Noncommutative Geometry, Quantum Fields and Motives Alain Connes

705 Pages·2007·6.44 MB·English

Noncommutative Geometry, Quantum Fields
and Motives
Alain Connes
Matilde Marcolli
A. Connes: Colle`ge 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¨r Mathematik, Vivatsgasse 7,
Bonn, D-53111 Germany
E-mail address: [email protected]

Contents
Preface 9
Chapter 1. Quantum fields, 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 effective 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. Superficial 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 affine 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 Birkhoff factorization 88
6.5. Diffeographisms and diffeomorphisms 93
6.6. The renormalization group 95
7. Renormalization and the Riemann–Hilbert correspondence 100
7.1. Counterterms and time-ordered exponentials 101
7.2. Flat equisingular connections 107
7.3. Equivariant principal bundles and the group G∗ = G(cid:111)G 116
m
7.4. Tannakian categories and affine group schemes 121
7.5. Differential Galois theory and the local Riemann–Hilbert
correspondence 124
7.6. Universal Hopf algebra and the Riemann–Hilbert correspondence 128
3
CONTENTS 4
8. Motives in a nutshell 136
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 fields 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 fixing 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 diffeomorphisms 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 fluctuations of the metric 194
11. The spectral action principle 196
11.1. Terms in Λ2 in the spectral action and scalar curvature 199
11.2. Seeley–DeWitt coefficients 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 finite noncommutative geometry 221
13.1. The subalgebra and the order one condition 224
13.2. The bimodule H and fermions 226
F
13.3. Unimodularity and hypercharges 229
13.4. The classification of Dirac operators 231
13.5. Moduli space of Dirac operators and Yukawa parameters 236
13.6. The intersection pairing of the finite geometry 239
14. The product geometry 241
14.1. The real part of the product geometry 242
15. Bosons as inner fluctuations 242
15.1. The local gauge transformations 243
15.2. Discrete part of the inner fluctuations and the Higgs field 244
15.3. Powers of D(0,1) 246
15.4. Continuous part of the inner fluctuations and gauge bosons 248
15.5. Independence of the boson fields 251
CONTENTS 5
15.6. The Dirac operator and its square 252
16. The spectral action and the Standard Model Lagrangian 253
16.1. The asymptotic expansion of the spectral action on M ×F 253
16.2. Fermionic action and Pfaffian 257
16.3. Fermion doubling, Pfaffian and Majorana fermions 259
17. The Standard Model Lagrangian from the spectral action 261
17.1. Change of variables in the asymptotic formula and unification 262
17.2. Coupling constants at unification 263
17.3. The coupling of fermions 264
17.4. The mass relation at unification 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 field 279
17.9. The Higgs field 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 286
18. Functional integral 287
18.1. Real orientation and volume form 289
18.2. The reconstruction of spin manifolds 291
18.3. Irreducible finite 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 flow 324
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 flow 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 ξ function 348
7. The ad`ele class space: finitely many degrees of freedom 351
7.1. Geometry of the semi-local ad`ele class space 352
CONTENTS 6
7.2. The Hilbert space L2(X ) and the trace formula 357
S
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
δ 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 field theory and the Kronecker–Weber theorem 429
4.6. KMS states and class field theory 432
4.7. The class field theory problem: algebras and fields 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 444
5.3. Time evolution and regular representation 449
5.4. Symmetries 451
6. The modular field 456
6.1. The modular field of level N = 1 457
6.2. Modular field of level N 459
CONTENTS 7
6.3. Modular functions and modular forms 465
6.4. Explicit computations for N = 2 and N = 4 467
6.5. The modular field F and Q-lattices 468
7. Arithmetic of the GL system 471
2
7.1. The arithmetic subalgebra: explicit elements 471
7.2. The arithmetic subalgebra: definition 474
7.3. Division relations in the arithmetic algebra 479
7.4. KMS states 484
7.5. Action of symmetries on KMS states 491
7.6. Low-temperature KMS states and Galois action 492
7.7. The high temperature range 494
7.8. The Shimura variety of GL 495
2
7.9. The noncommutative boundary of modular curves 496
7.10. Compatibility between the systems 498
8. KMS states and complex multiplication 500
8.1. 1-dimensional K-lattices 501
8.2. K-lattices and Q-lattices 502
8.3. Adelic description of K-lattices 503
8.4. Algebra and time evolution 504
8.5. K-lattices and ideals 506
8.6. Arithmetic subalgebra 508
8.7. Symmetries 508
8.8. Low-temperature KMS states and Galois action 511
8.9. High temperature KMS states 516
8.10. Comparison with other systems 519
9. Quantum statistical mechanics of Shimura varieties 520
Chapter 4. Endomotives, thermodynamics, and the Weil explicit formula 523
1. Morphisms and categories of noncommutative spaces 527
1.1. The KK-category 527
1.2. The cyclic category 530
1.3. The non-unital case 533
1.4. Cyclic (co)homology 533
2. Endomotives 535
2.1. Algebraic endomotives 537
2.2. Analytic endomotives 541
2.3. Galois action 543
2.4. Uniform systems and measured endomotives 545
2.5. Compatibility of endomotives categories 546
2.6. Self-maps of algebraic varieties 548
2.7. The Bost–Connes endomotive 549
3. Motives and noncommutative spaces: higher dimensional perspectives 552
3.1. Geometric correspondences 552
3.2. Algebraic cycles and K-theory 553
4. A thermodynamic “Frobenius” in characteristic zero 556
4.1. Tomita’s theory and the modular automorphism group 557
4.2. Regular extremal KMS states (cooling) 559
4.3. The dual system 562
CONTENTS 8
