Noncommutative Geometry, Quantum Fields and Motives Alain Connes Matilde Marcolli

685 Pages·2012·5.15 MB·English

Noncommutative Geometry, Quantum Fields
and Motives
Alain Connes
Matilde Marcolli
A. Connes: Coll(cid:18)ege de France, 3, rue d’Ulm, Paris, F-75005 France,
I.H.E.S. and Vanderbilt University
E-mail address: [email protected]
M. Marcolli: Max{Planck Institut fu(cid:127)r Mathematik, Vivatsgasse 7,
Bonn, D-53111 Germany
E-mail address: [email protected]

Contents
Preface 9
Chapter 1. Quantum (cid:12)elds, noncommutative spaces, and motives 17
1. Introduction 17
2. Basics of perturbative QFT 22
2.1. Lagrangian and Hamiltonian formalisms 23
2.2. Lagrangian and the Feynman integral 25
2.3. The Hamiltonian and canonical quantization 26
2.4. The simplest example 28
2.5. Green’s functions 31
2.6. Wick rotation and Euclidean Green’s functions 32
3. Feynman diagrams 35
3.1. The simplest case 36
3.2. The origins of renormalization 40
3.3. Feynman graphs and rules 43
3.4. Connected Green’s functions 47
3.5. The e(cid:11)ective action and one-particle irreducible graphs 48
3.6. Physically observable parameters 52
3.7. The physics idea of renormalization 54
4. Dimensional regularization 56
5. The graph by graph method of Bogoliubov{Parasiuk{Hepp{Zimmermann 62
5.1. The simplest example of subdivergence 63
5.2. Super(cid:12)cial degree of divergence 67
5.3. Subdivergences and preparation 68
6. The Connes{Kreimer theory of perturbative renormalization 74
6.1. Commutative Hopf algebras and a(cid:14)ne group schemes 75
6.2. The Hopf algebra of Feynman graphs: discrete part 78
6.3. The Hopf algebra of Feynman graphs: full structure 85
6.4. BPHZ as a Birkho(cid:11) factorization 87
6.5. Di(cid:11)eographisms and di(cid:11)eomorphisms 93
6.6. The renormalization group 95
7. Renormalization and the Riemann{Hilbert correspondence 100
7.1. Counterterms and time-ordered exponentials 100
7.2. Flat equisingular connections 107
7.3. Equivariant principal bundles and the group G = GoG 116
(cid:3) m
7.4. Tannakian categories and a(cid:14)ne group schemes 120
7.5. Di(cid:11)erential Galois theory and the local Riemann{Hilbert correspondence124
7.6. Universal Hopf algebra and the Riemann{Hilbert correspondence 128
8. Motives in a nutshell 136
3
CONTENTS 4
8.1. Algebraic varieties and motives 136
8.2. Pure motives 144
8.3. Mixed motives 148
8.4. Mixed Hodge structures 152
8.5. Tate motives, periods, and quantum (cid:12)elds 155
9. The Standard Model of elementary particles 156
9.1. Particles and interactions 158
9.2. Symmetries 159
9.3. Quark mixing: the CKM matrix 161
9.4. The Standard Model Lagrangian 161
9.5. Quantum level: anomalies, ghosts, gauge (cid:12)xing 164
9.6. Massive neutrinos 168
9.7. The Standard Model minimally coupled to gravity 172
9.8. Higher derivative terms in gravity 176
9.9. Symmetries as di(cid:11)eomorphisms 177
10. The framework of (metric) noncommutative geometry 178
10.1. Spectral geometry 179
10.2. Spectral triples 182
10.3. The real part of a real spectral triple 183
10.4. Hochschild and cyclic cohomology 184
10.5. The local index cocycle 188
10.6. Positivity in Hochschild cohomology and Yang-Mills action 191
10.7. Cyclic cohomology and Chern-Simons action 193
10.8. Inner (cid:13)uctuations of the metric 194
11. The spectral action principle 196
11.1. Terms in (cid:3)2 in the spectral action and scalar curvature 199
11.2. Seeley{DeWitt coe(cid:14)cients and Gilkey’s theorem 205
11.3. The generalized Lichnerowicz formula 206
11.4. The Einstein{Yang{Mills system 207
11.5. Scale independent terms in the spectral action 211
11.6. Spectral action with dilaton 215
12. Noncommutative geometry and the Standard Model 217
13. The (cid:12)nite noncommutative geometry 221
13.1. The subalgebra and the order one condition 224
13.2. The bimodule and fermions 226
F
H
13.3. Unimodularity and hypercharges 229
13.4. The classi(cid:12)cation of Dirac operators 231
13.5. Moduli space of Dirac operators and Yukawa parameters 236
