ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaFor es from Connes' GeometryThomas S h�u ker

Vienna, Preprint ESI 1237 (2002) November 15, 2002Supported by the Austrian Federal Ministry of Edu ation, S ien e and CultureAvailable via http://www.esi.a .at

CENTRE DE PHYSIQUE TH�EORIQUECNRS - Luminy, Case 90713288 Marseille Cedex 9FORCES FROM CONNES' GEOMETRYThomas SCH�UCKER 1Abstra tWe try to give a pedagogi al introdu tion to Connes' derivation of the standardmodel of ele tro-magneti , weak and strong for es from gravity.Le tures given at the Autumn S hool \Topology and Geometry in Physi s"of the Graduiertenkolleg `Physi al Systems with Many Degrees of Freedom'Universit�at HeidelbergSeptember 2001, Rot an der Rot, GermanyEditors: Eike Bi k & Frank Ste�enPACS-92: 11.15 Gauge �eld theoriesMSC-91: 81T13 Yang-Mills and other gauge theoriesCPT-01/P.4264hep-th/01112361 and Universit�e de Proven es hu ker� pt.univ-mrs.fr

Contents1 Introdu tion 32 Gravity from Riemannian geometry 42.1 First stroke: kinemati s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Se ond stroke: dynami s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Slot ma hines and the standard model 73.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3 The winner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Wi k rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Connes' non ommutative geometry 234.1 Motivation: quantum me hani s . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2 The alibrating example: Riemannian spin geometry . . . . . . . . . . . . . . . 244.3 Spin groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 The spe tral a tion 315.1 Repeating Einstein's derivation in the ommutative ase . . . . . . . . . . . . . 315.1.1 First stroke: kinemati s . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.1.2 Se ond stroke: dynami s . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.2 Almost ommutative geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 The minimax example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.4 A entral extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 Connes' do-it-yourself kit 446.1 Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.2 Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486.3 The standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.3.1 Spe tral triple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.3.2 Central harges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536.3.3 Flu tuating metri . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.3.4 Spe tral a tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.4 Beyond the standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 Outlook and on lusion 601

8 Appendix 628.1 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628.2 Group representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648.3 Semi-dire t produ t and Poin ar�e group . . . . . . . . . . . . . . . . . . . . . . 668.4 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2

1 Introdu tionStill today one of the major summits in physi s is the understanding of the spe trum of thehydrogen atom. The phenomenologi al formula by Balmer and Rydberg was a remarkable pre-summit on the way up. The true summit was rea hed by deriving this formula from quantumme hani s. We would like to ompare the standard model of ele tro-magneti , weak, andstrong for es with the Balmer-Rydberg formula [1℄ and review the present status of Connes'derivation of this model from non ommutative geometry, see table 1. This geometry extendsRiemannian geometry, and Connes' derivation is a natural extension of another major summit inphysi s: Einstein's derivation of general relativity from Riemannian geometry. Indeed, Connes'derivation uni�es gravity with the other three for es.atoms parti les and for esBalmer-Rydberg formula standard modelquantum me hani s non ommutative geometryTable 1: An analogyLet us brie y re all four nested, analyti geometries and their impa t on our understandingof for es and time, see table 2. Eu lidean geometry is underlying Newton's me hani s as spa eof positions. For es are des ribed by ve tors living in the same spa e and the Eu lidean s alarprodu t is needed to de�ne work and potential energy. Time is not part of geometry, it isabsolute. This point of view is abandoned in spe ial relativity unifying spa e and time intoMinkowskian geometry. This new point of view allows to derive the magneti �eld from theele tri �eld as a pseudo for e asso iated to a Lorentz boost. Although time has be omerelative, one an still imagine a grid of syn hronized lo ks, i.e. a universal time. The nextgeneralization is Riemannian geometry = urved spa etime. Here gravity an be viewed asthe pseudo for e asso iated to a uniformly a elerated oordinate transformation. At the sametime, universal time loses all meaning and we must ontent ourselves with proper time. Withtoday's pre ision in time measurement, this ompli ation of life be omes a bare ne essity, e.g.the global positioning system (GPS).Our last generalization is to Connes' non ommutative geometry = urved spa e(time) withun ertainty. It allows to understand some Yang-Mills and some Higgs for es as pseudo for esasso iated to transformations that extend the two oordinate transformations above to thenew geometry without points. Also, proper time omes with an un ertainty. This un ertaintyof some hundred Plan k times might be a essible to experiments through gravitational wavedete tors within the next ten years [2℄. 3

geometry for e timeEu lidean E = R ~F � d~x absoluteMinkowskian ~E; �0) ~B; �0 = 1�0 2 universalRiemannian Coriolis $ gravity proper, �non ommutative gravity ) YMH, � = 13g22 �� � 10�40 sTable 2: Four nested analyti geometriesPrerequisitesOn the physi al side, the reader is supposed to be a quainted with general relativity, e.g. [3℄,Dira spinors at the level of e.g. the �rst few hapters in [4℄ and Yang-Mills theory with spon-taneous symmetry break-down, for example the standard model, e.g. [5℄. I am not ashamed toadhere to the minimax prin iple: a maximum of pleasure with a minimum of e�ort. The e�ortis to do a al ulation, the pleasure is when its result oin ides with an experiment result. Con-sequently our mathemati al treatment is as low-te h as possible. We do need lo al di�erentialand Riemannian geometry at the level of e.g. the �rst few hapters in [6℄. Lo al means thatour spa es or manifolds an be thought of as open subsets of R4. Nevertheless, we sometimesuse ompa t spa es like the torus: only to simplify some integrals. We do need some grouptheory, e.g. [7℄, mostly matrix groups and their representations. We also need a few basi fa tson asso iative algebras. Most of them are re alled as we go along and an be found for instan ein [8℄. For the reader's onvenien e, a few simple de�nitions from groups and algebras are olle ted in the appendix. And, of ourse, we need some hapters of non ommutative geometrywhi h are developped in the text. For a more detailed presentation still with parti ular arefor the physi ist see [9, 10℄.2 Gravity from Riemannian geometryIn this se tion, we brie y review Einstein's derivation of general relativity from Riemanniangeometry. His derivation is in two strokes, kinemati s and dynami s.4

2.1 First stroke: kinemati sConsider at spa e(time) M in inertial or Cartesian oordinates ~x~�. Take as matter a free, lassi al point parti le. Its dynami s, Newton's free equation, �xes the traje tory ~x~�(p):d2~x~�dp2 = 0: (1)After a general oordinate transformation, x� = ��(~x), Newton's equation readsd2x�dp2 + ����(g) dx�dp dx�dp = 0: (2)Pseudo for es have appeared. They are oded in the Levi-Civita onne tion����(g) = 12g�� � ��x� g�� + ��x� g�� � ��x� g��� ; (3)where g�� is obtained by ` u tuating' the at metri ~�~�~� = diag(1;�1;�1;�1; ) with theJa obian of the oordinate transformation �:g��(x) = J (x)�1~�� �~�~� J (x)�1~�� ; J (~x)�~� := ���(~x)=� ~x~�: (4)For the oordinates of the rotating disk, the pseudo for es are pre isely the entrifugal andCoriolis for es. Einstein takes uniformly a elerated oordinates, t = ~t; z = ~z+ 12 g 2 ( ~t)2 withg = 9:81 m=s2. Then the geodesi equation (2) redu es to d2z=dt2 = �g. So far this gravityis still a pseudo for e whi h means that the urvature of its Levi-Civita onne tion vanishes.This onstraint is relaxed by the equivalen e prin iple: pseudo for es and true gravitationalfor es are oded together in a not ne essarily at onne tion �, that derives from a potential,the not ne essarily at metri g. The kinemati al variable to des ribe gravity is therefore theRiemannian metri . By onstru tion the dynami s of matter, the geodesi equation, is now ovariant under general oordinate transformations.2.2 Se ond stroke: dynami sNow that we know the kinemati s of gravity let us see how Einstein obtains its dynami s, i.e.di�erential equations for the metri tensor g�� . Of ourse Einstein wants these equations to be ovariant under general oordinate transformations and he wants the energy-momentum tensorT�� to be the sour e of gravity. From Riemannian geometry he knew that there is no ovariant,�rst order di�erential operator for the metri . But there are se ond order ones:Theorem: The most general tensor of degree 2 that an be onstru ted from the metri tensorg��(x) with at most two partial derivatives is�R�� + �Rg�� + �g�� ; �; �; � 2 R:: (5)5

Here are our onventions for the urvature tensors:Riemann tensor : R���� = ������ � ������ + �������� � ��������; (6)Ri i tensor : R�� = R����; (7) urvature s alar : R = R��g�� : (8)The mira le is that the tensor (5) is symmetri just as the energy-momentum tensor. How-ever, the latter is ovariantly onserved, D�T�� = 0, while the former one is onserved if andonly if � = �12�. Consequently, Einstein puts his equationR�� � 12Rg�� � � g�� = 8�G 4 T�� : (9)He hooses a vanishing osmologi al onstant, � = 0. Then for small stati mass density T00,his equation reprodu es Newton's universal law of gravity with G the Newton onstant. How-ever for not so small masses there are orre tions to Newton's law like pre ession of perihelia.Also Einstein's theory applies to massless matter and produ es the urvature of light. Ein-stein's equation has an agreeable formal property, it derives via the Euler-Lagrange variationalprin iple from an a tion, the famous Einstein-Hilbert a tion:SEH [g℄ = �116�G ZM R dV � 2� 16�G ZM dV; (10)with the invariant volume element dV := jdet g��j1=2 d4x:General relativity has a pre ise geometri origin: the left-hand side of Einstein's equation isa sum of some 80 000 terms in �rst and se ond partial derivatives of g�� and its matrix inverseg�� . All of these terms are ompletely �xed by the requirement of ovarian e under general oordinate transformations. General relativity is veri�ed experimentally to an extraordinarya ura y, even more, it has be ome a ornerstone of today's te hnology. Indeed length measure-ments had to be abandoned in favour of proper time measurements, e.g. the GPS. Nevertheless,the theory still leaves a few questions unanswered:� Einstein's equation is nonlinear and therefore does not allow point masses as sour e,in ontrast to Maxwell's equation that does allow point harges as sour e. Fromthis point of view it is not satisfying to onsider point-like matter.� The gravitational for e is oded in the onne tion �. Nevertheless we have a eptedits potential, the metri g, as kinemati al variable.� The equivalen e prin iple states that lo ally, i.e. on the traje tory of a point-likeparti le, one annot distinguish gravity from a pseudo for e. In other words, there is6

always a oordinate system, `the freely falling lift', in whi h gravity is absent. This isnot true for ele tro-magnetism and we would like to derive this for e (as well as theweak and strong for es) as a pseudo for e oming from a geometri transformation.� So far general relativity has resisted all attempts to re on ile it with quantum me- hani s.3 Slot ma hines and the standard modelToday we have a very pre ise phenomenologi al des ription of ele tro-magneti , weak, andstrong for es. This des ription, the standard model, works on a perturbative quantum leveland, as lassi al gravity, it derives from an a tion prin iple. Let us introdu e this a tion byanalogy with the Balmer-Rydberg formula.q

1

q

g

g

1

2 2

spectrumFigure 1: A slot ma hine for atomi spe traOne of the new features of atomi physi s was the appearan e of dis rete frequen ies andthe measurement of atomi spe tra be ame a highly developed art. It was natural to label thedis rete frequen ies � by natural numbers n. To �t the spe trum of a given atom, say hydrogen,let us try the ansatz � = g1nq11 + g2nq22 : (11)We view this ansatz as a slot ma hine. You input two bills, the integers q1, q2 and two oins,the two real numbers g1, g2, and ompare the output with the measured spe trum. (See Figure1.) If you are ri h enough, you play and replay on the slot ma hine until you win. The winneris the Balmer-Rydberg formula, i.e., q1 = q2 = �2 and g1 = �g2 = 3:289 1015 Hz, whi h is thefamous Rydberg onstant R. Then ame quantum me hani s. It explained why the spe trumof the hydrogen atom was dis rete in the �rst pla e and derived the exponents and the Rydberg onstant, R = me4�~3 e4(4��0)2 ; (12)7

from a non ommutativity, [x; p℄ = i~1.To ut short its long and ompli ated history we introdu e the standard model as the winnerof a parti ular slot ma hine. This ma hine, whi h has be ome popular under the names Yang,Mills and Higgs, has four slots for four bills. On e you have de ided whi h bills you hooseand entered them, a ertain number of small slots will open for oins. Their number dependson the hoi e of bills. You make your hoi e of oins, feed them in, and the ma hine startsworking. It produ es as output a Lagrange density. From this density, perturbative quantum�eld theory allows you to ompute a omplete parti le phenomenology: the parti le spe trumwith the parti les' quantum numbers, ross se tions, life times, and bran hing ratios. (SeeFigure 2.) You ompare the phenomenology to experiment to �nd out whether your input winsor loses.HHH

G

L

R

S

g

n

Y

l, m

gYMH

phenomenologyparticleFigure 2: The Yang-Mills-Higgs slot ma hine3.1 InputThe �rst bill is a �nite dimensional, real, ompa t Lie group G. The gauge bosons, spin 1, willlive in its adjoint representation whose Hilbert spa e is the omplexi� ation of the Lie algebrag ( f Appendix).The remaining bills are three unitary representations of G, �L; �R; �S , de�ned on the om-plex Hilbert spa es,HL; HR; HS . They lassify the left- and right-handed fermions, spin 12 , andthe s alars, spin 0. The group G is hosen ompa t to ensure that the unitary representationsare �nite dimensional, we want a �nite number of `elementary parti les' a ording to the redoof parti le physi s that parti les are orthonormal basis ve tors of the Hilbert spa es whi h arrythe representations. More generally, we might also admit multi-valued representations, `spinrepresentations', whi h would open the debate on harge quantization. More on this later.The oins are numbers, oupling onstants, more pre isely oeÆ ients of invariant polyno-mials. We need an invariant s alar produ t on g. The set of all these s alar produ ts is a one8

and the gauge ouplings are parti ular oordinates of this one. If the group is simple, sayG = SU(n), then the most general, invariant s alar produ t is(X;X 0) = 2g2n tr [X�X 0℄; X;X 0 2 su(n): (13)If G = U(1), we have (Y; Y 0) = 1g21 �Y Y 0; Y; Y 0 2 u(1): (14)We denote by �� the omplex onjugate and by �� the Hermitean onjugate. Mind the di�erentnormalizations, they are onventional. The gn are positive numbers, the gauge ouplings. Forevery simple fa tor of G there is one gauge oupling.Then we need the Higgs potential V ('). It is an invariant, fourth order, stable polynomialon HS 3 '. Invariant means V (�S(u)') = V (') for all u 2 G. Stable means bounded frombelow. For G = U(2) and the Higgs s alar in the fundamental or de�ning representation,' 2 HS = C 2 , �S(u) = u, we haveV (') = � ('�')2 � 12�2 '�': (15)The oeÆ ients of the Higgs potential are the Higgs ouplings, � must be positive for stability.We say that the potential breaks G spontaneously if no minimum of the potential is a trivialorbit under G. In our example, if � is positive, the minima of V (') lie on the 3-spherej'j = v := 12�=p�. v is alled va uum expe tation value and U(2) is said to break downspontaneously to its little group U(1) 3 � 1 00 ei�� : (16)The little group leaves invariant any given point of the minimum, e.g. ' = (v; 0)T . On the otherhand, if � is purely imaginary, then the minimum of the potential is the origin, no spontaneoussymmetry breaking and the little group is all of G.Finally, we need the Yukawa ouplings gY . They are the oeÆ ients of the most general,real, trilinear invariant on H�L HR (HS �H�S). For every 1-dimensional invariant subspa ein the redu tion of this tensor representation, we have one omplex Yukawa oupling. Forexample G = U(2), HL = C 2 , �L(u) L = (detu)qLu L; HR = C , �R(u) R = (detu)qR R;HS = C 2 , �S(u)' = (detu)qSu'. If �qL + qR + qS 6= 0 there is no Yukawa oupling, otherwisethere is one: ( L; R; ') = Re(gY �L R').If the symmetry is broken spontaneously, gauge and Higgs bosons a quire masses related togauge and Higgs ouplings, fermions a quire masses equal to the `va uum expe tation value' vtimes the Yukawa ouplings. 9

As explained in Jan-Willem van Holten's and Jean Zinn-Justin's le tures at this S hool [11,12℄, one must require for onsisten y of the quantum theory that the fermioni representationsbe free of Yang-Mills anomalies,tr ((~�L(X))3)� tr ((~�R(X))3) = 0; for all X 2 g: (17)We denote by ~� the Lie algebra representation of the group representation �. Sometimes onealso wants the mixed Yang-Mills-gravitational anomalies to vanish:tr ~�L(X)� tr ~�R(X) = 0; for all X 2 g: (18)3.2 RulesIt is time to open the slot ma hine and to see how it works. Its me hanism has �ve pie es:The Yang-Mills a tion: The a tor in this pie e is A = A�dx�, alled onne tion, gaugepotential, gauge boson or Yang-Mills �eld. It is a 1-form on spa etime M 3 x with values inthe Lie algebra g, A 2 1(M; g): We de�ne its urvature or �eld strength,F := dA+ 12 [A;A℄ = 12F��dx�dx� 2 2(M; g); (19)and the Yang-Mills a tion,SYM [A℄ = �12 ZM(F; �F ) = �12g2n ZM trF ���F ��dV: (20)The gauge group MG is the in�nite dimensional group of di�erentiable fun tions g : M ! Gwith pointwise multipli ation. �� is the Hermitean onjugate of matri es, �� is the Hodge starof di�erential forms. The spa e of all onne tions arries an aÆne representation ( f Appendix)�V of the gauge group: �V (g)A = gAg�1 + gdg�1: (21)Restri ted to x-independent (`rigid') gauge transformation, the representation is linear, theadjoint one. The �eld strength transforms homogeneously even under x-dependent (`lo al')gauge transformations, g :M ! G di�erentiable,�V (g)F = gFg�1; (22)and, as the s alar produ t (�; �) is invariant, the Yang-Mills a tion is gauge invariant,SY M [�V (g)A℄ = SYM [A℄ for all g 2 MG: (23)10

