.5.
Quantum level: anomalies, ghosts, gauge fixing
170
9.6.
Massive neutrinos
174
9.7.
The Standard Model minimally coupled to gravity
179
9.8.
Higher derivative terms in gravity
183
9.9.
Symmetries as diffeomorphisms
184
10.
The framework of (metric)
noncommutative
geometry
186
10.1.
Spectral geometry
187
10.2.
Spectral triples
190
10.3.
The real part of a real spectral triple
192
10.4. Hochschild
and cyclic cohomology
193
10.5.
The local index cocycle
198
10.6.
Positivity
in
Hochschild
cohomology and Yang-Mills action
201
10.7.
Cyclic cohomology and Chern-Simons action
202
10.8.
Inner fluctuations of the metric
203
11.
The spectral action principle
206
11.1.
Terms in
Л2
in the spectral action and scalar curvature
210
11.2.
Seeley-DeWitt coefficients and Gilkey's theorem
216
11.3.
The generalized Lichnerowicz formula
217
11.4.
The Einstein-Yang-Mills system
218
11.5.
Scale independent terms in the spectral action
223
11.6.
Spectral action with dilaton
227
12.
Noncommutative
geometry and the Standard Model
230
13.
The finite
noncommutative
geometry
234
13.1.
The
subalgebra
and the order one condition
238
13.2.
The bimodule
H f
and
fermions
240
13.3.
Unimodularity and hypercharges
243
13.4.
The classification of Dirac operators
246
13.5.
Moduli space of Dirac operators and Yukawa parameters
252
13.6.
The intersection pairing of the finite geometry
255
CONTENTS
vii
14.
The product geometry
257
14.1.
The real part of the product geometry
258
15.
Bosons as inner fluctuations
259
15.1.
The local gauge transformations
259
15.2.
Discrete part of the inner fluctuations and the Higgs field
260
15.3.
Powers of Z^0'1)
262
15.4.
Continuous part of the inner fluctuations and gauge
bosons
265
15.5.
Independence of the boson fields
269
15.6.
The Dirac operator and its square
269
16.
The spectral action and the Standard Model Lagrangian
271
16.1.
The asymptotic expansion of the spectral action on
M x F
271
16.2.
Fermionic action and Pfaffian
275
16.3.
Fermion doubling, Pfaffian and
Majorana
fermions
277
17.
The Standard Model Lagrangian from the spectral action
280
17.1.
Change of variables in the asymptotic formula and unification
281
17.2.
Coupling constants at unification
282
17.3.
The coupling of
fermions
284
17.4.
The mass relation at unification
292
17.5.
The see-saw mechanism
293
17.6.
The mass relation and the top quark mass
295
17.7.
The self-interaction of the gauge bosons
298
17.8.
The minimal coupling of the Higgs field
300
17.9.
The Higgs field self-interaction
302
17.10.
The Higgs scattering parameter and the Higgs mass
304
17.11.
The gravitational terms
306
17.12.
The parameters of the Standard Model
308
18.
Functional integral
309
18.1.
Real orientation and volume form
311
18.2.
The reconstruction of spin manifolds
313
18.3.
Irreducible finite geometries of
iťO-dimension
6 314
18.4.
The functional integral and open questions
316
19.
Dimensional regularization and
noncommutative
geometry
318
19.1.
Chir
al
anomalies
318
19.2.
The spaces Xz
322
19.3.
Chiral gauge transformations
325
19.4.
Finiteness of anomalous graphs and relation with
residues
326
19.5.
The simplest anomalous graphs
329
19.6.
Anomalous graphs in dimension
2
and the local index cocycle
335
Chapter
2.
The Riemann
zeta
function and
noncommutative
geometry
341
1.
Introduction
341
2.
Counting primes and the
zeta
function
345
3.
Classical and quantum mechanics of
zeta
351
viii CONTENTS
3.1.
Spectral
lines and the Riemann flow
352
3.2.
Symplectic volume and the scaling Hamiltonian
354
3.3.
Quantum system and prolate functions
356
4.
Principal values from the local trace formula
362
4.1.
Normalization of
Haar
measure on a modulated group
364
4.2.