13.6. The intersection pairing of the (cid:12)nite geometry 239
14. The product geometry 241
14.1. The real part of the product geometry 242
15. Bosons as inner (cid:13)uctuations 242
15.1. The local gauge transformations 243
15.2. Discrete part of the inner (cid:13)uctuations and the Higgs (cid:12)eld 244
15.3. Powers of D(0;1) 246
15.4. Continuous part of the inner (cid:13)uctuations and gauge bosons 248
15.5. Independence of the boson (cid:12)elds 251
15.6. The Dirac operator and its square 252
CONTENTS 5
16. The spectral action and the Standard Model Lagrangian 253
16.1. The asymptotic expansion of the spectral action on M F 253
(cid:2)
16.2. Fermionic action and Pfa(cid:14)an 257
16.3. Fermion doubling, Pfa(cid:14)an and Majorana fermions 259
17. The Standard Model Lagrangian from the spectral action 261
17.1. Change of variables in the asymptotic formula and uni(cid:12)cation 262
17.2. Coupling constants at uni(cid:12)cation 263
17.3. The coupling of fermions 264
17.4. The mass relation at uni(cid:12)cation 272
17.5. The see-saw mechanism 273
17.6. The mass relation and the top quark mass 275
17.7. The self-interaction of the gauge bosons 277
17.8. The minimal coupling of the Higgs (cid:12)eld 279
17.9. The Higgs (cid:12)eld self-interaction 281
17.10. The Higgs scattering parameter and the Higgs mass 282
17.11. The gravitational terms 284
17.12. The parameters of the Standard Model 287
18. Functional integral 287
18.1. Real orientation and volume form 289
18.2. The reconstruction of spin manifolds 291
18.3. Irreducible (cid:12)nite geometries of KO-dimension 6 292
18.4. The functional integral and open questions 294
19. Dimensional regularization and noncommutative geometry 295
19.1. Chiral anomalies 295
19.2. The spaces X 299
z
19.3. Chiral gauge transformations 301
19.4. Finiteness of anomalous graphs and relation with residues 303
19.5. The simplest anomalous graphs 305
19.6. Anomalous graphs in dimension 2 and the local index cocycle 310
Chapter 2. The Riemann zeta function and noncommutative geometry 315
1. Introduction 315
2. Counting primes and the zeta function 318
3. Classical and quantum mechanics of zeta 324
3.1. Spectral lines and the Riemann (cid:13)ow 325
3.2. Symplectic volume and the scaling Hamiltonian 326
3.3. Quantum system and prolate functions 329
4. Principal values from the local trace formula 333
4.1. Normalization of Haar measure on a modulated group 335
4.2. Principal values 337
5. Quantum states of the scaling (cid:13)ow 341
5.1. Quantized calculus 342
5.2. Proof of Theorem 2.18 346
6. The map E 347
6.1. Hermite{Weber approximation and Riemann’s (cid:24) function 348
7. The ad(cid:18)ele class space: (cid:12)nitely many degrees of freedom 351
7.1. Geometry of the semi-local ad(cid:18)ele class space 352
7.2. The Hilbert space L2(X ) and the trace formula 357
S
CONTENTS 6
8. Weil’s formulation of the explicit formulas 364
8.1. L-functions 364
8.2. Weil’s explicit formula 365
8.3. Fourier transform on C 366
K
8.4. Computation of the principal values 367
8.5. Reformulation of the explicit formula 372
9. Spectral realization of critical zeros of L-functions 373
9.1. L-functions and homogeneous distributions on A . 375
K
9.2. Approximate units in the Sobolev spaces L2(C ) 380
(cid:14) K
9.3. Proof of Theorem 2.47 382
10. A Lefschetz formula for Archimedean local factors 386
10.1. Archimedean local L-factors 387
10.2. Asymptotic form of the number of zeros of L-functions 389
10.3. Weil form of logarithmic derivatives of local factors 389
10.4. Lefschetz formula for complex places 392
10.5. Lefschetz formula for real places 393
10.6. The question of the spectral realization 396
10.7. Local factors for curves 398
10.8. Analogy with dimensional regularization 398
Chapter 3. Quantum statistical mechanics and Galois symmetries 400
1. Overview: three systems 400
2. Quantum statistical mechanics 404
2.1. Observables and time evolution 406
2.2. The KMS condition 407
2.3. Symmetries 410