Note that a mass term for the gauge bosons,12 ZM m2A(A; �A) = 1g2n ZM m2AtrA��A�dV; (24)is not gauge invariant be ause of the inhomogeneous term in the transformation law of a onne tion (21). Gauge invarian e for es the gauge bosons to be massless.In the Abelian ase G = U(1), the Yang-Mills Lagrangian is nothing but Maxwell's La-grangian, the gauge boson A is the photon and its oupling onstant g is e=p�0. Note however,that the Lie algebra of U(1) is iR and the ve tor potential is purely imaginary, while on-ventionally, in Maxwell's theory it is hosen real. Its quantum version is QED, quantumele tro-dynami s. For G = SU(3) and HL = HR = C 3 we have today's theory of strongintera tion, quantum hromo-dynami s, QCD.The Dira a tion: S hr�odinger's a tion is non-relativisti . Dira generalized it to be Lorentzinvariant, e.g. [4℄. The pri e to be paid is twofold. His generalization only works for spin12 parti les and requires that for every su h parti le there must be an antiparti le with samemass and opposite harges. Therefore, Dira 's wave fun tion (x) takes values in C 4 , spin up,spin down, parti le, antiparti le. Antiparti les have been dis overed and Dira 's theory was elebrated. Here it is in short for ( at) Minkowski spa e of signature + � ��, ��� = ��� =diag(+1;�1;�1;�1). De�ne the four Dira matri es, 0 = � 0 �12�12 0 � ; j = � 0 �j��j 0 � ; (25)for j = 1; 2; 3 with the three Pauli matri es,�1 = � 0 11 0� ; �2 = � 0 �ii 0 � ; �3 = � 1 00 �1� : (26)They satisfy the anti ommutation relations, � � + � � = 2���14: (27)In even spa etime dimensions, the hirality, 5 := � i4!����� � � � � = �i 0 1 2 3 = ��12 00 12� (28)is a natural operator and it paves the way to an understanding of parity violation in weakintera tions. The hirality is a unitary matrix of unit square, whi h anti ommutes with allfour Dira matri es. (1 � 5)=2 proje ts a Dira spinor onto its left-handed part, (1 + 5)=2proje ts onto the right-handed part. The two parts are alled Weyl spinors. A massless left-handed (right-handed) spinor, has its spin parallel (anti-parallel) to its dire tion of propagation.11

The hirality maps a left-handed spinor to a right-handed spinor. A spa e re e tion or paritytransformation hanges the sign of the velo ity ve tor and leaves the spin ve tor un hanged. Ittherefore has the same e�e t on Weyl spinors as the hirality operator. Similarly, there is the harge onjugation, an anti-unitary operator ( f Appendix) of unit square, that applied on aparti le produ es its antiparti leJ = 1i 0 2 Æ omplex onjugation = 0BB�0 �1 0 01 0 0 00 0 0 10 0 �1 01CCA Æ ; (29)i.e. J = 1i 0 2 � . Attention, here and for the last time � stands for the omplex onjugateof . In a few lines we will adopt a di�erent more popular onvention. The harge onjugation ommutes with all four Dira matri es. In at spa etime, the free Dira operator is simplyde�ned by, �= := i~ ���: (30)It is sometimes referred to as square root of the wave operator be ause �=2 = �ut. The ouplingof the Dira spinor to the gauge potential A = A�dx� is done via the ovariant derivative,and alled minimal oupling. In order to break parity, we write left- and right-handed partsindependently: SD[A; L; R℄ = ZM � L [ �=+ i~ �~�L(A�)℄ 1� 52 L dV+ZM � R [ �=+ i~ �~�R(A�)℄ 1 + 52 R dV: (31)The new a tors in this pie e are L and R, two multiplets of Dira spinors or fermions, that iswith values in HL and HR. We use the notations, � := � 0, where �� denotes the Hermitean onjugate with respe t to the four spinor omponents and the dual with respe t to the s alarprodu t in the (internal) Hilbert spa e HL or HR. The 0 is needed for energy reasons and forinvarian e of the pseudo{s alar produ t of spinors under lifted Lorentz transformations. The 0 is absent if spa etime is Eu lidean. Then we have a genuine s alar produ t and the squareintegrable spinors form a Hilbert spa e L2(S) = L2(R4) C 4 , the in�nite dimensional brotherof the internal one. The Dira operator is then self adjoint in this Hilbert spa e. We denote by~�L the Lie algebra representation in HL. The ovariant derivative, D� := ��+ ~�L(A�), deservesits name, [�� + ~�L(�V (g)A�)℄ (�L(g) L) = �L(g) [�� + ~�L(A�)℄ L; (32)for all gauge transformations g 2 MG. This ensures that the Dira a tion (31) is gauge invariant.12

If parity is onserved, HL = HR, we may add a mass term� ZM � Rm 1� 52 L dV � ZM � Lm 1 + 52 R dV = � ZM � m dV (33)to the Dira a tion. It gives identi al masses to all members of the multiplet. The fermionmasses are gauge invariant if all fermions in HL = HR have the same mass. For instan e QEDpreserves parity, HL = HR = C , the representation being hara terized by the ele tri harge,�1 for both the left- and right handed ele tron. Remember that gauge invarian e for es gaugebosons to be massless. For fermions, it is parity non-invarian e that for es them to be massless.Let us on lude by reviewing brie y why the Dira equation is the Lorentz invariant gener-alization of the S hr�odinger equation. Take the free S hr�odinger equation on ( at) R4. It is alinear di�erential equation with onstant oeÆ ients,� 2mi~ ��t ��� = 0: (34)We ompute its polynomial following Fourier and de Broglie,� 2m~ ! + k2 = � 2m~2 �E � p22m � : (35)Energy onservation in Newtonian me hani s is equivalent to the vanishing of the polynomial.Likewise, the polynomial of the free, massive Dira equation ( �=� m ) = 0 is~ ! 0 + ~ kj j � m1: (36)Putting it to zero implies energy onservation in spe ial relativity,(~ )2 !2 � ~2~k2 � 2m2 = 0: (37)In this sense, Dira 's equation generalizes S hr�odinger's to spe ial relativity. To see that Dira 'sequation is really Lorentz invariant we must lift the Lorentz transformations to the spa e ofspinors. We will ome ba k to this lift.So far we have seen the two noble pie es by Yang-Mills and Dira . The remaining threepie es are heap opies of the two noble ones with the gauge boson A repla ed by a s alar '.We need these three pie es to ure only one problem, give masses to some gauge bosons andto some fermions. These masses are forbidden by gauge invarian e and parity violation. Tosimplify the notation we will work from now on in units with = ~ = 1.The Klein-Gordon a tion: The Yang-Mills a tion ontains the kineti term for the gaugeboson. This is simply the quadrati term, (dA;dA), whi h by Euler-Lagrange produ es linear�eld equations. We opy this for our new a tor, a multiplet of s alar �elds or Higgs bosons,' 2 0(M;HS); (38)13

by writing the Klein-Gordon a tion,SKG[A;'℄ = 12 ZM(D')� �D' = 12 ZM (D�')�D�'dV; (39)with the ovariant derivative here de�ned with respe t to the s alar representation,D' := d'+ ~�S(A)': (40)Again we need this minimal oupling '�A' for gauge invarian e.The Higgs potential: The non-Abelian Yang-Mills a tion ontains intera tion terms for thegauge bosons, an invariant, fourth order polynomial, 2(dA; [A;A℄)+ ([A;A℄; [A;A℄). We mimi these intera tions for s alar bosons by adding the integrated Higgs potential RM �V (') to thea tion.The Yukawa terms: We also mimi the (minimal) oupling of the gauge boson to the fermions �A by writing all possible trilinear invariants,SY [ L; R; '℄ := ReZM � nXj=1 gY j ( �L; R; ')j + mXj=n+1 gY j ( �L; R; '�)j! : (41)In the standard model, there are 27 omplex Yukawa ouplings, m = 27.The Yang-Mills and Dira a tions, ontain three types of ouplings, a trilinear self ouplingAAA, a quadrilinear self oupling AAAA and the trilinear minimal oupling �A . The gaugeself ouplings are absent if the group G is Abelian, the photon has no ele tri harge, Maxwell'sequations are linear. The beauty of gauge invarian e is that if G is simple, all these ouplings are�xed in terms of one positive number, the gauge oupling g. To see this, take an orthonormalbasis Tb; b = 1; 2; :::dimG of the omplexi� ation gC of the Lie algebra with respe t to theinvariant s alar produ t and an orthonormal basis Fk; k = 1; 2; :::dimHL, of the fermioni Hilbert spa e, say HL, and expand the a tors,A =: Ab�Tbdx�; =: kFk: (42)Insert these expressions into the Yang-Mills and Dira a tions, then you get the followingintera tion terms, see Figure 3,g ��Aa�Ab�A � fab �����; g2Aa�Ab�A �Ad� fabefe d �����; g k�Ab� � ` tbk`; (43)with the stru ture onstants fabe, [Ta; Tb℄ =: fabeTe: (44)14

The indi es of the stru ture onstants are raised and lowered with the matrix of the invariants alar produ t in the basis Tb, that is the identity matrix. The tbk` is the matrix of the operator~�L(Tb) with respe t to the basis Fk. The di�eren e between the noble and the heap a tions isthat the Higgs ouplings, � and � in the standard model, and the Yukawa ouplings gY j arearbitrary, are neither onne ted among themselves nor onne ted to the gauge ouplings gi.A A

A

g

A

ψ ψ_

g

A A

AA

2g

ψ ψ_

ϕ ϕ

ϕϕ

λ

ϕ

gY

Figure 3: Tri- and quadrilinear gauge ouplings, minimal gauge oupling to fermions,Higgs self oupling and Yukawa oupling3.3 The winnerPhysi ists have spent some thirty years and billions of Swiss Fran s playing on the slot ma hineby Yang, Mills and Higgs. There is a winner, the standard model of ele tro-weak and strongfor es. Its bills are G = SU(2) � U(1)� SU(3)=(Z2�Z3); (45)HL = 3M1 �(2; 16; 3) � (2;�12 ; 1)� ; (46)HR = 3M1 �(1; 23; 3) � (1;�13 ; 3)� (1;�1; 1)� ; (47)HS = (2;�12 ; 1); (48)where (n2; y; n3) denotes the tensor produ t of an n2 dimensional representation of SU(2), ann3 dimensional representation of SU(3) and the one dimensional representation of U(1) withhyper harge y: �(exp(i�)) = exp(iy�). For histori al reasons the hyper harge is an integermultiple of 16. This is irrelevant: only the produ t of the hyper harge with its gauge ouplingis measurable and we do not need multi-valued representations, whi h are hara terized bynon-integer, rational hyper harges. In the dire t sum, we re ognize the three generations offermions, the quarks are SU(3) olour triplets, the leptons olour singlets. The basis of the15

fermion representation spa e is�ud�L ; � s�L ; � tb�L ; � �ee �L ; ���� �L ; � ��� �LuR;dR; R;sR; tR;bR; eR; �R; �RThe parentheses indi ate isospin doublets.The eight gauge bosons asso iated to su(3) are alled gluons. Attention, the U(1) is notthe one of ele tri harge, it is alled hyper harge, the ele tri harge is a linear ombinationof hyper harge and weak isospin, parameterized by the weak mixing angle �w to be introdu edbelow. This mixing is ne essary to give ele tri harges to theW bosons. The W+ and W� arepure isospin states, while the Z0 and the photon are (orthogonal) mixtures of the third isospingenerator and hyper harge.Be ause of the high degree of redu ibility in the bills, there are many oins, among them27 omplex Yukawa ouplings. Not all Yukawa ouplings have a physi al meaning and we onlyremain with 18 physi ally signi� ant, positive numbers [13℄, three gauge ouplings at energies orresponding to the Z mass,g1 = 0:3574 � 0:0001; g2 = 0:6518 � 0:0003; g3 = 1:218 � 0:01; (49)two Higgs ouplings, � and �, and 13 positive parameters from the Yukawa ouplings. TheHiggs ouplings are related to the boson masses:mW = 12g2 v = 80:419 � 0:056 GeV; (50)mZ = 12qg21 + g22 v = mW= os �w = 91:1882 � 0:0022 GeV; (51)mH = 2p2p� v > 98 GeV; (52)with the va uum expe tation value v := 12�=p� and the weak mixing angle �w de�ned bysin2 �w := g�22 =(g�22 + g�21 ) = 0:23117 � 0:00016: (53)For the standard model, there is a one{to{one orresponden e between the physi ally relevantpart of the Yukawa ouplings and the fermion masses and mixings,me = 0:510998902 � 0:000000021 MeV; mu = 3� 2 MeV; md = 6� 3 MeV;m� = 0:105658357 � 0:000000005 GeV; m = 1:25 � 0:1 GeV; ms = 0:125 � 0:05 GeV;m� = 1:77703 � 0:00003 GeV; mt = 174:3 � 5:1 GeV; mb = 4:2� 0:2 GeV:16

For simpli ity, we take massless neutrinos. Then mixing only o urs for quarks and is given bya unitary matrix, the Cabibbo-Kobayashi-Maskawa matrixCKM :=0�Vud Vus VubV d V s V bVtd Vts Vtb 1A : (54)For physi al purposes it an be parameterized by three angles �12, �23, �13 and one CP violatingphase Æ: CKM =0� 12 13 s12 13 s13e�iÆ�s12 23 � 12s23s13eiÆ 12 23 � s12s23s13eiÆ s23 13s12s23 � 12 23s13eiÆ � 12s23 � s12 23s13eiÆ 23 13 1A ; (55)with kl := os �kl, skl := sin �kl. The absolute values of the matrix elements in CKM are:0� 0:9750 � 0:0008 0:223 � 0:004 0:004 � 0:0020:222 � 0:003 0:9742 � 0:0008 0:040 � 0:0030:009 � 0:005 0:039 � 0:004 0:9992 � 0:00031A : (56)The physi al meaning of the quark mixings is the following: when a suÆ iently energeti W+de ays into a u quark, this u quark is produ ed together with a �d quark with probability jVudj2,together with a �s quark with probability jVusj2, together with a �b quark with probability jVubj2.The fermion masses and mixings together are an entity, the fermioni mass matrix or the matrixof Yukawa ouplings multiplied by the va uum expe tation value.Let us note six intriguing properties of the standard model.� The gluons ouple in the same way to left- and right-handed fermions, the gluon oupling is ve torial, the strong intera tion does not break parity.� The fermioni mass matrix ommutes with SU(3), the three olours of a given quarkhave the same mass.� The s alar is a olour singlet, the SU(3) part of G does not su�er spontaneoussymmetry break down, the gluons remain massless.� The SU(2) ouples only to left-handed fermions, its oupling is hiral, the weakintera tion breaks parity maximally.� The s alar is an isospin doublet, the SU(2) part su�ers spontaneous symmetry breakdown, the W� and the Z0 are massive.17

� The remaining olourless and neutral gauge boson, the photon, is massless and ou-ples ve torially. This is ertainly the most ad-ho feature of the standard model. In-deed the photon is a linear ombination of isospin, whi h ouples only to left-handedfermions, and of a U(1) generator, whi h may ouple to both hiralities. Thereforeonly the areful �ne tuning of the hyper harges in the three input representations(46-48) an save parity onservation and gauge invarian e of ele tro-magnetism,yuR = yqL � y`L ydR = yqL + y`L ; yeR = 2y`L ; y' = y`L; (57)The subs ripts label the multiplets, qL for the left-handed quarks, `L for the left-handed leptons, uR for the right-handed up-quarks and so forth and ' for the s alar.Nevertheless the phenomenologi al su ess of the standard model is phenomenal: with only ahandful of parameters, it reprodu es orre tly some millions of experimental numbers. Most ofthese numbers are measured with an a ura y of a few per ent and they an be reprodu ed by lassi al �eld theory, no ~ needed. However, the experimental pre ision has be ome so goodthat quantum orre tions annot be ignored anymore. At this point it is important to notethat the fermioni representations of the standard model are free of Yang-Mills (and mixed)anomalies. Today the standard model stands un ontradi ted.Let us ome ba k to our analogy between the Balmer-Rydberg formula and the standardmodel. One might obje t that the ansatz for the spe trum, equation (11), is ompletely ad ho ,while the lass of all (anomaly free) Yang-Mills-Higgs models is distinguished by perturbativerenormalizability. This is true, but this property was proved [14℄ only years after the ele tro-weak part of the standard model was published [15℄.By pla ing the hydrogen atom in an ele tri or magneti �eld, we know experimentallythat every frequen y `state' n, n = 1; 2; 3; :::, omes with n irredu ible unitary representationsof the rotation group SO(3). These representations are labelled by `, ` = 0; 1; 2; :::n � 1, ofdimensions 2`+1. An orthonormal basis of ea h representation ` is labelled by another integerm, m = �`;�` + 1; :::`. This experimental fa t has motivated the redo that parti les areorthonormal basis ve tors of unitary representations of ompa t groups. This redo is alsobehind the standard model. While SO(3) has a lear geometri interpretation, we are stilllooking for su h an interpretation of SU(2)� U(1) � SU(3)=[Z2�Z3℄:We lose this subse tion with Iliopoulos' joke [16℄ from 1976:18