4.4. Field extensions and duality of factors (an analogy) 563
4.5. Low temperature KMS states and scaling 566
4.6. The kernel of the dual trace 570
4.7. Holomorphic modules 573
4.8. The cooling morphism (distillation) 574
4.9. Distillation of the Bost–Connes endomotive 577
4.10. Spectral realization 585
5. A cohomological Lefschetz trace formula 587
5.1. The ad`ele class space of a global field 588
5.2. The cyclic module of the ad`ele class space 589
5.3. The restriction map to the id`ele class group 589
5.4. The Morita equivalence and cokernel for K = Q 590
5.5. The cokernel of ρ for general global fields 592
5.6. Trace pairing and vanishing 605
5.7. Weil’s explicit formula as a trace formula 605
5.8. Weil positivity and the Riemann Hypothesis 607
6. The Weil proof for function fields 608
6.1. Function fields and their zeta functions 609
6.2. Correspondences and divisors in C ×C 612
6.3. Frobenius correspondences and effective divisors 613
6.4. Positivity in the Weil proof 615
7. A noncommutative geometry perspective 618
7.1. Distributional trace of a flow 618
7.2. The periodic orbits of the action of C on X 622
K K
7.3. Frobenius (scaling) correspondence and the trace formula 623
7.4. The Fubini theorem and trivial correspondences 625
7.5. The curve inside the ad`ele class space 627
7.6. Vortex configurations (an analogy) 642
7.7. Building a dictionary 650
8. The analogy between QG and RH 652
8.1. KMS states and the electroweak phase transition 652
8.2. Observables in QG 656
8.3. Invertibility at low temperature 656
8.4. Spectral correspondences 658
8.5. Spectral cobordisms 658
8.6. Scaling action 659
8.7. Moduli spaces for Q-lattices and spectral correspondences 659
Appendix 661
9. Operator algebras 661
9.1. C∗-algebras 661
9.2. Von Neumann algebras 662
9.3. The passing of time 665
10. Galois theory 668
Bibliography 674
Index 685
Preface
The unifying theme, which the reader will encounter in different 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
• The set of prime numbers
One may be tempted to think that, looking from the vantage point of those who
sit atop the vast edifice 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 difficulty 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)
• The Riemann hypothesis (RH)
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
discoverthattherearedeepanalogiesbetweenthesetwoproblemswhich, 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. Thechapterisdedicatedtodiscussingtwomaintopics:
• Renormalization
• The Standard Model of high energy physics
We try to introduce the material as much as possible in a self-contained way, tak-
ing into consideration the fact that a significant number of mathematicians do not
necessarily have quantum field theory and particle physics as part of their cultural
background. Thus, the first half of the chapter is dedicated to giving a detailed
account of perturbative quantum field theory, presented in a manner that, we hope,
is palatable to the mathematician’s taste. In particular, we discuss basic tools,
such as the effective 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”, both because of its being the one most commonly used in the 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 first half of Chapter 1 we
9
PREFACE 10
give a new perspective on perturbative quantum field theory, which gives a clear
mathematical interpretation to the renormalization procedure used by physicists to
extract finite 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 Birkhoff factorization of a loop γ(z) ∈ G
associated to the unrenormalized theory evaluated in complex dimension D − z,
where D is the dimension of space-time and z (cid:54)= 0 is the complex parameter used
in dimensional regularization. The group G is defined through its Hopf algebra
of coordinates, which is the Hopf algebra of Feynman graphs of the theory. The
Birkhoff factorization of the loop gives a canonical way of removing the singularity
at z = 0 and obtaining the required finite result for the physical observables. This
gives renormalization a clear and well defined conceptual meaning.
The Birkhoff factorization of loops is a central tool in the construction of solutions
to the “Riemann-Hilbert problem”, which consists of finding a differential 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 differential geometric
nature, such as differential systems with specified 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 differential systems and differential
Galois theory.
The main new result of the first part of Chapter 1 is the explicit identification of the
Riemann–Hilbert correspondence secretly present in perturbative renormalization.
At the geometric level, the relevant category is that of equisingular flat vector bun-
dles. ThesearevectorbundlesoverabasespaceB whichisaprincipalG (C) = C∗-
m
bundle
π
G (C) → B −→ ∆
m
over an infinitesimal disk ∆. From the physical point of view, the complex number
z (cid:54)= 0 in the base space ∆ is the parameter of dimensional regularization, while the
parameter in the fiber is of the form (cid:126)µz, where (cid:126) is the Planck constant and µ
is a unit of mass. These vector bundles are endowed with a flat connection in the
complement of the fiber over 0∈ ∆. The fiberwise action of G (C) = C∗ is given by
m
(cid:126) ∂ . The equisingularity of the flat connection is a mathematical translation of the
∂(cid:126)
independence (in the minimal subtraction scheme) of the counterterms on the unit
of mass µ. It means that the singularity of the connection, restricted to a section
z ∈ ∆ (cid:55)→ σ(z) ∈ B ofthebundleB, onlydependsuponthevalueσ(0)ofthesection.
We show that the category of equisingular flat vector bundles is a Tannakian cate-
gory and we identify explicitly the corresponding group (more precisely, affine group
scheme) that encodes, through its category of finite dimensional linear representa-
tions,theRiemann-Hilbertcorrespondenceunderlyingperturbativerenormalization.
This is a very specific proalgebraic group of the formU∗ = U(cid:111)G , whose unipotent
m