2.4. Warming up and cooling down 412
2.5. Pushforward of KMS states 413
3. Q-lattices and commensurability 413
4. 1-dimensional Q-lattices 415
4.1. The Bost{Connes system 418
4.2. Hecke algebras 419
4.3. Symmetries of the BC system 421
4.4. The arithmetic subalgebra 422
4.5. Class (cid:12)eld theory and the Kronecker{Weber theorem 429
4.6. KMS states and class (cid:12)eld theory 432
4.7. The class (cid:12)eld theory problem: algebras and (cid:12)elds 434
4.8. The Shimura variety of G 437
m
4.9. QSM and QFT of 1-dimensional Q-lattices 439
5. 2-dimensional Q-lattices 440
5.1. Elliptic curves and Tate modules 443
5.2. Algebras and groupoids 445
5.3. Time evolution and regular representation 450
5.4. Symmetries 451
6. The modular (cid:12)eld 457
6.1. The modular (cid:12)eld of level N = 1 457
6.2. Modular (cid:12)eld of level N 460
6.3. Modular functions and modular forms 466
CONTENTS 7
6.4. Explicit computations for N = 2 and N = 4 467
6.5. The modular (cid:12)eld F and Q-lattices 468
7. Arithmetic of the GL system 472
2
7.1. The arithmetic subalgebra: explicit elements 472
7.2. The arithmetic subalgebra: de(cid:12)nition 475
7.3. Division relations in the arithmetic algebra 480
7.4. KMS states 485
7.5. Action of symmetries on KMS states 492
7.6. Low-temperature KMS states and Galois action 493
7.7. The high temperature range 495
7.8. The Shimura variety of GL 496
2
7.9. The noncommutative boundary of modular curves 497
7.10. Compatibility between the systems 499
8. KMS states and complex multiplication 502
8.1. 1-dimensional K-lattices 502
8.2. K-lattices and Q-lattices 503
8.3. Adelic description of K-lattices 504
8.4. Algebra and time evolution 506
8.5. K-lattices and ideals 508
8.6. Arithmetic subalgebra 509
8.7. Symmetries 510
8.8. Low-temperature KMS states and Galois action 512
8.9. High temperature KMS states 517
8.10. Comparison with other systems 520
9. Quantum statistical mechanics of Shimura varieties 522
Chapter 4. Endomotives, thermodynamics, and the Weil explicit formula 524
1. Morphisms and categories of noncommutative spaces 528
1.1. The KK-category 528
1.2. The cyclic category 531
1.3. The non-unital case 534
1.4. Cyclic (co)homology 534
2. Endomotives 536
2.1. Algebraic endomotives 538
2.2. Analytic endomotives 542
2.3. Galois action 544
2.4. Uniform systems and measured endomotives 546
2.5. Compatibility of endomotives categories 547
2.6. Self-maps of algebraic varieties 549
2.7. The Bost{Connes endomotive 550
3. Motives and noncommutative spaces: higher dimensional perspectives 553
3.1. Geometric correspondences 553
3.2. Algebraic cycles and K-theory 554
4. A thermodynamic \Frobenius" in characteristic zero 557
4.1. Tomita’s theory and the modular automorphism group 558
4.2. Regular extremal KMS states (cooling) 560
4.3. The dual system 563
4.4. Field extensions and duality of factors (an analogy) 565
CONTENTS 8
4.5. Low temperature KMS states and scaling 567
4.6. The kernel of the dual trace 571
4.7. Holomorphic modules 574
4.8. The cooling morphism (distillation) 575
4.9. Distillation of the Bost{Connes endomotive 578
4.10. Spectral realization 587
5. A cohomological Lefschetz trace formula 589
5.1. The ad(cid:18)ele class space of a global (cid:12)eld 589
5.2. The cyclic module of the ad(cid:18)ele class space 590
5.3. The restriction map to the id(cid:18)ele class group 591
5.4. The Morita equivalence and cokernel for K = Q 592
5.5. The cokernel of (cid:26) for general global (cid:12)elds 594
5.6. Trace pairing and vanishing 606
5.7. Weil’s explicit formula as a trace formula 606
5.8. Weil positivity and the Riemann Hypothesis 608
6. The Weil proof for function (cid:12)elds 610
6.1. Function (cid:12)elds and their zeta functions 610
6.2. Correspondences and divisors in C C 613
(cid:2)
6.3. Frobenius correspondences and e(cid:11)ective divisors 615
6.4. Positivity in the Weil proof 616
7. A noncommutative geometry perspective 619
7.1. Distributional trace of a (cid:13)ow 620