Do-it-yourself kit for gauge models:1) Choose a gauge group G.2) Choose the �elds of the \elementary parti les" you want to introdu e, and theirrepresentations. Do not forget to in lude enough �elds to allow for the Higgs me h-anism.3) Write the most general renormalizable Lagrangian invariant under G. At this stagegauge invarian e is still exa t and all ve tor bosons are massless.4) Choose the parameters of the Higgs s alars so that spontaneous symmetry breakingo urs. In pra ti e, this often means to hoose a negative value [positive in ournotations℄ for the parameter �2.5) Translate the s alars and rewrite the Lagrangian in terms of the translated �elds.Choose a suitable gauge and quantize the theory.6) Look at the properties of the resulting model. If it resembles physi s, even remotely,publish it.7) GO TO 1.Meanwhile his joke has be ome experimental reality.3.4 Wi k rotationEu lidean signature is te hni ally easier to handle than Minkowskian. What is more, in Connes'geometry it will be vital that the spinors form a Hilbert spa e with a true s alar produ tand that the Dira a tion takes the form of a s alar produ t. We therefore put together theEinstein-Hilbert and Yang-Mills-Higgs a tions with emphasis on the relative signs and indi atethe hanges ne essary to pass from Minkowskian to Eu lidean signature.In 1983 the meter disappeared as fundamental unit of s ien e and te hnology. The on ep-tual revolution of general relativity, the abandon of length in favour of time, had made its wayup to the domain of te hnology. Said di�erently, general relativity is not really geo-metry, but hrono-metry. Hen e our hoi e of Minkowskian signature is +���.With this hoi e the ombined Lagrangian reads,f� 2� 16�G � 116�G R � 12g2 tr (F ���F ��) + 1g2m2Atr (A��A�)+ 12 (D�')�D�' � 12 m2'j'j2 + 12 �2j'j2 � �j'j4+ � 0 [i �D� � m 14℄ g jdet g��j1=2: (58)19

This Lagrangian is real if we suppose that all �elds vanish at in�nity. The relative oeÆ ientsbetween kineti terms and mass terms are hosen as to reprodu e the orre t energy momentumrelations from the free �eld equations using Fourier transform and the de Broglie relations asexplained after equation (34). With the hiral de omposition L = 1� 52 ; R = 1+ 52 ; (59)the Dira Lagrangian reads � 0 [i �D� � m 14℄ = �L 0 i �D� L + �R 0 i �D� R � m �L 0 R � m �R 0 L: (60)The relativisti energy momentum relations are quadrati in the masses. Therefore the sign ofthe fermion mass m is onventional and merely re e ts the hoi e: who is parti le and whois antiparti le. We an even adopt one hoi e for the left-handed fermions and the opposite hoi e for the right-handed fermions. Formally this an be seen by the hange of �eld variable( hiral transformation): := exp(i� 5) 0: (61)It leaves invariant the kineti term and the mass term transforms as,�m 0� 0[ os(2�) 14 + i sin(2�) 5℄ 0: (62)With � = ��=4 the Dira Lagrangian be omes: 0� 0[ i �D� + im 5℄ 0= 0�L 0 i �D� 0L + 0�R 0 i �D� 0R + m 0�L 0i 5 0R + m 0�R 0i 5 0L= 0�L 0 i �D� 0L + 0�R 0 i �D� 0R + im 0�L 0 0R � im 0�R 0 0L: (63)We have seen that gauge invarian e forbids massive gauge bosons, mA = 0, and that parityviolation forbids massive fermions, m = 0. This is �xed by spontaneous symmetry breaking,where we take the s alar mass term with wrong sign, m' = 0; � > 0. The shift of the s alarthen indu es masses for the gauge bosons, the fermions and the physi al s alars. These massesare al ulable in terms of the gauge, Yukawa, and Higgs ouplings.The other relative signs in the ombined Lagrangian are �xed by the requirement that theenergy density of the non-gravitational part T00 be positive (up to a osmologi al onstant)and that gravity in the Newtonian limit be attra tive. In parti ular this implies that the Higgspotential must be bounded from below, � > 0. The sign of the Einstein-Hilbert a tion mayalso be obtained from an asymptoti ally at spa e of weak urvature, where we an de�ne20

gravitational energy density. Then the requirement is that the kineti terms of all physi albosons, spin 0, 1, and 2, be of the same sign. Take the metri of the formg�� = ��� + h��; (64)h�� small. Then the Einstein-Hilbert Lagrangian be omes [17℄,� 116�G R jdet g��j1=2 = 116�Gf14��h����h�� � 18��h����h��� [��h�� � 12��h��℄[��0h��0 � 12��h�0�0 ℄ + O(h3)g: (65)Here indi es are raised with ���. After an appropriate hoi e of oordinates, `harmoni oor-dinates', the bra ket ���h�� � 12��h��� vanishes and only two independent omponents of h��remain, h11 = �h22 and h12. They represent the two physi al states of the graviton, heli ity�2. Their kineti terms are both positive, e.g.:+ 116�G 14��h12��h12: (66)Likewise, by an appropriate gauge transformation, we an a hieve ��A� = 0, `Lorentz gauge',and remain with only two `transverse' omponents A1; A2 of heli ity �1. They have positivekineti terms, e.g.: + 12g2 tr (��A�1��A1): (67)Finally, the kineti term of the s alar is positive:+12��'���': (68)An old re ipe from quantum �eld theory, `Wi k rotation', amounts to repla ing spa etime bya Riemannian manifold with Eu lidean signature. Then ertain al ulations be ome feasible oreasier. One of the reasons for this is that Eu lidean quantum �eld theory resembles statisti alme hani s, the imaginary time playing formally the role of the inverse temperature. Only atthe end of the al ulation the result is `rotated ba k' to real time. In some ases, this re ipe an be justi�ed rigorously. The pre ise formulation of the re ipe is that the n-point fun tions omputed from the Eu lidean Lagrangian be the analyti ontinuations in the omplex timeplane of the Minkowskian n-point fun tions. We shall indi ate a hand waving formulation ofthe re ipe, that is suÆ ient for our purpose: In a �rst stroke we pass to the signature �+++.In a se ond stroke we repla e t by it and repla e all Minkowskian s alar produ ts by the orresponding Eu lidean ones.The �rst stroke amounts simply to repla ing the metri by its negative. This leaves in-variant the Christo�el symbols, the Riemann and Ri i tensors, but reverses the sign of the21

urvature s alar. Likewise, in the other terms of the Lagrangian we get a minus sign for every ontra tion of indi es, e.g.: ��'���' = ��'���0'g��0 be omes ��'���0'(�g��0) = ���'���'.After multipli ation by a onventional overall minus sign the ombined Lagrangian reads now,f 2� 16�G � 116�G R + 12g2 tr (F ���F ��) + 1g2m2Atr (A��A�)+ 12 (D�')�D�' + 12 m2'j'j2 � 12 �2j'j2 + �j'j4+ � 0[ i �D� + m 14 ℄ g jdet g��j1=2: (69)To pass to the Eu lidean signature, we multiply time, energy and mass by i. This amountsto ��� = Æ�� in the s alar produ t. In order to have the Eu lidean anti ommutation relations, � � + � � = 2Æ��14; (70)we hange the Dira matri es to the Eu lidean ones, 0 = � 0 �12�12 0 � ; j = 1i � 0 �j��j 0 � ; (71)All four are now self adjoint. For the hirality we take 5 := 0 1 2 3 = ��12 00 12� : (72)The Minkowskian s alar produ t for spinors has a 0. This 0 is needed for the orre t physi alinterpretation of the energy of antiparti les and for invarian e under lifted Lorentz transforma-tions, Spin(1; 3). In the Eu lidean, there is no physi al interpretation and we an only retainthe requirement of a Spin(4) invariant s alar produ t. This s alar produ t has no 0. But thenwe have a problem if we want to write the Dira Lagrangian in terms of hiral spinors as above.For instan e, for a purely left-handed neutrino, R = 0 and �L i �D� L vanishes identi allybe ause 5 anti ommutes with the four �. The standard tri k of Eu lidean �eld theoreti ians[12℄ is fermion doubling, L and R are treated as two independent, four omponent spinors.They are not hiral proje tions of one four omponent spinor as in the Minkowskian, equation(59). The spurious degrees of freedom in the Eu lidean are kept all the way through the al u-lation. They are proje ted out only after the Wi k rotation ba k to Minkowskian, by imposing 5 L = � L; 5 R = R.In non ommutative geometry the Dira operator must be self adjoint, whi h is not the ase for the Eu lidean Dira operator i �D� + im 14 we get from the Lagrangian (69) aftermultipli ation of the mass by i. We therefore prefer the primed spinor variables 0 produ ingthe self adjoint Eu lidean Dira operator i �D� + m 5. Dropping the prime, the ombinedLagrangian in the Eu lidean then reads:f 2� 16�G � 116�G R + 12g2 tr (F ���F ��) + 1g2m2Atr (A��A�)22

+ 12 (D�')�D�' + 12 m2'j'j2 � 12 �2j'j2 + �j'j4+ �L i �D� L + �R i �D� R + m �L 5 R + m �R 5 Lg (det g��)1=2: (73)4 Connes' non ommutative geometryConnes equips Riemannian spa es with an un ertainty prin iple. As in quantum me hani s,this un ertainty prin iple is derived from non ommutativity.4.1 Motivation: quantum me hani sConsider the lassi al harmoni os illator. Its phase spa e is R2 with points labelled by positionx and momentum p. A lassi al observable is a di�erentiable fun tion on phase spa e su h asthe total energy p2=(2m) + kx2. Observables an be added and multiplied, they form thealgebra C1(R2), whi h is asso iative and ommutative. To pass to quantum me hani s, thisalgebra is rendered non ommutative by means of the following non ommutation relation forthe generators x and p, [x; p℄ = i~1: (74)Let us all A the resulting algebra `of quantum observables'. It is still asso iative, has aninvolution �� (the adjoint or Hermitean onjugation) and a unit 1. Let us brie y re all thede�ning properties of an involution: it is a linear map from the real algebra into itself thatreverses the produ t, (ab)� = b�a�, respe ts the unit, 1� = 1, and is su h that a�� = a.-

p6 x~=2rFigure 4: The �rst example of non ommutative geometryOf ourse, there is no spa e anymore of whi h A is the algebra of fun tions. Nevertheless,we talk about su h a `quantum phase spa e' as a spa e that has no points or a spa e with an un-23

ertainty relation. Indeed, the non ommutation relation (74) implies Heisenberg's un ertaintyrelation �x�p � ~=2 (75)and tells us that points in phase spa e lose all meaning, we an only resolve ells in phase spa eof volume ~=2, see Figure 4. To de�ne the un ertainty �a for an observable a 2 A, we needa faithful representation of the algebra on a Hilbert spa e, i.e. an inje tive homomorphism� : A ! End(H) ( f Appendix). For the harmoni os illator, this Hilbert spa e is H = L2(R).Its elements are the wave fun tions (x), square integrable fun tions on on�guration spa e.Finally, the dynami s is de�ned by a self adjoint observable H = H� 2 A via S hr�odinger'sequation �i~ ��t � �(H)� (t; x) = 0: (76)Usually the representation is not written expli itly. Sin e it is faithful, no onfusion shouldarise from this abuse. Here time is onsidered an external parameter, in parti ular, time is not onsidered an observable. This is di�erent in the spe ial relativisti setting where S hr�odinger'sequation is repla ed by Dira 's equation, �= = 0: (77)Now the wave fun tion is the four- omponent spinor onsisting of left- and right-handed,parti le and antiparti le wave fun tions. The Dira operator is not in A anymore, but �= 2End(H). The Dira operator is only formally self adjoint be ause there is no positive de�nites alar produ t, whereas in Eu lidean spa etime it is truly self adjoint, �=� = �=:Connes' geometries are des ribed by these three purely algebrai items, (A;H; �=), with A areal, asso iative, possibly non ommutative involution algebra with unit, faithfully representedon a omplex Hilbert spa e H, and �= is a self adjoint operator on H.4.2 The alibrating example: Riemannian spin geometryConnes' geometry [18℄ does to spa etime what quantum me hani s does to phase spa e. Of ourse, the �rst thing we have to learn is how to re onstru t the Riemannian geometry from thealgebrai data (A;H; �=) in the ase where the algebra is ommutative. We start the easy wayand onstru t the triple (A;H; �=) given a four dimensional, ompa t, Eu lidean spa etimeM .As beforeA = C1(M) is the real algebra of omplex valued di�erentiable fun tions on spa etimeand H = L2(S) is the Hilbert spa e of omplex, square integrable spinors on M . Lo ally, in24

any oordinate neighborhood, we write the spinor as a olumn ve tor, (x) 2 C 4; x 2M . Thes alar produ t of two spinors is de�ned by( ; 0) = ZM �(x) 0(x) dV; (78)with the invariant volume form dV := jdet g��j1=2 d4x de�ned with the metri tensor,g�� = g� ��x� ; ��x� � ; (79)that is the matrix of the Riemannian metri g with respe t to the oordinates x�, � = 0; 1; 2; 3:Note { and this is important { that with Eu lidean signature the Dira a tion is simply as alar produ t, SD = ( ; �= ). The representation is de�ned by pointwise multipli ation,(�(a) )(x) := a(x) (x); a 2 A. For a start, it is suÆ ient to know the Dira operator on a at manifold M and with respe t to inertial or Cartesian oordinates ~x~� su h that ~g~�~� = Æ~�~�.Then we use Dira 's original de�nition,D = �= = i ~��=�~x~�; (80)with the self adjoint -matri es 0 = � 0 �12�12 0 � ; j = 1i � 0 �j��j 0 � ; (81)with the Pauli matri es�1 = � 0 11 0� ; �2 = � 0 �ii 0 � ; �3 = � 1 00 �1� : (82)We will onstru t the general urved Dira operator later.When the dimension of the manifold is even like in our ase, the representation � is redu ible.Its Hilbert spa e de omposes into left- and right-handed spa es,H = HL �HR; HL = 1 � �2 H; HR = 1 + �2 H: (83)Again we make use of the unitary hirality operator,� = 5 := 0 1 2 3 = ��12 00 12� : (84)We will also need the harge onjugation or real stru ture, the anti-unitary operator:J = C := 0 2 Æ omplex onjugation = 0BB�0 �1 0 01 0 0 00 0 0 10 0 �1 01CCA Æ ; (85)25

that permutes parti les and antiparti les.The �ve items (A;H;D; J; �) form what Connes alls an even, real spe tral triple [19℄.A is a real, asso iative involution algebra with unit, represented faithfully by bounded operatorson the Hilbert spa e H.D is an unbounded self adjoint operator on H.J is an anti-unitary operator,� a unitary one.They enjoy the following properties:� J2 = �1 in four dimensions (J2 = 1 in zero dimensions).� [�(a); J�(~a)J�1℄ = 0 for all a; ~a 2 A.� DJ = JD, parti les and antiparti les have the same dynami s.� [D; �(a)℄ is bounded for all a 2 A and [[D; �(a)℄; J�(~a)J�1℄ = 0 for all a; ~a 2 A.This property is alled �rst order ondition be ause in the alibrating example itstates that the genuine Dira operator is a �rst order di�erential operator.� �2 = 1 and [�; �(a)℄ = 0 for all a 2 A. These properties allow the de ompositionH = HL �HR.� J� = �J .� D� = ��D, hirality does not hange under time evolution.� There are three more properties, that we do not spell out, orientability, whi h relatesthe hirality to the volume form, Poin ar�e duality and regularity, whi h states thatour fun tions a 2 A are di�erentiable.Connes promotes these properties to the axioms de�ning an even, real spe tral triple. Theseaxioms are justi�ed by hisRe onstru tion theorem (Connes 1996 [20℄): Consider an (even) spe tral triple (A;H;D; J; (�))whose algebra A is ommutative. Then here exists a ompa t, Riemannian spin manifoldM (ofeven dimensions), whose spe tral triple (C1(M);L2(S); �=; C; ( 5)) oin ides with (A;H;D; J; (�)).For details on this theorem and non ommutative geometry in general, I warmly re ommendthe Costa Ri a book [10℄. Let us try to get a feeling of the lo al information ontained in thistheorem. Besides des ribing the dynami s of the spinor �eld , the Dira operator �= en odesthe dimension of spa etime, its Riemannian metri , its di�erential forms and its integration,that is all the tools that we need to de�ne a Yang-Mills-Higgs model. In Minkowskian signature,26

the square of the Dira operator is the wave operator, whi h in 1+2 dimensions governs thedynami s of a drum. The deep question: `Can you hear the shape of a drum?' has been raised.This question on erns a global property of spa etime, the boundary. Can you re onstru t itfrom the spe trum of the wave operator?The dimension of spa etime is a lo al property. It an be retrieved from the asymp-toti behaviour of the spe trum of the Dira operator for large eigenvalues. Sin eM is ompa t, the spe trum is dis rete. Let us order the eigenvalues, :::�n�1 ��n � �n+1::: Then Weyl's spe tral theorem states that the eigenvalues grow asymp-toti ally as n1=dimM . To explore a lo al property of spa etime we only need thehigh energy part of the spe trum. This is in ni e agreement with our intuition fromquantum me hani s and motivates the name `spe tral triple'.The metri an be re onstru ted from the ommutative spe tral triple by Connes dis-tan e formula (86) below. In the ommutative ase a point x 2M is re onstru tedas the pure state. The general de�nition of a pure state of ourse does not use the ommutativity. A state Æ of the algebra A is a linear form on A, that is normalized,Æ(1) = 1, and positive, Æ(a�a) � 0 for all a 2 A. A state is pure if it annot bewritten as a linear ombination of two states. For the alibrating example, thereis a one-to-one orresponden e between points x 2 M and pure states Æx de�nedby the Dira distribution, Æx(a) := a(x) = RM Æx(y)a(y)d4y. The geodesi distan ebetween two points x and y is re onstru ted from the triple as:sup fjÆx(a)� Æy(a)j; a 2 C1(M) su h that jj[ �=; �(a)℄jj � 1g : (86)For the alibrating example, [ �=; �(a)℄ is a bounded operator. Indeed, [ �=; �(a)℄ =i ���(a )� ia ��� = i �(��a) , and ��a is bounded as a di�erentiable fun tionon a ompa t spa e.For a general spe tral triple this operator is bounded by axiom. In any ase, theoperator norm jj[ �=; �(a)℄jj in the distan e formula is �nite.Consider the ir le, M = S1, of ir umferen e 2� with Dira operator �= = id=dx.A fun tion a 2 C1(S1) is represented faithfully on a wavefun tion 2 L2(S1)by pointwise multipli ation, (�(a) )(x) = a(x) (x). The ommutator [ �=; �(a)℄ =i�(a0) is familiar from quantum me hani s. Its operator norm is jj[ �=; �(a)℄jj :=sup j[ �=; �(a)℄ j=j j = supx ja0(x)j, with j j2 = R 2�0 � (x) (x) dx. Therefore, thedistan e between two points x and y on the ir le issupa fja(x)� a(y)j; supx ja0(x)j � 1g = jx� yj: (87)27