Principal values
366
5.
Quantum states of the scaling flow
370
5.1.
Quantized calculus
372
5.2.
Proof of Theorem
2.18 375
6.
The map
<£ 377
6.1.
Hermite-Weber approximation and Riemann's
ξ
function
378
7.
The
adèle
class space: finitely many degrees of freedom
381
7.1.
Geometry of the semi-local
adèle
class space
383
7.2.
The Hubert space L2(XS) and the trace formula
388
8.
Weil's formulation of the explicit formulas
396
8.1.
L-functions
396
8.2.
Weil's explicit formula
398
8.3.
Fourier transform on
Ск
399
8.4.
Computation of the principal values
400
8.5.
Reformulation of the explicit formula
406
9.
Spectral realization of critical zeros of L-functions
407
9.1.
L-functions and homogeneous distributions on
Ак
409
9.2.
Approximate units in the Sobolev spaces
L¿(Ck)
414
9.3.
Proof of Theorem
2.47 416
10.
A Lefschetz formula for Archimedean local factors
421
10.1.
Archimedean local L-factors
422
10.2.
Asymptotic form of the number of zeros of L-functions
423
10.3.
Weil form of logarithmic derivatives of local factors
424
10.4.
Lefschetz formula for complex places
427
10.5.
Lefschetz formula for real places
428
10.6.
The question of the spectral realization
431
10.7.
Local factors for curves
434
10.8.
Analogy with dimensional regularization
435
Chapter
3.
Quantum statistical mechanics and Galois symmetries
437
1.
Overview: three systems
437
2.
Quantum statistical mechanics
442
2.1.
Observables
and time evolution
444
2.2.
The KMS condition
445
2.3.
Symmetries
449
2.4.
Warming up and cooling down
451
2.5.
Pushforward of KMS states
451
3.
Q-lattices and commensurability
452
4.
1-dimensional Q-lattices
454
4.1.
The Bost-Connes system
458
CONTENTS ix
4.2. Hecke
algebras
459
4.3.
Symmetries of the
ВС
system
461
4.4.
The arithmetic
subalgebra
462
4.5.
Class field theory and the
Kronecker-
Weber theorem
470
4.6.
KMS states and class field theory
474
4.7.
The class field theory problem: algebras and fields
476
4.8.
The Shimura variety of Gm
479
4.9.
QSM and QFT of 1-dimensional Q-lattices
481
5.
2-dimensional Q-lattices
483
5.1.
Elliptic curves and
Tate
modules
486
5.2.
Algebras and groupoids
488
5.3.
Time evolution and regular representation
493
5.4.
Symmetries
495
6.
The modular field
501
6.1.
The modular field of level
N =1 502
6.2.
Modular field of level
N 504
6.3.
Modular functions and modular forms
511
6.4.
Explicit computations for
N = 2
and
N = 4 513
6.5.
The modular field
F
and Q-lattices
514
7.
Arithmetic of the GL2 system
518
7.1.
The arithmetic
subalgebra:
explicit elements
518
7.2.
The arithmetic
subalgebra:
definition
521
7.3.
Division relations in the arithmetic algebra
527
7.4.
KMS states
532
7.5.
Action of symmetries on KMS states
541
7.6.
Low-temperature KMS states and Galois action
542
7.7.
The high temperature range
543
7.8.
The Shimura variety of GL2
545
7.9.
The
noncommutative
boundary of modular curves
546
7.10.
Compatibility between the systems
549
8.
KMS states and complex multiplication
551
8.1.
1-dimensional K-lattices
551
8.2.
K-lattices and Q-lattices
553
8.3.
Adelic description of K-lattices
554
8.4.
Algebra and time evolution
556
8.5.
K-lattices and ideals
558
8.6.
Arithmetic
subalgebra
559
8.7.
Symmetries
560
8.8.
Low-temperature KMS states and Galois action
563
8.9.
High temperature KMS states
569
8.10.
Comparison with other systems
572
9.
Quantum statistical mechanics of Shimura varieties
574
Chapter
4.
Endomotives, thermodynamics, and the Weil explicit
formula
577
CONTENTS
1. Morphisms
and categories of
noncommutative
spaces
582
1.1.