7.2. The periodic orbits of the action of C on X 624
K K
7.3. Frobenius (scaling) correspondence and the trace formula 625
7.4. The Fubini theorem and trivial correspondences 627
7.5. The curve inside the ad(cid:18)ele class space 628
7.6. Vortex con(cid:12)gurations (an analogy) 644
7.7. Building a dictionary 651
8. The analogy between QG and RH 653
8.1. KMS states and the electroweak phase transition 653
8.2. Observables in QG 657
8.3. Invertibility at low temperature 657
8.4. Spectral correspondences 659
8.5. Spectral cobordisms 659
8.6. Scaling action 660
8.7. Moduli spaces for Q-lattices and spectral correspondences 660
Appendix 662
9. Operator algebras 662
9.1. C -algebras 662
(cid:3)
9.2. Von Neumann algebras 663
9.3. The passing of time 666
10. Galois theory 669
Bibliography 675
Preface
The unifying theme, which the reader will encounter in di(cid:11)erent guises throughout
the book, is the interplay between noncommutative geometry and number theory,
the latter especially in its manifestation through the theory of motives. For us, this
interwoven texture of noncommutative spaces and motives will become a tool in the
exploration of two spaces, whose role is central to many developments of modern
mathematics and physics:
Space-time
(cid:15)
The set of prime numbers
(cid:15)
One may be tempted to think that, looking from the vantage point of those who
sit atop the vast edi(cid:12)ce of our accumulated knowledge of such topics as space and
numbers, we ought to know a great deal about these two spaces. However, there
are two fundamental problems whose di(cid:14)culty is a clear reminder of our limited
knowledge, and whose solution would require a more sophisticated understanding
than the one currently within our immediate grasp:
The construction of a theory of quantum gravity (QG)
(cid:15)
The Riemann hypothesis (RH)
(cid:15)
The purpose of this book is to explain the relevance of noncommutative geometry
(NCG) in dealing with these two problems. Quite surprisingly, in so doing we shall
discover thattherearedeepanalogies betweenthesetwo problemswhich, ifproperly
exploited, are likely to enhance our grasp of both of them.
Although the book is perhaps more aimed at mathematicians than at physicists,
or perhaps precisely for that reason, we choose to begin our account in Chapter 1
squarelyonthephysicsside. Thechapterisdedicatedtodiscussingtwo maintopics:
Renormalization
(cid:15)
The Standard Model of high energy physics
(cid:15)
We try to introduce the material as much as possible in a self-contained way, tak-
ing into consideration the fact that a signi(cid:12)cant number of mathematicians do not
necessarily have quantum (cid:12)eld theory and particle physics as part of their cultural
background. Thus, the (cid:12)rst half of the chapter is dedicated to giving a detailed
account of perturbative quantum (cid:12)eld theory, presented in a manner that, we hope,
is palatable to the mathematician’s taste. In particular, we discuss basic tools,
such as the e(cid:11)ective action and the perturbative expansion in Feynman graphs, as
well as the regularization procedures used to evaluate the corresponding Feynman
integrals. In particular, we concentrate on the procedure known as \dimensional
regularization", bothbecause ofits beingthe one most commonly usedinthe actual
calculations of particle physics, and because of the fact that it admits a very nice
and conceptually simple interpretation in terms of noncommutative geometry, as we
will come to see towards the end of the chapter. In this (cid:12)rst half of Chapter 1 we
9
PREFACE 10
give a new perspective on perturbative quantum (cid:12)eld theory, which gives a clear
mathematical interpretation to the renormalization procedure used by physicists to
extract (cid:12)nite values from the divergent expressions obtained from the evaluations
of the integrals associated to Feynman diagrams. This viewpoint is based on the
Connes{Kreimer theory and then on more recent results by the authors.