Note that Connes' distan e formula ontinues to make sense for non- onne ted man-ifolds, like dis rete spa es of dimension zero, i.e. olle tions of points.Di�erential forms, for example of degree one like da for a fun tion a 2 A, are re on-stru ted as (�i)[ �=; �(a)℄. This is again motivated from quantum me hani s. Indeedin a 1+0 dimensional spa etime da is just the time derivative of the `observable' aand is asso iated with the ommutator of the Hamilton operator with a.Motivated from quantum me hani s, we de�ne a non ommutative geometry by a real spe traltriple with non ommutative algebra A.4.3 Spin groupsLet us go ba k to quantum me hani s of spin and re all how a spa e rotation a ts on a spin 12parti le. For this we need group homomorphisms between the rotation group SO(3) and theprobability preserving unitary group SU(2). We onstru t �rst the group homomorphismp : SU(2) �! SO(3)U 7�! p(U):With the help of the auxiliary fun tionf : R3 �! su(2)~x = 0�x1x2x31A 7�! �12ixj�j ;we de�ne the rotation p(U) by p(U)~x := f�1(Uf(~x)U�1): (88)The onjugation by the unitary U will play an important role and we give it a spe ial name,iU(w) := UwU�1, i for inner. Sin e i(�U) = iU , the proje tion p is two to one, Ker(p) = f�1g.Therefore the spin lift L : SO(3) �! SU(2)R = exp(!) 7�! exp(18!jk[�j; �k℄) (89)is double-valued. It is a lo al group homomorphism and satis�es p(L(R)) = R. Its double-valuedness is a essible to quantum me hani al experiments: neutrons have to be rotatedthrough an angle of 720Æ before interferen e patterns repeat [21℄.28

AutH(A) - Di�(M)nMSpin(1; 3) - SO(1; 3) � Spin(1; 3) - SO(3) � SU(2)Aut(A) - Di�(M) - SO(1; 3) - SO(3)p ? CCCCO 6L p ? CCCCO 6L p ? CCCCO 6L p ? CCCCO 6LFigure 5: The nested spin lifts of Connes, Cartan, Dira , and PauliThe lift L was generalized by Dira to the spe ial relativisti setting, e.g. [4℄, and by E.Cartan [22℄ to the general relativisti setting. Connes [23℄ generalizes it to non ommutativegeometry, see Figure 5. The transformations we need to lift are Lorentz transformations inspe ial relativity, and general oordinate transformations in general relativity, i.e. our alibrat-ing example. The latter transformations are the lo al elements of the di�eomorphism groupDi�(M). In the setting of non ommutative geometry, this group is the group of algebra au-tomorphisms Aut(A). Indeed, in the alibrating example we have Aut(A)=Di�(M). In orderto generalize the spin group to spe tral triples, Connes de�nes the re epta le of the group of`lifted automorphisms',AutH(A) := fU 2 End(H); UU� = U�U = 1; UJ = JU; U� = �U; iU 2 Aut(�(A))g: (90)The �rst three properties say that a lifted automorphism U preserves probability, harge on-jugation, and hirality. The fourth, alled ovarian e property, allows to de�ne the proje tionp : AutH(A) �! Aut(A) by p(U) = ��1iU� (91)We will see that the ovarian e property will prote t the lo ality of �eld theory. For the ali-brating example of a four dimensional spa etime, a lo al al ulation, i.e. in a oordinate pat h,that we still denote byM , yields the semi-dire t produ t ( f Appendix) of di�eomorphisms withlo al or gauged spin transformations, AutL2(S)(C1(M)) = Di�(M)n MSpin(4). We say re ep-ta le be ause already in six dimensions, AutL2(S)(C1(M)) is larger than Di�(M)n MSpin(6).However we an use the lift L with p(L(�)) = �, � 2Aut(A) to orre tly identify the spingroup in any dimension of M . Indeed we will see that the spin group is the image of the spinlift L(Aut(A)), in general a proper subgroup of the re epta le AutH(A).Let � be a di�eomorphism lose to the identity. We interpret � as oordinate transformation,all our al ulations will be lo al,M standing for one hart, on whi h the oordinate systems ~x~�and x� = (�(~x))� are de�ned. We will work out the lo al expression of a lift of � to the Hilbertspa e of spinors. This lift U = L(�) will depend on the metri and on the initial oordinatesystem ~x~�. 29

In a �rst step, we onstru t a group homomorphism � : Di�(M) ! Di�(M) n MSO(4)into the group of lo al `Lorentz' transformations, i.e. the group of di�erentiable fun tionsfrom spa etime into SO(4) with pointwise multipli ation. Let (~e�1(~x))~�a = (~g�1=2(~x))~�a bethe inverse of the square root of the positive matrix ~g of the metri with respe t to the initial oordinate system ~x~�. Then the four ve tor �elds ~ea, a = 0; 1; 2; 3, de�ned by~ea := (~e�1)~�a ��~x~� (92)give an orthonormal frame of the tangent bundle. This frame de�nes a omplete gauge �xing ofthe Lorentz gauge group MSO(4) be ause it is the only orthonormal frame to have symmetri oeÆ ients (~e�1)~�a with respe t to the oordinate system ~x~�. We all this gauge the symmetri gauge for the oordinates ~x~�: Now let us perform a lo al hange of oordinates, x = �(~x). Theholonomi frame with respe t to the new oordinates is related to the former holonomi one bythe inverse Ja obian matrix of ���x� = �~x~��x� ��~x~� = (J �1)~�� ��~x~� ; �J �1(x)�~�� = �~x~��x� : (93)The matrix g of the metri with respe t to the new oordinates reads,g��(x) := g� ��x� ; ��x� �����x = �J �1T (x)~g(��1(x))J �1(x)��� ; (94)and the symmetri gauge for the new oordinates x is the new orthonormal frameeb = e�1�b ��x� = g�1=2�bJ �1 ~�� ��~x~� = �J �1pJ ~g�1J T�~�b ��~x~� : (95)New and old orthonormal frames are related by a Lorentz transformation �, eb = ��1 ab~ea, with�(�)j~x = pJ �1T ~gJ �1����(~x) J j~x p~g�1���~x = pgJp~g�1: (96)If M is at and ~x~� are `inertial' oordinates, i.e. ~g~�~� = Æ~�~� , and � is a lo al isometry thenJ (~x) 2 SO(4) for all ~x and �(�) = J . In spe ial relativity, therefore, the symmetri gaugeties together Lorentz transformations in spa etime with Lorentz transformations in the tangentspa es.In general, if the oordinate transformation � is lose to the identity, so is its Lorentztransformation �(�) and it an be lifted to the spin group,S : SO(4) �! Spin(4)� = exp! 7�! exp �14!ab ab� (97)30

with ! = �!T 2 so(4) and ab := 12 [ a; b℄. With our hoi e (81) for the matri es, we have 0j = i���j 00 �j � ; jk = i�jk`��` 00 �`� ; j; k = 1; 2; 3; �123 = 1: (98)We an write the lo al expression [24℄ of the lift L : Di�(M)! Di�(M)n MSpin(4),(L(�) ) (x) = S (�(�))j��1(x) (��1(x)): (99)L is a double-valued group homomorphism. For any � lose to the identity, L(�) is unitary, om-mutes with harge onjugation and hirality, satis�es the ovarian e property, and p(L(�)) = �.Therefore, we have lo allyL(Di�(M)) � Di�(M) n MSpin(4) = AutL2(S)(C1(M)): (100)The symmetri gauge is a omplete gauge �xing and this redu tion follows Einstein's spirit inthe sense that the only arbitrary hoi e is the one of the initial oordinate system ~x~� as willbe illustrated in the next se tion. Our omputations are deliberately lo al. The global pi ture an be found in referen e [25℄.5 The spe tral a tion5.1 Repeating Einstein's derivation in the ommutative aseWe are ready to parallel Einstein's derivation of general relativity in Connes' language ofspe tral triples. The asso iative algebra C1(M) is ommutative, but this property will neverbe used. As a by-produ t, the lift L will re on ile Einstein's and Cartan's formulations of generalrelativity and it will yield a self ontained introdu tion to Dira 's equation in a gravitational�eld a essible to parti le physi ists. For a omparison of Einstein's and Cartan's formulationsof general relativity see for example [6℄.5.1.1 First stroke: kinemati sInstead of a point-parti le, Connes takes as matter a �eld, the free, massless Dira parti le (~x)in the at spa etime of spe ial relativity. In inertial oordinates ~x~�, its dynami s is given bythe Dira equation, ~�= = iÆ~�a a ��~x~� = 0: (101)We have written Æ~�a a instead of ~� to stress that the matri es are ~x-independent. ThisDira equation is ovariant under Lorentz transformations. Indeed if � is a lo al isometry thenL(�)~�=L(�)�1 = �= = iÆ�a a ��x� : (102)31

To prove this spe ial relativisti ovarian e, one needs the identity S(�) aS(�)�1 = ��1 ab bfor Lorentz transformations � 2 SO(4) lose to the identity. Take a general oordinate trans-formation � lose to the identity. Now omes a long, but straight-forward al ulation. It is auseful exer ise requiring only matrix multipli ation and standard al ulus, Leibniz and hainrules. Its result is the Dira operator in urved oordinates,L(�)~�=L(�)�1 = �= = ie�1�a a � ��x� + s(!�)� ; (103)where e�1 = pJJ T is a symmetri matrix,s : so(4) �! spin(4)! 7�! 14!ab ab (104)is the Lie algebra isomorphism orresponding to the lift (97) and!�(x) = �j��1(x) �� ��1��x : (105)The `spin onne tion' ! is the gauge transform of the Levi-Civita onne tion �, the latteris expressed with respe t to the holonomi frame ��, the former is written with respe t tothe orthonormal frame ea = e�1�a��. The gauge transformation passing between them ise 2 MGL4, ! = e�e�1 + ede�1: (106)We re over the well known expli it expression!ab�(e) = 12 [(��ea�)� (��ea�) + em�(��em�)e�1�a℄ e�1�b � [a$ b℄ (107)of the spin onne tion in terms of the �rst derivatives of ea� = pga�: Again the spin onne tionhas zero urvature and the equivalen e prin iple relaxes this onstraint. But now equation (103)has an advantage over its analogue (2). Thanks to Connes' distan e formula (86), the metri an be read expli itly in (103) from the matrix of fun tions e�1�a, while in (2) �rst derivatives ofthe metri are present. We are used to this nuan e from ele tro-magnetism, where the lassi alparti le feels the for e while the quantum parti le feels the potential. In Einstein's approa h,the zero onne tion u tuates, in Connes' approa h, the at metri u tuates. This meansthat the onstraint e�1 = pJJ T is relaxed and e�1 now is an arbitrary symmetri matrixdepending smoothly on x.Let us mention two experiments with neutrons on�rming the `minimal oupling' of theDira operator to urved oordinates, equation (103). The �rst takes pla e in at spa etime.32

The neutron interferometer is mounted on a loud speaker and shaken periodi ally [26℄. Theresulting pseudo for es oded in the spin onne tion do shift the interferen e patterns observed.The se ond experiment takes pla e in a true gravitational �eld in whi h the neutron interfer-ometer is pla ed [27℄. Here shifts of the interferen e patterns are observed that do depend onthe gravitational potential, ea� in equation (103).5.1.2 Se ond stroke: dynami sThe se ond stroke, the ovariant dynami s for the new lass of Dira operators �= is due toChamseddine & Connes [28℄. It is the elebrated spe tral a tion. The beauty of their approa hto general relativity is that it works pre isely be ause the Dira operator �= plays two rolessimultaneously, it de�nes the dynami s of matter and the kinemati s of gravity. For a dis ussionof the transformation passing from the metri to the Dira operator I re ommend the arti le[29℄ by Landi & Rovelli.The starting point of Chamseddine & Connes is the simple remark that the spe trumof the Dira operator is invariant under di�eomorphisms interpreted as general oordinatetransformations. From �=� = ���= we know that the spe trum of �= is even. Indeed, forevery eigenve tor of �= with eigenvalue E, � is eigenve tor with eigenvalue �E. We maytherefore onsider only the spe trum of the positive operator �=2=�2 where we have divided bya �xed arbitrary energy s ale to make the spe trum dimensionless. If it was not divergent thetra e tr �=2=�2 would be a general relativisti a tion fun tional. To make it onvergent, take adi�erentiable fun tion f : R+! R+ of suÆ iently fast de rease su h that the a tionSCC := tr f( �=2=�2) (108) onverges. It is still a di�eomorphism invariant a tion. The following theorem, also known asheat kernel expansion, is a lo al version of an index theorem [30℄, that as explained in JeanZinn-Justin's le tures [12℄ is intimately related to Feynman graphs with one fermioni loop.Theorem: Asymptoti ally for high energies, the spe tral a tion isSCC = ZM [ 2� 16�G � 116�GR+ a(5R2 � 8Ri i2 � 7Riemann2)℄ dV + O(��2); (109)where the osmologi al onstant is � = 6f0f2 �2, Newton's onstant is G = 3�f2��2 and a = f45760�2 .On the right-hand side of the theorem we have omitted surfa e terms, that is terms that donot ontribute to the Euler-Lagrange equations. The Chamseddine-Connes a tion is universalin the sense that the ` ut o�' fun tion f only enters through its �rst three `moments', f0 :=R10 uf(u)du, f2 := R10 f(u)du and f4 = f(0).If we take for f a di�erentiable approximation of the hara teristi fun tion of the unitinterval, f0 = 1=2, f2 = f4 = 1, then the spe tral a tion just ounts the number of eigenvalues33

of the Dira operator whose absolute values are below the ` ut o�' �. In four dimensions, theminimax example is the at 4-torus with all ir umferen es measuring 2�. Denote by B(x),B = 1; 2; 3; 4, the four omponents of the spinor. The Dira operator is�= = 0BB� 0 0 �i�0 + �3 �1 � i�20 0 �1 + i�2 �i�0 � �3�i�0 � �3 ��1 + i�2 0 0��1 � i�2 �i�0 + �3 0 0 1CCA : (110)After a Fourier transform B(x) =: Xj0;:::;j32Z B(j0; :::; j3) exp(�ij�x�); B = 1; 2; 3; 4 (111)the eigenvalue equation �= = � reads0BB� 0 0 �j0 � ij3 �ij1 � j20 0 �ij1 + j2 �j0 + ij3�j0 + ij3 ij1 + j2 0 0ij1 � j2 �j0 � ij3 0 0 1CCA0BB� 1 2 3 41CCA = �0BB� 1 2 3 41CCA : (112)Its hara teristi equation is [�2 � (j20 + j21 + j22 + j23)℄2 = 0 and for �xed j�, ea h eigenvalue� = �pj20 + j21 + j22 + j23 has multipli ity two. Therefore asymptoti ally for large � there are4B4�4 eigenvalues ( ounted with their multipli ity) whose absolute values are smaller than �.B4 = �2=2 denotes the volume of the unit ball in R4. En passant, we he k Weyl's spe traltheorem. Let us arrange the absolute values of the eigenvalues in an in reasing sequen e andnumber them by naturals n, taking due a ount of their multipli ities. For large n, we havej�nj � � n2�2�1=4 : (113)The exponent is indeed the inverse dimension. To he k the heat kernel expansion, we omputethe right-hand side of equation (109):SCC = ZM � 8�G dV = (2�)4 f04�2�4 = 2�2�4; (114)whi h agrees with the asymptoti ount of eigenvalues, 4B4�4. This example was the at torus.Curvature will modify the spe trum and this modi� ation an be used to measure the urvature= gravitational �eld, exa tly as the Zeemann or Stark e�e t measures the ele tro-magneti �eldby observing how it modi�es the spe tral lines of an atom.In the spe tral a tion, we �nd the Einstein-Hilbert a tion, whi h is linear in urvature.In addition, the spe tral a tion ontains terms quadrati in the urvature. These terms ansafely be negle ted in weak gravitational �elds like in our solar system. In homogeneous,34