The
ÜT/f-category 582
1.2.
The cyclic category
585
1.3.
The non-unital case
588
1.4.
Cyclic (co)homology
589
2.
Endomotives
591
2.1.
Algebraic endomotives
594
2.2.
Analytic endomotives
598
2.3.
Galois action
600
2.4.
Uniform systems and measured endomotives
603
2.5.
Compatibility of endomotives categories
604
2.6.
Self-maps of algebraic varieties
606
2.7.
The Bost-Connes endomotive
607
3.
Motives and noncommutative spaces: higher dimensional
perspectives
610
3.1.
Geometric correspondences
610
3.2.
Algebraic cycles and
.ří-theory
612
4.
A thermodynamic "Frobenius" in characteristic zero
615
4.1.
Tomita's theory and the modular automorphism group
616
4.2.
Regular extremal KMS states (cooling)
618
4.3.
The dual system
622
4.4.
Field extensions and duality of factors (an analogy)
623
4.5.
Low temperature KMS states and scaling
626
4.6.
The kernel of the dual trace
631
4.7.
Holomorphic modules
634
4.8.
The cooling morphism (distillation)
636
4.9.
Distillation of the Bost-Connes endomotive
638
4.10.
Spectral realization
648
5.
A cohomological Lefschetz trace formula
650
5.1.
The
adèle
class space of a global field
651
5.2.
The cyclic module of the
adèle
class space
652
5.3.
The restriction map to the
idèle
class group
653
5.4.
The Morita equivalence and cokernel for
К
=
Q
654
5.5.
The cokernel of
p
for general global fields
656
5.6.
Trace pairing and vanishing
670
5.7.
Weil's explicit formula as a trace formula
671
5.8.
Weil
positivity
and the Riemann Hypothesis
672
6.
The Weil proof for function fields
674
6.1.
Function fields and their
zeta
functions
675
6.2.
Correspondences and divisors in
C x C
678
6.3.
Frobenius correspondences and effective divisors
680
6.4.
Positivity
in the Weil proof
682
7.
A noncommutative geometry perspective
685
7.1.
Distributional trace of a flow
686
7.2.
The periodic orbits of the action of CK on XK
690
CONTENTS xi
7.3. Probenius
(scaling) correspondence and the trace formula
691
7.4.
The Fubini theorem and trivial correspondences
693
7.5.
The curve inside the
adèle
class space
694
7.6.
Vortex configurations (an analogy)
712
7.7.
Building a dictionary
721
8.
The analogy between QG and RH
723
8.1.
KMS states and the electroweak phase transition
723
8.2.
Observables
in QG
727
8.3.
Invertibility at low temperature
729
8.4.
Spectral correspondences
730
8.5.
Spectral cobordisms
730
8.6.
Scaling action
730
8.7.
Moduli spaces for Q-lattices and spectral correspondences
731
Appendix
733
1.
Operator algebras
733
1.1.
C*-algebras
733
1.2. Von
Neumann algebras
734
1.3.
The passing of time
738
2.