Throughout this discussion, we always assume that we work with the procedure
known in physics as \dimensional regularization and minimal substraction". The
basic result of the Connes{Kreimer theory is then to show that the renormalization
procedure corresponds exactly to the Birkho(cid:11) factorization of a loop (cid:13)(z) G
2
associated to the unrenormalized theory evaluated in complex dimension D z,
(cid:0)
where D is the dimension of space-time and z = 0 is the complex parameter used
6
in dimensional regularization. The group G is de(cid:12)ned through its Hopf algebra
of coordinates, which is the Hopf algebra of Feynman graphs of the theory. The
Birkho(cid:11) factorization of the loop gives a canonical way of removing the singularity
at z = 0 and obtaining the required (cid:12)nite result for the physical observables. This
gives renormalization a clear and well de(cid:12)ned conceptual meaning.
The Birkho(cid:11) factorization of loops is a central tool in the construction of solutions
to the \Riemann-Hilbert problem", which consists of (cid:12)nding a di(cid:11)erential equation
with prescribed monodromy. With time, out of this original problem a whole area
of mathematics developed, under the name of \Riemann{Hilbert correspondence".
Broadly speaking, this denotes a way of encoding objects of di(cid:11)erential geometric
nature, such as di(cid:11)erential systems with speci(cid:12)ed types of singularities, in terms of
group representations. In its most general form, the Riemann{Hilbert correspon-
dence is formulated as an equivalence of categories between the two sides. It relies
on the \Tannakian formalism" to reconstruct the group from its category of repre-
sentations. We give a general overview of all these notions, including the formalism
of Tannakian categories and its application to di(cid:11)erential systems and di(cid:11)erential
Galois theory.
Themain new resultof the (cid:12)rstpart of Chapter1 is the explicit identi(cid:12)cation of the
Riemann{Hilbert correspondence secretly present in perturbative renormalization.
At the geometric level, the relevant category is that of equisingular (cid:13)at vector bun-
dles. ThesearevectorbundlesoverabasespaceB whichisaprincipalG (C) =C -
m (cid:3)
bundle
(cid:25)
G (C) B (cid:1)
m
! (cid:0)!
over an in(cid:12)nitesimal disk (cid:1). From the physical point of view, the complex number
z = 0 in the base space (cid:1) is the parameter of dimensional regularization, while the
6
parameter in the (cid:12)ber is of the form ~(cid:22)z, where ~ is the Planck constant and (cid:22)
is a unit of mass. These vector bundles are endowed with a (cid:13)at connection in the
complement of the (cid:12)berover 0 (cid:1). The(cid:12)berwiseaction ofG (C)=C is given by
m (cid:3)
2
~ @ . The equisingularity of the (cid:13)at connection is a mathematical translation of the
@~
independence (in the minimal subtraction scheme) of the counterterms on the unit
of mass (cid:22). It means that the singularity of the connection, restricted to a section
z (cid:1) (cid:27)(z) B ofthebundleB,onlydependsuponthevalue(cid:27)(0) of thesection.
2 7! 2
We show that the category of equisingular (cid:13)at vector bundles is a Tannakian cate-
gory andweidentify explicitly the correspondinggroup(more precisely, a(cid:14)ne group
scheme) that encodes, through its category of (cid:12)nite dimensional linear representa-
tions,theRiemann-Hilbertcorrespondenceunderlyingperturbativerenormalization.
Thisisa veryspeci(cid:12)cproalgebraic groupofthe formU = UoG , whoseunipotent
(cid:3) m