isotropi osmologies, these terms are a surfa e term and do not modify Einstein's equa-tion. Nevertheless the quadrati terms render the (Eu lidean) Chamseddine-Connes a tionpositive. Therefore this a tion has minima. For instan e, the 4-sphere with a radius of(11f4)1=2(90�(1 � (1 � 11=15 f0f4f�22 )1=2))�1=2 times the Plan k length pG is a minimum, a`ground state'. This minimum breaks the di�eomorphism group spontaneously [23℄ down tothe isometry group SO(5). The little group is the isometry group, onsisting of those liftedautomorphisms that ommute with the Dira operator �=. Let us anti ipate that the sponta-neous symmetry breaking via the Higgs me hanism will be a mirage of this gravitational breakdown. Physi ally this ground state seems to regularize the initial osmologi al singularity withits ultra strong gravitational �eld in the same way in whi h quantum me hani s regularizes theCoulomb singularity of the hydrogen atom.We lose this subse tion with a te hni al remark. We noti ed that the matrix e�1�a inequation (103) is symmetri . A general, not ne essarily symmetri matrix e�1�a an be obtainedfrom a general Lorentz transformation � 2 MSO(4):e�1�a�ab = e�1�b; (115)whi h is nothing but the polar de omposition of the matrix e�1. These transformations arethe gauge transformations of general relativity in Cartan's formulation. They are invisible inEinstein's formulation be ause of the omplete (symmetri ) gauge �xing oming from the initial oordinate system ~x~�.5.2 Almost ommutative geometryWe are eager to see the spe tral a tion in a non ommutative example. Te hni ally the simplestnon ommutative examples are almost ommutative. To onstru t the latter, we need a naturalproperty of spe tral triples, ommutative or not: The tensor produ t of two even spe tral triplesis an even spe tral triple. If both are ommutative, i.e. des ribing two manifolds, then theirtensor produ t simply des ribes the dire t produ t of the two manifolds.Let (Ai;Hi;Di; Ji; �i), i = 1; 2 be two even, real spe tral triples of even dimensions d1 andd2. Their tensor produ t is the triple (At;Ht;Dt; Jt; �t) of dimension d1 + d2 de�ned byAt = A1 A2; Ht = H1 H2;Dt = D1 12 + �1 D2;Jt = J1 J2; �t = �1 �2:The other obvious hoi e for the Dira operator, D1 �2 + 11 D2, is unitarily equivalentto the �rst one. By de�nition, an almost ommutative geometry is a tensor produ t of two35

spe tral triples, the �rst triple is a 4-dimensional spa etime, the alibrating example,�C1(M);L2(S); �=; C; 5� ; (116)and the se ond is 0-dimensional. In a ordan e with Weyl's spe tral theorem, a 0-dimensionalspe tral triple has a �nite dimensional algebra and a �nite dimensional Hilbert spa e. We willlabel the se ond triple by the subs ript �f (for �nite) rather than by �2. The origin of the wordalmost ommutative is lear: we have a tensor produ t of an in�nite dimensional ommutativealgebra with a �nite dimensional, possibly non ommutative algebra.This tensor produ t is, in fa t, already familiar to you from the quantum me hani s of spin,whose Hilbert spa e is the in�nite dimensional Hilbert spa e of square integrable fun tionson on�guration spa e tensorized with the 2-dimensional Hilbert spa e C 2 on whi h a ts thenon ommutative algebra of spin observables. It is the algebra H of quaternions, 2� 2 omplexmatri es of the form�x ��yy �x � x; y 2 C . A basis of H is given by f12; i�1; i�2; i�3g, the identitymatrix and the three Pauli matri es (82) times i. The group of unitaries of H is SU(2), thespin over of the rotation group, the group of automorphisms of H is SU(2)=Z2, the rotationgroup.A ommutative 0-dimensional or �nite spe tral triple is just a olle tion of points, forexamples see [31℄. The simplest example is the two-point spa e,Af = C L � C R 3 (aL; aR); Hf = C 4 ; �f (aL; aR) = 0BB�aL 0 0 00 aR 0 00 0 �aR 00 0 0 �aR1CCA ; (117)Df =0BB� 0 m 0 0�m 0 0 00 0 0 �m0 0 m 0 1CCA ; m 2 C ; Jf = � 0 1212 0 � Æ ; �f = 0BB��1 0 0 00 1 0 00 0 �1 00 0 0 11CCA :(118)The algebra has two points = pure states, ÆL and ÆR, ÆL(aL; aR) = aL. By Connes' formula(86), the distan e between the two points is 1=jmj. On the other hand Dt = �=14 + 5Df ispre isely the free massive Eu lidean Dira operator. It des ribes one Dira spinor of mass jmjtogether with its antiparti le. The tensor produ t of the alibrating example and the two pointspa e is the two-sheeted universe, two identi al spa etimes at onstant distan e. It was the�rst example in non ommutative geometry to exhibit spontaneous symmetry breaking [32, 33℄.One of the major advantages of the algebrai des ription of spa e in terms of a spe traltriple, ommutative or not, is that ontinuous and dis rete spa es are in luded in the samepi ture. We an view almost ommutative geometries as Kaluza-Klein models [34℄ whose �fth36

dimension is dis rete. Therefore we will also all the �nite spe tral triple `internal spa e'. Innon ommutative geometry, 1-forms are naturally de�ned on dis rete spa es where they playthe role of onne tions. In almost ommutative geometry, these dis rete, internal onne tionswill turn out to be the Higgs s alars responsible for spontaneous symmetry breaking.Almost ommutative geometry is an ideal play-ground for the physi ist with low ulture inmathemati s that I am. Indeed Connes' re onstru tion theorem immediately redu es the in�-nite dimensional, ommutative part to Riemannian geometry and we are left with the internalspa e, whi h is a essible to anybody mastering matrix multipli ation. In parti ular, we aneasily make pre ise the last three axioms of spe tral triples: orientability, Poin ar�e duality andregularity. In the �nite dimensional ase { let us drop the �f from now on { orientability meansthat the hirality an be written as a �nite sum,� =Xj �(aj)J�(~aj)J�1; aj; ~aj 2 A: (119)The Poin ar�e duality says that the interse tion form\ij := tr ���(pi)J�(pj)J�1� (120)must be non-degenerate, where the pj are a set of minimal proje tors of A. Finally, thereis the regularity ondition. In the alibrating example, it ensures that the algebra elements,the fun tions on spa etimeM , are not only ontinuous but di�erentiable. This ondition is of ourse empty for �nite spe tral triples.Let us ome ba k to our �nite, ommutative example. The two-point spa e is orientable,� = �(�1; 1)J�(�1; 1)J�1. It also satis�es Poin ar�e duality, there are two minimal proje tors,p1 = (1; 0), p2 = (0; 1), and the interse tion form is \ = � 0 �1�1 2 �.5.3 The minimax exampleIt is time for a non ommutative internal spa e, a mild variation of the two point spa e:A = H � C 3 (a; b); H = C 6; �(a; b) = 0BB� a 0 0 00 �b 0 00 0 b12 00 0 0 b1CCA ; (121)~D = 0BB� 0 M 0 0M� 0 0 00 0 0 �M0 0 �M� 0 1CCA ; M = � 0m� ; m 2 C ; (122)37

J = � 0 1313 0 � Æ ; � = 0BB��12 0 0 00 1 0 00 0 �12 00 0 0 11CCA : (123)The unit is (12; 1) and the involution is (a; b)� = (a�;�b); where a� is the Hermitean onjugate ofthe quaternion a. The Hilbert spa e now ontains one massless, left-handed Weyl spinor andone Dira spinor of mass jmj and M is the fermioni mass matrix. We denote the anoni albasis of C 6 symboli ally by (�; e)L; eR; (� ; e )L; e R. The spe tral triple still des ribes two points,ÆL(a; b) = 12tr a and ÆR(a; b) = b separated by a distan e 1=jmj. There are still two minimalproje tors, p1 = (12; 0), p2 = (0; 1) and the interse tion form \ = � 0 �2�2 2 � is invertible.Our next task is to lift the automorphisms to the Hilbert spa e and u tuate the ` at'metri ~D. All automorphisms of the quaternions are inner, the omplex numbers onsidered as2-dimensional real algebra only have one non-trivial automorphism, the omplex onjugation.It is dis onne ted from the identity and we may negle t it. ThenAut(A) = SU(2)=Z2 3 ��u; ��u(a; b) = (uau�1; b): (124)The re epta le group, subgroup of U(6) is readily al ulated,AutH(A) = U(2)� U(1) 3 U = 0BB�U2 0 0 00 U1 0 00 0 �U2 00 0 0 �U11CCA ; U2 2 U(2); U1 2 U(1): (125)The ovarian e property is ful�lled, iU�(a; b) = �(iU2a; b) and the proje tion, p(U) =� (detU2)�1=2U2, has kernel Z2. The lift,L(�u) = �(�u; 1)J�(�u; 1)J�1 = 0BB��u 0 0 00 1 0 00 0 ��u 00 0 0 11CCA ; (126)is double-valued. The spin group is the image of the lift, L(Aut(A)) = SU(2), a propersubgroup of the re epta le AutH(A) = U(2) � U(1). The u tuated Dira operator isD := L(�u) ~DL(�u)�1 = 0BB� 0 �uM 0 0(�uM)� 0 0 00 0 0 �uM0 0 (�uM)� 0 1CCA : (127)An absolutely remarkable property of the u tuated Dira operator in internal spa e is that it an be written as the at Dira operator plus a 1-form:D = ~D + �(�u; 1) [D; �(�u�1; 1)℄ + J �(�u; 1) [D; �(�u�1; 1)℄J�1: (128)38

The anti-Hermitean 1-form(�i)�(�u; 1) [D; �(�u�1; 1)℄ = (�i)0BB� 0 h 0 0h� 0 0 00 0 0 00 0 0 01CCA ; h := �uM�M (129)is the internal onne tion. The u tuated Dira operator is the ovariant one with respe t tothis onne tion. Of ourse, this onne tion is at, its �eld strength = urvature 2-form vanishes,a onstraint that is relaxed by the equivalen e prin iple. The result an be stated without goinginto the details of the re onstru tion of 2-forms from the spe tral triple: h be omes a general omplex doublet, not ne essarily of the form �uM�M.Now we are ready to tensorize the spe tral triple of spa etime with the internal one and ompute the spe tral a tion. The algebra At = C1(M) A des ribes a two-sheeted universe.Let us all again its sheets `left' and `right'. The Hilbert spa e Ht = L2(S)H des ribes theneutrino and the ele tron as genuine �elds, that is spa etime dependent. The Dira operator~Dt = ~�= 16 + 5 ~D is the at, free, massive Dira operator and it is impatient to u tuate.The automorphism group lose to the identity,Aut(At) = [Di�(M)n MSU(2)=Z2℄ � Di�(M) 3 ((�L; ��u); �R); (130)now ontains two independent oordinate transformations �L and �R on ea h sheet and agauged, that is spa etime dependent, internal transformation ��u. The gauge transformationsare inner, they a t by onjugation i�u. The re epta le group isAutHt(At) = Di�(M)n M (Spin(4)� U(2) � U(1)): (131)It only ontains one oordinate transformation, a point on the left sheet travels together withits right shadow. Indeed the ovarian e property forbids to lift an automorphism with �L 6= �R.Sin e the mass term multiplies left- and right-handed ele tron �elds, the ovarian e propertysaves the lo ality of �eld theory, whi h postulates that only �elds at the same spa etime point an be multiplied. We have seen examples where the re epta le has more elements than theautomorphism group, e.g. six-dimensional spa etime or the present internal spa e. Now wehave an example of automorphisms that do not �t into the re epta le. In any ase the spingroup is the image of the ombined, now 4-valued lift Lt(�; ��u),Lt(Aut(At)) = Di�(M)n M (Spin(4)� SU(2)): (132)The u tuating Dira operator isDt = Lt(�; ��u) ~DtLt(�; ��u)�1 = 0BB� �=L 5' 0 0 5'� �=R 0 00 0 C �=LC�1 5 �'0 0 5 �'� C �=RC�11CCA ; (133)39

with e�1 =pJJ T ; �=L = ie�1�a a[�� + s(!(e)�) +A�℄; (134)A� = �� u��(�u�1); �=R = ie�1�a a[�� + s(!(e)�)℄; (135)' = �uM: (136)Note that the sign ambiguity in �u drops out from the su(2)-valued 1-form A = A�dx� onspa etime. This is not the ase for the ambiguity in the `Higgs' doublet ' yet, but this am-biguity does drop out from the spe tral a tion. The variable ' is the homogeneous vari-able orresponding to the aÆne variable h = ' � M in the onne tion 1-form on internalspa e. The u tuating Dira operator Dt is still at. This onstraint has now three parts,e�1 = pJ (�)J (�)T ; A = �ud(u�1); and ' = �uM. A ording to the equivalen e prin iple,we will take e to be any symmetri , invertible matrix depending di�erentiably on spa etime,A to be any su(2)-valued 1-form on spa etime and ' any omplex doublet depending di�eren-tiably on spa etime. This de�nes the new kinemati s. The dynami s of the spinors = matteris given by the u tuating Dira operator Dt, whi h is ovariant with respe t to i.e. minimally oupled to gravity, the gauge bosons and the Higgs boson. This dynami s is equivalently givenby the Dira a tion ( ;Dt ) and this a tion delivers the awkward Yukawa ouplings for free.The Higgs boson ' enjoys two geometri interpretations, �rst as onne tion in the dis rete di-re tion. The se ond derives from Connes' distan e formula: 1=j'(x)j is the { now x-dependent{ distan e between the two sheets. The al ulation behind the se ond interpretation makesexpli it use of the Kaluza-Klein nature of almost ommutative geometries [35℄.As in pure gravity, the dynami s of the new kinemati s derives from the Chamseddine-Connes a tion,SCC [e;A; '℄ = tr f(D2t =�2)= ZM [ 2� 16�G � 116�GR+ a(5R2 � 8Ri i2 � 7Riemann2)12g22 trF ���F �� + 12(D�')�D�'�j'j4 � 12�2j'j2 + 112j'j2R ℄ dV + O(��2); (137)where the oupling onstants are� = 6f0f2 �2; G = �2f2 ��2; a = f4960�2 ; g22 = 6�2f4 ; � = �23f4 ; �2 = 2f2f4 �2: (138)Note the presen e of the onformal oupling of the s alar to the urvature s alar, + 112j'j2R.From the u tuation of the Dira operator, we have derived the s alar representation, a omplexdoublet '. Geometri ally, it is a onne tion on the �nite spa e and as su h uni�ed with the40

Yang-Mills bosons, whi h are onne tions on spa etime. As a onsequen e, the Higgs self oupling � is related to the gauge oupling g2 in the spe tral a tion, g22 = 18�. Furthermorethe spe tral a tion ontains a negative mass square term for the Higgs �12�2j'j2 implying anon-trivial ground state or va uum expe tation value j'j = v = �(4�)�1=2 in at spa etime.Reshifting to the inhomogeneous s alar variable h = '� v, whi h vanishes in the ground state,modi�es the osmologi al onstant by V (v) and Newton's onstant from the term 112v2R:� = 6�3f0f2 � f2f4��2; G = 3�2f2��2: (139)Now the osmologi al onstant an have either sign, in parti ular it an be zero. This is wel omebe ause experimentally the osmologi al onstant is very lose to zero, � < 10�119=G. Onthe other hand, in spa etimes of large urvature, like for example the ground state, the positive onformal oupling of the s alar to the urvature dominates the negative mass square term�12�2j'j2. Therefore the va uum expe tation value of the Higgs vanishes, the gauge symmetryis unbroken and all parti les are massless. It is only after the big bang, when spa etime losesits strong urvature that the gauge symmetry breaks down spontaneously and parti les a quiremasses.The omputation of the spe tral a tion is long, let us set some waypoints. The square of the u tuating Dira operator is D2t = ��+E, where � is the ovariant Lapla ian, in oordinates:� = g�~� �� ��x�14 1H + 14!ab� ab 1H + 14 [�(A�) + J�(A�)J�1℄� Æ� ~� � �� ~��14 1H�� � ��x� 14 1H + 14!ab� ab 1H + 14 [�(A�) + J�(A�)J�1℄� ; (140)and where E, for endomorphism, is a zero order operator, that is a matrix of size 4 dimH whoseentries are fun tions onstru ted from the bosoni �elds and their �rst and se ond derivatives,E = 12 [ � � 1H℄R�� (141)+ 0BB� 14 ''� �i 5 � D�' 0 0�i 5 � (D�')� 14 '�' 0 00 0 14 ''� �i 5 � D�'0 0 �i 5 � (D�')� 14 '�' 1CCA :R is the total urvature, a 2-form with values in the (Lorentz � internal) Lie algebra representedon (spinors H). It ontains the urvature 2-form R = d! + !2 and the �eld strength 2-formF = dA+A2, in omponentsR�� = 14Rab�� a b 1H + 14 [�(F��) + J�(F��)J�1℄: (142)41

The �rst term in equation (142) produ es the urvature s alar, whi h we also (!) denote by R,12 �e�1� e�1 �d d� 14Rab�� a b = 14R14: (143)We have also used the possibly dangerous notation � = e�1�a a. Finally D is the ovariantderivative appropriate for the representation of the s alars. The above formula for the squareof the Dira operator is also known as Li hn�erowi z formula. The Li hn�erowi z formula witharbitrary torsion an be found in [36℄.Let f : R+! R+ be a positive, smooth fun tion with �nite moments,f0 = R10 uf(u) du; f2 = R10 f(u) du; f4 = f(0); (144)f6 = �f 0(0); f8 = f 00(0); ::: (145)Asymptoti ally, for large �, the distribution fun tion of the spe trum is given in terms of theheat kernel expansion [37℄:S = tr f(D2t =�2) = 116�2 ZM [�4f0a0 + �2f2a2 + f4a4 + ��2f6a6 + :::℄ dV; (146)where the aj are the oeÆ ients of the heat kernel expansion of the Dira operator squared [30℄,a0 = tr (14 1H); (147)a2 = 16R tr (14 1H)� trE; (148)a4 = 172R2tr (14 1H)� 1180R��R��tr (14 1H) + 1180R����R����tr (14 1H)+ 112tr (R��R��)� 16R trE + 12trE2 + surfa e terms: (149)As already noted, for large � the positive fun tion f is universal, only the �rst three moments,f0; f2 and f4 appear with non-negative powers of �. For the minimax model, we get (moredetails an be found in [38℄):a0 = 4dimH = 4� 6; (150)trE = dimHR+ 16j'j2; (151)a2 = 23 dimHR� dimHR � 16j'j2 = �13 dimHR� 16j'j2; (152)tr �12[ a; b℄12[ ; d℄� = 4 �ÆadÆb � Æa Æbd� ; (153)tr fR��R��g = �12 dimHR����R�����4 tr f[�(F��) + J�(F��)J�1℄�[�(F ��) + J�(F ��)J�1℄g= �12 dimHR����R���� � 8 tr f�(F��)��(F ��)g; (154)trE2 = 14 dimHR2 + 4 tr f�(F��)��(F ��)g+16j'j4 + 16(D�')�(D�') + 8j'j2R; (155)42