Galois theory
741
Bibliography
749
Index
763 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Connes, Alain 1947- Marcolli, Matilde 1969- |
| author_GND | (DE-588)112614760 (DE-588)141750545 |
| author_facet | Connes, Alain 1947- Marcolli, Matilde 1969- |
| author_role | aut aut |
| author_sort | Connes, Alain 1947- |
| author_variant | a c ac m m mm |
| building | Verbundindex |
| bvnumber | BV035016066 |
| callnumber-first | Q - Science |
| callnumber-label | QC20 |
| callnumber-raw | QC20.7.D52 |
| callnumber-search | QC20.7.D52 |
| callnumber-sort | QC 220.7 D52 |
| callnumber-subject | QC - Physics |
| classification_rvk | SK 370 |
| ctrlnum | (OCoLC)165958837 (DE-599)BVBBV035016066 |
| dewey-full | 512/.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.55 |
| dewey-search | 512/.55 |
| dewey-sort | 3512 255 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV035016066 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:45:24Z |
| indexdate | 2024-07-09T21:20:15Z |
| institution | BVB |
| isbn | 9780821842102 |
| language | English |
| lccn | 2007060843 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016685231 |
| oclc_num | 165958837 |
| open_access_boolean | |
| owner | DE-703 DE-20 DE-634 DE-83 DE-11 DE-188 DE-384 DE-29T DE-19 DE-BY-UBM |
| owner_facet | DE-703 DE-20 DE-634 DE-83 DE-11 DE-188 DE-384 DE-29T DE-19 DE-BY-UBM |
| physical | XXII, 785 Seiten Illustrationen, Diagramme |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | American Mathematical Society Colloquium publications |
| series2 | American Mathematical Society Colloquium publications |
| spelling | Connes, Alain 1947- Verfasser (DE-588)112614760 aut Noncommutative geometry, quantum fields and motives Alain Connes ; Matilde Marcolli Providence, Rhode Island American Mathematical Society 2008 © 2008 XXII, 785 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier American Mathematical Society Colloquium publications volume 55 Literaturverzeichnis: Seiten 749 - 761 Análise global larpcal Corpos globais larpcal Geometria algébrica larpcal Mecânica quântica larpcal Teoria analítica dos números larpcal Teoria dos números larpcal Álgebra larpcal Noncommutative differential geometry Quantum field theory Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd rswk-swf Motiv Mathematik (DE-588)4197596-0 gnd rswk-swf Quantenfeldtheorie (DE-588)4047984-5 gnd rswk-swf Nichtkommutative Differentialgeometrie (DE-588)4311174-9 s Quantenfeldtheorie (DE-588)4047984-5 s Motiv Mathematik (DE-588)4197596-0 s DE-604 Marcolli, Matilde 1969- Verfasser (DE-588)141750545 aut Erscheint auch als Online-Ausgabe 978-1-4704-3201-0 American Mathematical Society Colloquium publications volume 55 (DE-604)BV035417609 55 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016685231&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Connes, Alain 1947- Marcolli, Matilde 1969- Noncommutative geometry, quantum fields and motives American Mathematical Society Colloquium publications Análise global larpcal Corpos globais larpcal Geometria algébrica larpcal Mecânica quântica larpcal Teoria analítica dos números larpcal Teoria dos números larpcal Álgebra larpcal Noncommutative differential geometry Quantum field theory Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd Motiv Mathematik (DE-588)4197596-0 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd |
| subject_GND | (DE-588)4311174-9 (DE-588)4197596-0 (DE-588)4047984-5 |
| title | Noncommutative geometry, quantum fields and motives |
| title_auth | Noncommutative geometry, quantum fields and motives |
| title_exact_search | Noncommutative geometry, quantum fields and motives |
| title_exact_search_txtP | Noncommutative geometry, quantum fields and motives |
| title_full | Noncommutative geometry, quantum fields and motives Alain Connes ; Matilde Marcolli |
| title_fullStr | Noncommutative geometry, quantum fields and motives Alain Connes ; Matilde Marcolli |
| title_full_unstemmed | Noncommutative geometry, quantum fields and motives Alain Connes ; Matilde Marcolli |
| title_short | Noncommutative geometry, quantum fields and motives |
| title_sort | noncommutative geometry quantum fields and motives |
| topic | Análise global larpcal Corpos globais larpcal Geometria algébrica larpcal Mecânica quântica larpcal Teoria analítica dos números larpcal Teoria dos números larpcal Álgebra larpcal Noncommutative differential geometry Quantum field theory Nichtkommutative Differentialgeometrie (DE-588)4311174-9 gnd Motiv Mathematik (DE-588)4197596-0 gnd Quantenfeldtheorie (DE-588)4047984-5 gnd |
| topic_facet | Análise global Corpos globais Geometria algébrica Mecânica quântica Teoria analítica dos números Teoria dos números Álgebra Noncommutative differential geometry Quantum field theory Nichtkommutative Differentialgeometrie Motiv Mathematik Quantenfeldtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016685231&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035417609 |
| work_keys_str_mv | AT connesalain noncommutativegeometryquantumfieldsandmotives AT marcollimatilde noncommutativegeometryquantumfieldsandmotives |