Finally we have up to surfa e terms,a4 = 1360 dimH (5R2 � 8Ri i2 � 7Riemann2) + 43tr �(F��)��(F ��)+8j'j4 + 8(D�')�(D�') + 43j'j2R: (156)We arrive at the spe tral a tion with its onventional normalization, equation (137), after a�nite renormalization j'j2! �2f4 j'j2.Our �rst timid ex ursion into gravity on a non ommutative geometry produ ed a ratherunexpe ted dis overy. We stumbled over a Yang-Mills-Higgs model, whi h is pre isely theele tro-weak model for one family of leptons but with the U(1) of hyper harge amputated.The s epti al reader suspe ting a sleight of hand is en ouraged to try and �nd a simpler,non ommutative �nite spe tral triple.5.4 A entral extensionWe will see in the next se tion the te hni al reason for the absen e of U(1)s as automorphisms:all automorphisms of �nite spe tral triples onne ted to the identity are inner, i.e. onjugationby unitaries. But onjugation by entral unitaries is trivial. This explains that in the minimaxexample,A = H �C , the omponent of the automorphism group onne ted to the identity wasSU(2)=Z2 3 (�u; 1). It is the domain of de�nition of the lift, equation (126),L(�u; 1) = �(�u; 1)J�(�u; 1)J�1 = 0BB��u 0 0 00 1 0 00 0 ��u 00 0 0 11CCA : (157)It is tempting to entrally extend the lift to all unitaries of the algebra:L(w; v) = �(w; v)J�(w; v)J�1 =0BB� �vw 0 0 00 �v2 0 00 0 v �w 00 0 0 v21CCA ; (w; v) 2 SU(2) � U(1): (158)An immediate onsequen e of this extension is en ouraging: the extended lift is single-valuedand after tensorization with the one from Riemannian geometry, the multi-valuedness willremain two.Then redoing the u tuation of the Dira operator and re omputing the spe tral a tionyields gravity oupled to the omplete ele tro-weak model of the ele tron and its neutrino witha weak mixing angle of sin2 �w = 1=4. 43

6 Connes' do-it-yourself kitOur �rst example of gravity on an almost ommutative spa e leaves us wondering what otherexamples will look like. To play on the Yang-Mills-Higgs ma hine, one must know the lassi-� ation of all real, ompa t Lie groups and their unitary representations. To play on the newma hine, we must know all �nite spe tral triples. The �rst good news is that the list of algebrasand their representations is in�nitely shorter than the one for groups. The other good news isthat the rules of Connes' ma hine are not made up opportunisti ally to suit the phenomenol-ogy of ele tro-weak and strong for es as in the ase of the Yang-Mills-Higgs ma hine. On the ontrary, as developed in the last se tion, these rules derive naturally from geometry.6.1 InputOur �rst input item is a �nite dimensional, real, asso iative involution algebra with unit andthat admits a �nite dimensional faithful representation. Any su h algebra is a dire t sum ofsimple algebras with the same properties. Every su h simple algebra is an algebra of n � nmatri es with real, omplex or quaternioni entries, A = Mn(R), Mn(C ) or Mn(H ). Theirunitary groups U(A) := fu 2 A; uu� = u�u = 1g are O(n), U(n) and USp(n). Note thatUSp(1) = SU(2). The entre Z of an algebra A is the set of elements z 2 A that om-mute with all elements a 2 A. The entral unitaries form an abelian subgroup of U(A).Let us denote this subgroup by U (A) := U(A) \ Z. We have U (Mn(R)) = Z2 3 �1n,U (Mn(C )) = U(1) 3 exp(i�)1n, � 2 [0; 2�), U (Mn(H )) = Z2 3 �12n. All automorphisms ofthe real, omplex and quaternioni matrix algebras are inner with one ex eption, Mn(C ) hasone outer automorphism, omplex onjugation, whi h is dis onne ted from the identity auto-morphism. An inner automorphism � is of the form �(a) = uau�1 for some u 2 U(A) and for alla 2 A. We will denote this inner automorphism by � = iu and we will write Int(A) for the groupof inner automorphisms. Of ourse a ommutative algebra, e.g. A = C , has no inner auto-morphism. We have Int(A) = U(A)=U (A), in parti ular Int(Mn(R)) = O(n)=Z2; n = 2; 3; :::;Int(Mn(C )) = U(n)=U(1) = SU(n)=Zn; n = 2; 3; :::; Int(Mn(H )) = USp(n)=Z2; n = 1; 2; ::.Note the apparent injusti e: the ommutative algebra C1(M) has the nonAbelian automor-phism group Di�(M) while the non ommutative algebraM2(R) has the Abelian automorphismgroup O(2)=Z2. All ex eptional groups are missing from our list of groups. Indeed they areautomorphism groups of non-asso iative algebras, e.g. G2 is the automorphism group of theo tonions.The se ond input item is a faithful representation � of the algebra A on a �nite dimensional, omplex Hilbert spa eH. Any su h representation is a dire t sum of irredu ible representations.44

Mn(R) has only one irredu ible representation, the fundamental one on Rn,Mn(C ) has two, thefundamental one and its omplex onjugate. Both are de�ned on H = C n 3 by �(a) = a and by �(a) = �a . Mn(H ) has only one irredu ible representation, the fundamental onede�ned on C 2n . For example, while U(1) has an in�nite number of inequivalent irredu iblerepresentations, hara terized by an integer ` harge', its algebra C has only two with harge plusand minus one. While SU(2) has an in�nite number of inequivalent irredu ible representations hara terized by its spin, 0; 12 ; 1; :::, its algebra H has only one, spin 12 . The main reasonbehind this multitude of group representation is that the tensor produ t of two representationsof one group is another representation of this group, hara terized by the sum of harges forU(1) and by the sum of spins for SU(2). The same is not true for two representations of oneasso iative algebra whose tensor produ t fails to be linear. (Attention, the tensor produ t oftwo representations of two algebras does de�ne a representation of the tensor produ t of thetwo algebras. We have used this tensor produ t of Hilbert spa es to de�ne almost ommutativegeometries.)The third input item is the �nite Dira operator D or equivalently the fermioni massmatrix, a matrix of size dimHL�dimHR.These three items an however not be hosen freely, they must still satisfy all axioms ofthe spe tral triple [39℄. I do hope you have onvin ed yourself of the nontriviality of thisrequirement for the ase of the minimax example.The minimax example has taught us something else. If we want abelian gauge �elds fromthe u tuating metri , we must entrally extend the spin lift, an operation, that at the sametime may redu e the multivaluedness of the original lift. Central extensions are by no meansunique, its hoi e is our last input item [40℄.To simplify notations, we on entrate on omplex matrix algebras Mn(C ) in the followingpart. Indeed the others, Mn(R) and Mn(H ), do not have entral unitaries lose to the identity.We have already seen that it is important to separate the ommutative and non ommutativeparts of the algebra:A = CM � NMk=1 Mnk (C ) 3 a = (b1; :::bM; 1; :::; N); nk � 2: (159)Its group of unitaries isU(A) = U(1)M � N�k = 1U(nk) 3 u = (v1; :::; vM; w1; :::; wN) (160)and its group of entral unitariesU (A) = U(1)M+N 3 u = (v 1; :::; v M; w 11n1 ; :::; w N1nN ): (161)45

All automorphisms onne ted to the identity are inner, there are outer automorphisms, the omplex onjugation and, if there are identi al summands in A, their permutations. In om-plian e with the minimax prin iple, we disregard the dis rete automorphisms. Multiplying aunitary u with a entral unitary u of ourse does not a�e t its inner automorphism iu u = iu.This ambiguity distinguishes between `harmless' entral unitaries v 1; :::; v M and the others,w 1; :::; w N, in the sense that Int(A) = Un(A)=Un (A); (162)where we have de�ned the group of non ommutative unitariesUn(A) := N�k = 1U(nk) 3 w (163)and Un (A) := Un(A) \ U (A) 3 w . The mapi : Un(A) �! Int(A)w 7�! iw (164)has kernel Ker i = Un (A).The lift of an inner automorphism to the Hilbert spa e has a simple losed form [19℄,L = L Æ i�1 with L(w) = �(1; w)J�(1; w)J�1: (165)It satis�es p(L(w)) = i(w). If the kernel of i is ontained in the kernel of L, then the lift is wellde�ned, as e.g. for A = H , Un (H ) =Z2.AutH(A)p ? CCCCO 6L AAAAK L HHHHHHHY ` (166)Int(A) i � Un(A) --det Un (A)For more ompli ated real or quaternioni algebras, Un (A) is �nite and the lift L is multi-valued with a �nite number of values. For non ommutative, omplex algebras, their ontinuousfamily of entral unitaries annot be eliminated ex ept for very spe ial representations and wefa e a ontinuous in�nity of values. The solution of this problem follows an old strategy: `Ifyou an't beat them, adjoin them'. Who is them? The harmful entral unitaries w 2 Un (A)46

and adjoining means entral extending. The entral extension (158), only on erned a dis retegroup and a harmless U(1). Nevertheless it generalizes naturally to the present setting:L : Int(A)� Un (A) �! AutH(A)(w�; w ) 7�! (L Æ i�1)(w�) `(w ) (167)with (w ) := � NYj1=1(w j1)q1;j1 ; :::; NYjM=1(w jM )qM;jM ; (168)NYjM+1=1(w jM+1)qM+1;jM+11n1 ; :::; NYjM+N=1(w jM+N )qM+N;jM+N 1nN1AJ�(:::)J�1with the (M+N)�N matrix of harges qkj . The extension satis�es indeed p(`(w )) = 1 2 Int(A)for all w 2 Un (A).Having adjoined the harmful, ontinuous entral unitaries, we may now stream line ournotations and write the group of inner automorphisms asInt(A) = 0� N�k = 1SU(nk)1A =� 3 [w�℄ = [(w�1; :::; w�N)℄ mod ; (169)where � is the dis rete group� = N�k = 1Znk 3 (z11n1 ; :::; zN1nN ); zk = exp[�mk2�i=nk℄; mk = 0; :::; nk � 1 (170)and the quotient is fa tor by fa tor. This way to write inner automorphisms is onvenient for omplex matri es, but not available for real and quaternioni matri es. Equation (162) remainsthe general hara terization of inner automorphisms.The lift L(w�) = (LÆi�1)(w�), w� = w mod Un (A), is multi-valued with, depending on therepresentation, up to j�j =QNj=1 nj values. More pre isely the multi-valuedness of L is indexedby the elements of the kernel of the proje tion p restri ted to the image L(Int(A)). Dependingon the hoi e of the harge matrix q, the entral extension ` may redu e this multi-valuedness.Extending harmless entral unitaries is useless for any redu tion. With the multi-valued grouphomomorphism(h�; h ) : Un(A) �! Int(A)� Un (A)(wj) 7�! ((w�j ; w j)) = ((wj(detwj)�1=nj ; (detwj)1=nj)); (171)47

we an write the two lifts L and ` together in losed form L : Un(A)! AutH(A):L(w) = L(h�(w)) `(h (w))= � NYj1=1(detwj1)~q1;j1 ; :::; NYjM=1(detwjM )~qM;jM ;w1 NYjM+1=1(detwjM+1)~qM+1;jM+1 ; :::; wN NYjN+M=1(detwjN+M )~qN+M;jN+M1A�J�(:::)J�1: (172)We have set ~q :=0�q �0� 0M�N1N�N 1A1A0�n1 . . . nN 1A�1 : (173)Due to the phase ambiguities in the roots of the determinants, the extended lift L is multi-valued in general. It is single-valued if the matrix ~q has integer entries, e.g. q = � 01N �, then~q = 0 and L(w) = L(w). On the other hand, q = 0 gives L(w) = L(i�1(h�(w))), not alwayswell de�ned as already noted. Unlike the extension (158), and unlike the map i, the extendedlift L is not ne essarily even. We do impose this symmetry L(�u) = L(u), whi h translatesinto onditions on the harges, onditions that depend on the details of the representation �.Let us note that the lift L is simply a representation up to a phase and as su h it is notthe most general lift. We ould have added harmless entral unitaries if any present, and, ifthe representation � is redu ible, we ould have hosen di�erent harge matri es in di�erentirredu ible omponents. If you are not happy with entral extensions, then this is a sign of goodtaste. Indeed ommutative algebras like the alibrating example have no inner automorphismsand a huge entre. Truly non ommutative algebras have few outer automorphism and a small entre. We believe that almost ommutative geometries with their entral extensions are onlylow energy approximations of a truly non ommutative geometry where entral extensions arenot an issue.6.2 OutputFrom the input data of a �nite spe tral triple, the entral harges and the three moments ofthe spe tral fun tion, non ommutative geometry produ es a Yang-Mills-Higgs model oupledto gravity. Its entire Higgs se tor is omputed from the input data, Figure 6. The Higgs48

H

A j

l, m

NCG

l, m

f

YMH

gravity,HS

,

f

ff , q

D~

Figure 6: Connes' slot ma hinerepresentation derives from the u tuating metri and the Higgs potential from the spe trala tion.To see how the Higgs representation derives in general from the u tuating Dira operator D,we must write it as ` at' Dira operator ~D plus internal 1-form H like we have done in equation(128) for the minimax example without extension. Take the extended lift L(w) = �(w)J�(w)J�1with the unitaryw = NYj1=1(detwj1)~q1j1 ; :::; NYjM=1(detwjM )~qMjM ;w1 NYjM+1=1(detwjM+1)~qM+1;jM+1 ; :::; wN NYjN+M=1(detwjN+M )~qN+M;jN+M : (174)ThenD = L ~DL�1= ��(w)J�(w)J�1� ~D ��(w)J�(w)J�1��1 = �(w)J�(w)J�1 ~D �(w�1)J�(w�1)J�1= �(w)J�(w)J�1(�(w�1) ~D + [ ~D; �(w�1)℄)J�(w�1)J�1= J�(w)J�1 ~DJ�(w�1)J�1 + �(w)[ ~D; �(w�1)℄ = J�(w) ~D�(w�1)J�1 + �(w)[ ~D; �(w�1)℄= J(�(w)[ ~D; �(w�1)℄ + ~D)J�1 + �(w)[ ~D; �(w�1)℄= ~D + H + JHJ�1; (175)with the internal 1-form, the Higgs s alar, H = �(w)[ ~D; �(w�1)℄. In the hain (175) we haveused su essively the following three axioms of spe tral triples, [�(a); J�(~a)J�1℄ = 0, the �rstorder ondition [[ ~D; �(a)℄; J�(~a)J�1℄ = 0 and [ ~D; J ℄ = 0. Note that the unitaries, whose repre-sentation ommutes with the internal Dira operator, drop out from the Higgs, it transformsas a singlet under their subgroup. 49

Yang-Mills-Higgs

left-right symm.

GUT

supersymm.

NCG

standard modelFigure 7: Pseudo for es from non ommutative geometryThe onstraints from the axioms of non ommutative geometry are so tight that only veryfew Yang-Mills-Higgs models an be derived from non ommutative geometry as pseudo for es.No left-right symmetri model an [41℄, no Grand Uni�ed Theory an [42℄, for instan e theSU(5) model needs 10-dimensional fermion representations, SO(10) 16-dimensional ones, E6 isnot the group of an asso iative algebra. Moreover the last two models are left-right symmetri .Mu h e�ort has gone into the onstru tion of a supersymmetri model from non ommutativegeometry, in vain [43℄. The standard model on the other hand �ts perfe tly into Connes'pi ture, Figure 7.6.3 The standard modelThe �rst non ommutative formulation of the standard model was published by Connes & Lott[33℄ in 1990. Sin e then it has evolved into its present form [18, 19, 20, 28℄ and triggered quitean amount of literature [44℄.6.3.1 Spe tral tripleThe internal algebra A is hosen as to reprodu e SU(2)� U(1)� SU(3) as subgroup of U(A),A = H � C �M3(C ) 3 (a; b; ): (176)The internal Hilbert spa e is opied from the Parti le Physi s Booklet [13℄,HL = �C 2 C N C 3� � �C 2 C N C � ; (177)HR = �C C N C 3� � �C C N C 3� � �C C N C � : (178)50

In ea h summand, the �rst fa tor denotes weak isospin doublets or singlets, the se ond denotesN generations, N = 3, and the third denotes olour triplets or singlets. Let us hoose thefollowing basis of the internal Hilbert spa e, ounting fermions and antifermions (indi ated bythe supers ript � for ` harge onjugated') independently, H = HL �HR �H L �H R = C 90:�ud�L ; � s�L ; � tb�L ; � �ee �L ; � ��� �L ; � ��� �L ;uR;dR; R;sR; tR;bR; eR; �R; �R;�ud� L ; � s� L ; � tb� L ; � �ee � L ; � ��� � L ; � ��� � L ;u R;d R; R;s R; t R;b R; e R; � R; � R:This is the urrent eigenstate basis, the representation � a ting on H by�(a; b; ) := 0BB� �L 0 0 00 �R 0 00 0 �� L 00 0 0 �� R1CCA (179)with �L(a) := � a 1N 13 00 a 1N � ; �R(b) := 0� b1N 13 0 00 �b1N 13 00 0 �b1N 1A ; (180)� L(b; ) := � 12 1N 00 �b12 1N � ; � R(b; ) :=0� 1N 0 00 1N 00 0 �b1N 1A : (181)The apparent asymmetry between parti les and antiparti les { the former are subje t to weak,the latter to strong intera tions { will disappear after appli ation of the lift L withJ = � 0 115N115N 0 � Æ omplex onjugation: (182)For the sake of ompleteness, we re ord the hirality as matrix� =0BB��18N 0 0 00 17N 0 00 0 �18N 00 0 0 17N 1CCA : (183)51

The internal Dira operator ~D = 0BB� 0 M 0 0M� 0 0 00 0 0 �M0 0 �M� 0 1CCA (184)is made of the fermioni mass matrix of the standard model,M = 0BB�� 1 00 0�Mu 13 + � 0 00 1�Md 13 00 � 01�Me1CCA ; (185)with Mu := 0�mu 0 00 m 00 0 mt1A ; Md := CKM 0�md 0 00 ms 00 0 mb1A ; (186)Me :=0�me 0 00 m� 00 0 m�1A : (187)From the booklet we know that all indi ated fermion masses are di�erent from ea h otherand that the Cabibbo-Kobayashi-Maskawa matrix CKM is non-degenerate in the sense that noquark is simultaneously mass and weak urrent eigenstate.We must a knowledge the fa t { and this is far from trivial { that the �nite spe tral tripleof the standard model satis�es all of Connes' axioms:� It is orientable, � = �(�12; 1; 13)J�(�12; 1; 13)J�1:� Poin ar�e duality holds. The standard model has three minimal proje tors,p1 = (12; 0; 0); p2 = (0; 1; 0); p3 =0�0; 0;0� 1 0 00 0 00 0 01A1A (188)and the interse tion form \ = �2N 0� 0 1 11 �1 �11 �1 0 1A ; (189)is non-degenerate. We note that Majorana masses are forbidden be ause of the axiom ~D� =�� ~D: On the other hand if we wanted to give Dira masses to all three neutrinos we wouldhave to add three right-handed neutrinos to the standard model. Then the interse tion form,\ = �2N 0� 0 1 11 �2 �11 �1 0 1A ; (190)52

would be ome degenerate and Poin ar�e duality would fail.� The �rst order axiom is satis�ed pre isely be ause of the �rst two of the six ad ho propertiesof the standard model re alled in subse tion 3.3, olour ouples ve torially and ommutes withthe fermioni mass matrix, [D; �(12; 1; )℄ = 0. As an immediate onsequen e the Higgs s alar= internal 1-form will be a olour singlet and the gluons will remain massless, the third ad ho property of the standard model in its onventional formulation.� There seems to be some arbitrariness in the hoi e of the representation under C 3 b. In fa tthis is not true, any hoi e di�erent from the one in equations (180,181) is either in ompatiblewith the axioms of spe tral triples or it leads to harged massless parti les in ompatible withthe Lorentz for e or to a symmetry breaking with equal top and bottom masses. Therefore,the only exibility in the fermioni harges is from the hoi e of the entral harges [40℄.6.3.2 Central hargesThe standard model has the following groups,U(A) = SU(2)� U(1) � U(3) 3 u = (u0; v; w); (191)U (A) = Z2� U(1) � U(1) 3 u = (u 0; v ; w 13); (192)Un(A) = SU(2) � U(3) 3 (u0; w); (193)Un (A) = Z2 � U(1) 3 (u 0; w 13); (194)Int(A) = [SU(2) � SU(3)℄=� 3 u� = (u�0; w�); (195)� = Z2 � Z3 3 = (exp[�m02�i=2℄; exp[�m22�i=3℄); (196)with m0 = 0; 1 and m2 = 0; 1; 2. Let us ompute the re epta le of the lifted automorphisms,AutH(A) = [U(2)L � U(3) �U(N)qL � U(N)`L � U(N)uR � U(N)dR℄=[U(1)� U(1)℄�U(N)eR: (197)The subs ripts indi ate on whi h multiplet the U(N)s a t. The kernel of the proje tion downto the automorphism group Aut(A) iskerp = [U(1)� U(1) �U(N)qL � U(N)`L � U(N)uR � U(N)dR℄=[U(1)� U(1)℄�U(N)eR; (198)and its restri tions to the images of the lifts arekerp \ L(Int(A)) =Z2�Z3; ker p \ L(Un(A)) =Z2� U(1): (199)The kernel of i is Z2� U(1) in sharp ontrast to the kernel of L, whi h is trivial. The isospinSU(2)L and the olour SU(3) are the image of the lift L. If q 6= 0; the image of ` onsists of53

one U(1) 3 w = exp[i�℄ ontained in the �ve avour U(N)s. Its embedding depends on q:L(12; 1; w 13) = `(w ) = (200)diag ( uqL12 1N 13; u`L12 1N ; uuR1N 13; udR1N 13; ueR1N ;�uqL12 1N 13; �u`L12 1N ; �uuR1N 13; �udR1N 13; �ueR1N )with uj = exp[iyj�℄ andyqL = q2; y`L = �q1; yuR = q1 + q2; ydR = �q1 + q2; yeR = �2q1: (201)Independently of the embedding, we have indeed derived the three fermioni onditions ofthe hyper harge �ne tuning (57). In other words, in non ommutative geometry the masslessele troweak gauge boson ne essarily ouples ve torially.Our goal is now to �nd the minimal extension ` that renders the extended lift symmetri ,L(�u0;�w) = L(u0; w), and that renders L(12; w) single-valued. The �rst requirement meansf ~q1 = 1 and ~q2 = 0 g modulo 2, with� ~q1~q2� = 13 �� q1q2��� 01�� : (202)The se ond requirement means that ~q has integer oeÆ ients.The �rst extension whi h omes to mind has q = 0, ~q = � 0�1=3�. With respe t to theinterpretation (169) of the inner automorphisms, one might obje t that this is not an extensionat all. With respe t to the generi hara terization (162), it ertainly is a non-trivial extension.Anyhow it fails both tests. The most general extension that passes both tests has the form~q = � 2z1 + 12z2 � ; q = � 6z1 + 36z2 + 1� ; z1; z2 2Z: (203)Consequently, y`L = �q1 annot vanish, the neutrino omes out ele tri ally neutral in ompli-an e with the Lorentz for e. As ommon pra tise, we normalize the hyper harges to y`L = �1=2and ompute the last remaining hyper harge yqL,yqL = q22q1 = 16 + z21 + 2z1 : (204)We an hange the sign of yqL by permuting u with d and d with u . Therefore it is suÆ ient totake z1 = 0; 1; 2; ::: The minimal su h extension, z1 = z2 = 0, re overs nature's hoi e yqL = 16.Its lift, L(u0; w) = �(u0;detw;w)J�(u0;detw;w)J�1; (205)54

is the anomaly free fermioni representation of the standard model onsidered as SU(2)�U(3)Yang-Mills-Higgs model. The double-valuedness of L omes from the dis rete group Z2 of entral quaternioni unitaries (�12; 13) 2 Z2 � � � Un (A). On the other hand,O'Raifeartaigh's [5℄ Z2 in the group of the standard model (45), �(12; 13) 2 Z2 � Un (A),is not a subgroup of �. It re e ts the symmetry of L.6.3.3 Flu tuating metri The stage is set now for u tuating the metri by means of the extended lift. This algorithmanswers en passant a long standing question in Yang-Mills theories: To gauge or not to gauge?Given a fermioni Lagrangian, e.g. the one of the standard model, our �rst re ex is to omputeits symmetry group. In non ommutative geometry, this group is simply the internal re epta le(197). The painful question in Yang-Mills theory is what subgroup of this symmetry groupshould be gauged? For us, this question is answered by the hoi es of the spe tral triple and ofthe spin lift. Indeed the image of the extended lift is the gauge group. The u tuating metri promotes its generators to gauge bosons, the W�, the Z, the photon and the gluons. At thesame time, the Higgs representation is derived, equation (175):H = �(u0;detw;w)[ ~D; �(u0;detw;w)�1℄ =0BB� 0 H 0 0H� 0 0 00 0 0 00 0 0 01CCA (206)with H =0BB��h1Mu ��h2Mdh2Mu �h1Md � 13 00 ���h2Me�h1Me �1CCA (207)and �h1 ��h2h2 �h1 � = �u0�detw 00 det �w�� 12: (208)The Higgs is hara terized by one omplex doublet, (h1; h2)T . Again it will be onvenient topass to the homogeneous Higgs variable,D = L ~DL�1 = ~D +H + JHJ�1= �+ J�J�1 =0BB� 0 � 0 0�� 0 0 00 0 0 ��0 0 ��� 0 1CCA (209)55

with � =0BB��'1Mu � �'2Md'2Mu �'1Md � 13 00 �� �'2Me�'1Me �1CCA = �L(�)M (210)and � = �'1 � �'2'2 �'1 � = �u0�detw 00 det �w� : (211)In order to satisfy the �rst order ondition, the representation ofM3(C ) 3 had to ommutewith the Dira operator. Therefore the Higgs is a olour singlet and the gluons will remainmassless. The �rst two of the six intriguing properties of the standard model listed in subse tion3.3 have a geometri raison d'etre, the �rst order ondition. In turn, they imply the thirdproperty: we have just shown that the Higgs ' = ('1; '2)T is a olour singlet. At the sametime the �fth property follows from the fourth: the Higgs of the standard model is an isospindoublet be ause of the parity violating ouplings of the quaternions H . Furthermore, this Higgshas hyper harge y' = �12 and the last �ne tuning of the sixth property (57) also derives fromConnes' algorithm: the Higgs has a omponent with vanishing ele tri harge, the physi alHiggs, and the photon will remain massless.In on lusion, in Connes version of the standard model there is only one intriguing inputproperty, the fourth: expli it parity violation in the algebra representation HL �HR, the �veothers are mathemati al onsequen es.6.3.4 Spe tral a tionComputing the spe tral a tion SCC = f(D2t =�2) in the standard model is not more diÆ ultthan in the minimax example, only the matri es are a little bigger,Dt = Lt ~DtL�1t = 0BBB� �=L 5� 0 0 5�� �=R 0 00 0 C �=LC�1 5 ��0 0 5 ��� C �=RC�11CCCA : (212)The tra e of the powers of � are omputed from the identities � = �L(�)M and ��� = ��� =(j'1j2 + j'2j2)12 = j'j212 by using that �L as a representation respe ts multipli ation andinvolution.The spe tral a tion produ es the omplete a tion of the standard model oupled to gravitywith the following relations for oupling onstants:g23 = g22 = 9N �: (213)56

Our hoi e of entral harges, ~q = (1; 0)T , entails a further relation, g21 = 35g22, i.e. sin2 �w = 3=8.However only produ ts of the Abelian gauge oupling g1 and the hyper harges yj appear in theLagrangian. By res aling the entral harges, we an res ale the hyper harges and onsequentlythe Abelian oupling g1. It seems quite moral that non ommutative geometry has nothing tosay about Abelian gauge ouplings.0.2

0.4

0.6

0.8

1

1.2

1.4 g

LmZ

2

3

910 GeV

E

g

(3l)1/2Figure 8: Running oupling onstantsExperiment tells us that the weak and strong ouplings are unequal, equation (49) at energies orresponding to the Z mass, g2 = 0:6518� 0:0003; g3 = 1:218� 0:01: Experiment also tells usthat the oupling onstants are not onstant, but that they evolve with energy. This evolution an be understood theoreti ally in terms of renormalization: one an get rid of short distan edivergen ies in perturbative quantum �eld theory by allowing energy depending gauge, Higgs,and Yukawa ouplings where the theoreti al evolution depends on the parti le ontent of themodel. In the standard model, g2 and g3 ome together with in reasing energy, see Figure8. They would be ome equal at astronomi al energies, � = 1017 GeV, if one believed thatbetween presently explored energies, 102 GeV, and the `uni� ation s ale' �, no new parti lesexist. This hypothesis has be ome popular under the name `big desert' sin e Grand Uni�edTheories. It was believed that new gauge bosons, `lepto-quarks' with masses of order � existed.The lepto-quarks together with the W�, the Z, the photon and the gluons generate the simplegroup SU(5), with only one gauge oupling, g25 := g23 = g22 = 53g21 at �. In the minimal SU(5)model, these lepto-quarks would mediate proton de ay with a half life that today is ex ludedexperimentally.If we believe in the big desert, we an imagine that { while almost ommutative at presentenergies { our geometry be omes truly non ommutative at time s ales of ~=� � 10�41 s. Sin ein su h a geometry smaller time intervals annot be resolved, we expe t the oupling onstantsto be ome energy independent at the orresponding energy s ale �. We remark that the57

�rst motivation for non ommutative geometry in spa etime goes ba k to Heisenberg and waspre isely the regularization of short distan e divergen ies in quantum �eld theory, see e.g. [45℄.The big desert is an opportunisti hypothesis and remains so in the ontext of non ommutativegeometry. But in this ontext, it has at least the merit of being onsistent with three otherphysi al ideas:Plan k time: There is an old hand waving argument ombiningHeisenberg's un ertaintyrelation of phase spa e with the S hwarzs hild horizon to �nd an un ertainty relationin spa etime with a s ale � smaller than the Plan k energy (~ 5=G)1=2 � 1019 GeV:To measure a position with a pre ision �x we need, following Heisenberg, at least amomentum ~=�x or, by spe ial relativity, an energy ~ =�x. A ording to generalrelativity, su h an energy reates an horizon of size G~ �3=�x. If this horizonex eeds �x all information on the position is lost. We an only resolve positionswith �x larger than the Plan k length, �x > (~G= 3)1=2 � 10�35 m. Or we anonly resolve time with �t larger than the Plan k time, �t > (~G= 5)1=2 � 10�43 s.This is ompatible with the above time un ertainty of ~=� � 10�41 s.Stability: We want the Higgs self oupling � to remain positive [46℄ during its perturba-tive evolution for all energies up to �. A negative Higgs self oupling would meanthat no ground state exists, the Higgs potential is unstable. This requirement ismet for the self oupling given by the onstraint (213) at energy �, see Figure 8.Triviality: We want the Higgs self oupling � to remain perturbatively small [46℄ duringits evolution for all energies up to � be ause its evolution is omputed from aperturbative expansion. This requirement as well is met for the self oupling givenby the onstraint (213), see Figure 8. If the top mass was larger than 231 GeV orif there were N = 8 or more generations this riterion would fail.Sin e the big desert gives a minimal and onsistent pi ture we are urious to know its numeri alimpli ation. If we a ept the onstraint (213) with g2 = 0:5170 at the energy � = 0:968 1017GeV and evolve it down to lower energies using the perturbative renormalization ow of thestandard model, see Figure 8, we retrieve the experimental nonAbelian gauge ouplings g2 andg3 at the Z mass by onstru tion of �. For the Higgs oupling, we obtain� = 0:06050 � 0:0037 at E = mZ: (214)The indi ated error omes from the experimental error in the top mass, mt = 174:3� 5:1 GeV,whi h a�e ts the evolution of the Higgs oupling. From the Higgs oupling at low energies we58

ompute the Higgs mass,mH = 4p2 p�g2 mW = 171:6 � 5 GeV: (215)For details of this al ulation see [47℄.6.4 Beyond the standard modelA so ial reason, that made the Yang-Mills-Higgs ma hine popular, is that it is an inexhaustiblesour e of employment. Even after the standard model, physi ists ontinue to play on thema hine and try out extensions of the standard model by adding new parti les, `let the desertbloom'. These parti les an be gauge bosons oupling only to right-handed fermions in orderto restore left-right symmetry. The added parti les an be lepto-quarks for grand uni� ation orsupersymmetri parti les. These models are arefully tuned not to upset the phenomenologi alsu ess of the standard model. This means in pra ti e to hoose Higgs representations andpotentials that give masses to the added parti les, large enough to make them undete tablein present day experiments, but not too large so that experimentalists an propose biggerma hines to test these models. Independently there are always short lived deviations from thestandard model predi tions in new experiments. They never miss to trigger new, short livedmodels with new parti les to �t the `anomalies'. For instan e, the literature ontains hundredsof superstring inspired Yang-Mills-Higgs models, ea h of them with hundreds of parameters, oins, waiting for the standard model to fail.Of ourse, we are trying the same game in Connes' do-it-yourself kit. So far, we have notbeen able to �nd one single onsistent extension of the standard model [41, 42, 43, 48℄. Thereason is lear, we have no handle on the Higgs representation and potential, whi h are onthe output side, and, in general, we meet two problems: light physi al s alars and degeneratefermion masses in irredu ible multiplets. The extended standard model with arbitrary numbersof quark generations, Nq � 0, of lepton generations, N` � 1, and of olours N , somehowmanages to avoid both problems and we are trying to prove that it is unique as su h. Theminimax model has Nq = 0; N` = 1; N = 0. The standard model has Nq = N` =: N andN = 3 to avoid Yang-Mills anomalies [12℄. It also has N = 3 generations. So far, the onlyrealisti extension of the standard model that we know of in non ommutative geometry, is theaddition of right-handed neutrinos and of Dira masses in one or two generations. These mightbe ne essary to a ount for observed neutrino os illations [13℄.59

7 Outlook and on lusionNon ommutative geometry re on iles Riemannian geometry and un ertainty and we expe t itto re on ile general relativity with quantum �eld theory. We also expe t it to improve ourstill in omplete understanding of quantum �eld theory. On the perturbative level su h animprovement is happening right now: Connes, Mos ovi i, and Kreimer dis overed a subtle linkbetween a non ommutative generalization of the index theorem and perturbative quantum �eldtheory. This link is a Hopf algebra relevant to both theories [49℄.In general, Hopf algebras play the same role in non ommutative geometry as Lie groupsplay in Riemannian geometry and we expe t new examples of non ommutative geometry fromits merging with the theory of Hopf algebras. Referen e [50℄ ontains a simple example wherequantum group te hniques an be applied to non ommutative parti le models.The running of oupling onstants from perturbative quantum �eld theory must be takeninto a ount in order to perform the high pre ision test of the standard model at present dayenergies. We have invoked an extrapolation of this running to astronomi al energies to makethe onstraint g2 = g3 from the spe tral a tion ompatible with experiment. This extrapolationis still based on quantum loops in at Minkowski spa e. While a eptable at energies below thes ale � where gravity and the non ommutativity of spa e seem negligible, this approximationis unsatisfa tory from a on eptual point of view and one would like to see quantum �elds onstru ted on a non ommutative spa e. At the end of the nineties �rst examples of quan-tum �elds on the ( at) non ommutative torus or its non- ompa t version, the Moyal plane,were published [51℄. These examples ame straight from the spe tral a tion. The non om-mutative torus is motivated from quantum me hani al phase spa e and was the �rst exampleof a non ommutative spe tral triple [52℄. Bellissard [53℄ has shown that the non ommutativetorus is relevant in solid state physi s: one an understand the quantum Hall e�e t by takingthe Brillouin zone to be non ommutative. Only re ently other examples of non ommutativespa es like non ommutative spheres where un overed [54℄. Sin e 1999, quantum �elds on thenon ommutative torus are being studied extensively in luding the �elds of the standard model[56℄. So far, its internal part is not treated as a non ommutative geometry and Higgs bosonsand potentials are added opportunisti ally. This problem is avoided naturally by onsideringthe tensor produ t of the non ommutative torus with a �nite spe tral triple, but I am surethat the axioms of non ommutative geometry an be redis overed by playing long enough withmodel building.In quantum me hani s and in general relativity, time and spa e play radi ally di�erentroles. Spa ial position is an observable in quantum me hani s, time is not. In general relativity,spa ial position loses all meaning and only proper time an be measured. Distan es are then60

measured by a parti ular observer as (his proper) time of ight of photons going ba k and forthmultiplied by the speed of light, whi h is supposed to be universal. This de�nition of distan esis operational thanks to the high pre ision of present day atomi lo ks, for example in theGPS. The `Riemannian' de�nition of the meter, the forty millionth part of a omplete geodesi on earth, had to be abandoned in favour of a quantum me hani al de�nition of the se ondvia the spe trum of an atom. Connes' de�nition of geometry via the spe trum of the Dira operator is the pre ise ounter part of today's experimental situation. Note that the meter sti kis an extended (rigid ?) obje t. On the other hand an atomi lo k is a pointlike obje t andexperiment tells us that the atom is sensitive to the potentials at the lo ation of the lo k, thepotentials of all for es, gravitational, ele tro-magneti , ... The spe ial role of time remains tobe understood in non ommutative geometry [55℄ as well as the notion of spe tral triples withLorentzian signature and their 1+3 split [57℄.Let us ome ba k to our initial laim: Connes derives the standard model of ele tro-magneti , weak and strong for es from non ommutative geometry and, at the same time, uni�esthem with gravity. If we say that the Balmer-Rydberg formula is derived from quantum me- hani s, then this laim has three levels:Explain the nature of the variables: The hoi e of the dis rete variables nj, ontains al-ready a { at the time revolutionary { pie e of physi s, energy quantization. Where does it omefrom?Explain the ansatz: Why should one take the power law (11)?Explain the experimental �t: The ansatz omes with dis rete parameters, the `bills' qj, and ontinuous parameters, the ` oins' gj , whi h are determined by an experimental �t. Where dothe �tted values, `the winner', ome from?How about deriving gravity from Riemannian geometry? Riemannian geometry has onlyone possible variable, the metri g. The minimax prin iple di tates the Lagrangian ansatz:S[g℄ = ZM [� � 116�GRq℄ dV: (216)Experiment rules on the parameters: q = 1, G = 6:670 �10�11 m3s�2kg, Newton's onstant, and� � 0. Riemannian geometry remains silent on the third level. Nevertheless, there is generalagreement, gravity derives from Riemannian geometry.Non ommutative geometry has only one possible variable, the Dira operator, whi h inthe ommutative ase oin ides with the metri . Its u tuations explain the variables of theadditional for es, gauge and Higgs bosons. The minimax prin iple di tates the Lagrangianansatz: the spe tral a tion. It reprodu es the Einstein-Hilbert a tion and the ansatz of Yang,Mills and Higgs, see Table 3. On the third level, non ommutative geometry is not silent, it61

produ es lots of onstraints, all ompatible with the experimental �t. And their exploration isnot �nished yet.Riemanniangeometry -Einstein gravity� �?non ommutativegeometry -Connes gravity + Yang-Mills-HiggsConnesTable 3: Deriving some YMH for es from gravityI hope to have onvin ed one or the other reader that non ommutative geometry ontainselegant solutions of long standing problems in fundamental physi s and that it proposes on retestrategies to ta kle the remaining ones. I would like to on lude our outlook with a senten eby Plan k who tells us how important the opinion of our young, unbiased olleagues is. Plan ksaid, a new theory is a epted, not be ause the others are onvin ed, be ause they die.It is a pleasure to thank Eike Bi k and Frank Ste�en for the organization of a splendid S hool. Ithank the parti ipants for their unbiased riti ism and Kurus h Ebrahimi-Fard, Volker S hatz,and Frank Ste�en for a areful reading of the manus ript.8 Appendix8.1 GroupsGroups are an extremely powerful tool in physi s. Most symmetry transformations form agroup. Invarian e under ontinuous transformation groups entails onserved quantities, likeenergy, angular momentum or ele tri harge.A group G is a set equipped with an asso iative, not ne essarily ommutative (or `Abelian')multipli ation law that has a neutral element 1. Every group element g is supposed to have aninverse g�1:We denote by Zn the y li group of n elements. You an either think of Zn as the setf0; 1; :::; n � 1g with multipli ation law being addition modulo n and neutral element 0. Or62

equivalently, you an take the set f1; exp(2�i=n); exp(4�i=n); :::; exp((n� 1)2�i=n)g with mul-tipli ation and neutral element 1. Zn is an Abelian subgroup of the permutation group on nobje ts.Other immediate examples are matrix groups: The general linear groups GL(n; C ) andGL(n;R) are the sets of omplex (real), invertible n � n matri es. The multipli ation law ismatrix multipli ation and the neutral element is the n � n unit matrix 1n. There are manyimportant subgroups of the general linear groups: SL(n; �), � = R or C , onsist only of matri eswith unit determinant. S stands for spe ial and will always indi ate that we add the onditionof unit determinant. The orthogonal group O(n) is the group of real n�n matri es g satisfyingggT = 1n. The spe ial orthogonal group SO(n) des ribes the rotations in the Eu lidean spa eRn. The Lorentz group O(1; 3) is the set of real 4 � 4 matri es g satisfying g�gT = �, with� =diagf1;�1;�1;�1g. The unitary group U(n) is the set of omplex n�nmatri es g satisfyinggg� = 1n. The unitary symple ti group USp(n) is the group of omplex 2n � 2n matri es gsatisfying gg� = 12n and gIgT = I withI := 0BBBB�� 0 1�1 0� � � � 0... . . . ...0 � � � � 0 1�1 0�1CCCCA : (217)The enter Z(G) of a group G onsists of those elements in G that ommute with allelements in G, Z(G) = fz 2 G; zg = gz for all g 2 Gg. For example, Z(U(n)) = U(1) 3exp(i�) 1n; Z(SU(n)) =Zn 3 exp(2�ik=n) 1n.All matrix groups are subsets of R2n2 and therefore we an talk about ompa tness ofthese groups. Re all that a subset of RN is ompa t if and only if it is losed and bounded. Forinstan e, U(1) is a ir le in R2 and therefore ompa t. The Lorentz group on the other handis unbounded be ause of the boosts.The matrix groups are Lie groups whi h means that they ontain in�nitesimal elements X lose to the neutral element: expX = 1 +X +O(X2) 2 G: For instan e,X = 0� 0 � 0�� 0 00 0 01A ; � small; (218)des ribes an in�nitesimal rotation around the z-axis by an in�nitesimal angle �. IndeedexpX =0� os � sin � 0� sin � os � 00 0 11A 2 SO(3); 0 � � < 2�; (219)63

is a rotation around the z-axis by an arbitrary angle �. The in�nitesimal transformations X ofa Lie group G form its Lie algebra g. It is losed under the ommutator [X;Y ℄ = XY � Y X.For the above matrix groups the Lie algebras are denoted by lower ase letters. For example,the Lie algebra of the spe ial unitary group SU(n) is written as su(n). It is the set of omplexn�nmatri esX satisfying X+X� = 0 and trX = 0. Indeed, 1n = (1n+X+:::)(1n+X+:::)� =1n + X + X� + O(X2) and 1 = det expX = exp trX. Attention, although de�ned in termsof omplex matri es, su(n) is a real ve tor spa e. Indeed, if a matrix X is anti-Hermitean,X +X� = 0, then in general, its omplex s alar multiple iX is no longer anti-Hermitean.However, in real ve tor spa es, eigenve tors do not always exist and we will have to om-plexify the real ve tor spa e g: Take a basis of g. Then g onsists of linear ombinationsof these basis ve tors with real oeÆ ients. The omplexi� ation gC of g onsits of linear ombinations with omplex oeÆ ients.The translation group of Rn is Rn itself. The multipli ation law now is ve tor addition andthe neutral element is the zero ve tor. As the ve tor addition is ommutative, the translationgroup is Abelian.The di�eomorphism group Di�(M) of an open subset M of Rn (or of a manifold) is the setof di�erentiable maps � fromM into itself that are invertible (for the omposition Æ) and su hthat its inverse is di�erentiable. (Attention, the last ondition is not automati , as you see bytaking M = R 3 x and �(x) = x3.) By virtue of the hain rule we an take the omposition asmultipli ation law. The neutral element is the identity map on M , � = 1M with 1M (x) = x forall x 2M .8.2 Group representationsWe said that SO(3) is the rotation group. This needs a little explanation. A rotation is givenby an axis, that is a unit eigenve tor with unit eigenvalue, and an angle. Two rotations anbe arried out one after the other, we say ` omposed'. Note that the order is important, wesay that the 3-dimensional rotation group is nonAbelian. If we say that the rotations form agroup, we mean that the omposition of two rotations is a third rotation. However, it is noteasy to ompute the multipli ation law, i.e., ompute the axis and angle of the third rotation asa fun tion of the axes and angles of the two initial rotations. The equivalent `representation' ofthe rotation group as 3� 3 matri es is mu h more onvenient be ause the multipli ation law issimply matrix multipli ation. There are several `representations' of the 3-dimensional rotationgroup in terms of matri es of di�erent sizes, say N�N . It is sometimes useful to know all theserepresentations. The N �N matri es are linear maps, `endomorphisms', of the N -dimensionalve tor spa e RN into itself. Let us denote by End(RN) the set of all these matri es. By64

de�nition, a representation of the group G on the ve tor spa e RN is a map � : G! End(RN)reprodu ing the multipli ation law as matrix multipli ation or in nobler terms as ompositionof endomorphisms. This means �(g1g2) = �(g1) �(g2) and �(1) = 1N . The representation is alled faithful if the map � is inje tive. By the minimax prin iple we are interested in thefaithful representations of lowest dimension. Although not always unique, physi ists all themfundamental representations. The fundamental representation of the 3-dimensional rotationgroup is de�ned on the ve tor spa e R3. Two N -dimensional representations �1 and �2 of agroup G are equivalent if there is an invertible N �N matrix C su h that �2(g) = C�1(g)C�1for all g 2 G. C is interpreted as des ribing a hange of basis in RN. A representation is alledirredu ible if its ve tor spa e has no proper invariant subspa e, i.e. a subspa e W � RN, withW 6= RN; f0g and �(g)W � W for all g 2 G.Representations an be de�ned in the same manner on omplex ve tor spa es, C N . Thenevery �(g) is a omplex, invertible matrix. It is often useful, e.g. in quantum me hani s, to rep-resent a group on a Hilbert spa e, we put a s alar produ t on the ve tor spa e, e.g. the standards alar produ t on C N 3 v;w, (v;w) := v�w. A unitary representation is a representation whosematri es �(g) all respe t the s alar produ t, whi h means that they are all unitary. In quantumme hani s, unitary representations are important be ause they preserve probability. For exam-ple, take the adjoint representation of SU(n) 3 g. Its Hilbert spa e is the omplexi� ationof its Lie algebra su(n)C 3 X;Y with s alar produ t (X;Y ) := tr (X�Y ). The representa-tion is de�ned by onjugation, �(g)X := gXg�1, and it is unitary, (�(g)X; �(g)Y ) = (X;Y ).In Yang-Mills theories, the gauge bosons live in the adjoint representation. In the Abelian ase, G = U(1), this representation is 1-dimensional, there is one gauge boson, the photon,A 2 u(1)C = C : The photon has no ele tri harge, whi h means that it transforms trivially,�(g)A = A for all g 2 U(1).Unitary equivalen e of representations is de�ned by hange of orthonormal bases. ThenC is a unitary matrix. A key theorem for parti le physi s states that all irredu ible unitaryrepresentations of any ompa t group are �nite dimensional. If we a ept the de�nition of ele-mentary parti les as orthonormal basis ve tors of unitary representations, then we understandwhy Yang and Mills only take ompa t groups. They only want a �nite number of elementaryparti les. Unitary equivalen e expresses the quantum me hani al superposition prin iple ob-served for instan e in the K0 � �K0 system. The unitary matrix C is sometimes referred to asmixing matrix.Bound states of elementary parti les are des ribed by tensor produ ts: the tensor produ tof two unitary representations �1 and �2 of one group de�ned on two Hilbert spa es H1 and H2is the unitary representation �1 �2 de�ned on H1H2 3 1 2 by (�1 �2)(g) ( 1 2) :=�1(g) 1 �2(g) 2. In the ase of ele tro-magnetism, G = U(1) 3 exp(i�) we know that65

all irredu ible unitary representations are 1-dimensional, H = C 3 and hara terized bythe ele tri harge q, �(exp(i�)) = exp(iq ) . Under tensorization the ele tri harges areadded. For G = SU(2), the irredu ible unitary representations are hara terized by the spin,` = 0; 12; 1; ::: The addition of spin from quantum me hani s is pre isely tensorization of theserepresentations.Let � be a representation of a Lie group G on a ve tor spa e and let g be the Lie algebra ofG. We denote by ~� the Lie algebra representation of the group representation �. It is de�nedon the same ve tor spa e by �(expX) = exp(~�(X)). The ~�(X)s are not ne essarily invertibleendomorphisms. They satisfy ~�([X;Y ℄) = [~�(X); ~�(Y )℄ := ~�(X)~�(Y )� ~�(Y )~�(X):An aÆne representation is the same onstru tion as above, but we allow the �(g)s to beinvertible aÆne maps, i.e. linear maps plus onstants.8.3 Semi-dire t produ t and Poin ar�e groupThe dire t produ t G � H of two groups G and H is again a group with multipli ation law:(g1; h1)(g2; h2) := (g1g2; h1h2): In the dire t produ t, all elements of the �rst fa tor ommutewith all elements of the se ond fa tor: (g; 1H)(1G; h) = (1G; h)(g; 1H): We write 1H for theneutral element of H. Warning, you sometimes see the misleading notation G H for thedire t produ t.To be able to de�ne the semi-dire t produ t GnH we must have an a tion of G on H, thatis a map � : G! Di�(H) satisfying �g(h1h2) = �g(h1) �g(h2), �g(1H) = 1H , �g1g2 = �g1 Æ�g2 and�1G = 1H . If H is a ve tor spa e arrying a representation or an aÆne representation � of thegroup G, we an view � as an a tion by onsidering H as translation group. Indeed, invertiblelinear maps and aÆne maps are di�eomorphisms on H. As a set, the semi-dire t produ t GnHis the dire t produ t, but the multipli ation law is modi�ed by help of the a tion:(g1; h1)(g2; h2) := (g1g2; h1 �g1(h2)): (220)We retrieve the dire t produ t if the a tion is trivial, �g = 1H for all g 2 G. Our �rstexample is the invarian e group of ele tro-magnetism oupled to gravity Di�(M) n MU(1):A di�eomorphism �(x) a ts on a gauge fun tion g(x) by ��(g) := g Æ ��1 or more expli itly(��(g))(x) := g(��1(x)). Other examples ome with other gauge groups like SU(n) or spingroups.Our se ond example is the Poin ar�e group, O(1; 3) n R4, whi h is the isometry group ofMinkowski spa e. The semi-dire t produ t is important be ause Lorentz transformations donot ommute with translations. Sin e we are talking about the Poin ar�e group, let us mentionthe theorem behind the de�nition of parti les as orthonormal basis ve tors of unitary repre-sentations: The irredu ible, unitary representations of the Poin ar�e group are hara terized66

by mass and spin. For �xed mass M � 0 and spin `, an orthonormal basis is labelled by themomentum ~p with E2= 2� ~p2 = 2M2, = exp(i(Et� ~p � ~x)=~) and the z- omponent m of thespin with jmj � `, = Y`;m(�; ').8.4 AlgebrasObservables an be added, multiplied and multiplied by s alars. They form naturally an asso- iative algebra A, i.e. a ve tor spa e equipped with an asso iative produ t and neutral elements0 and 1. Note that the multipli ation does not always admit inverses, a�1, e.g. the neutralelement of addition, 0, is not invertible. In quantum me hani s, observables are self adjoint.Therefore, we need an involution �� in our algebra. This is an anti-linear map from the algebrainto itself, (�a + b)� = ��a� + b�; � 2 C ; a; b 2 A; that reverses the produ t, (ab)� = b�a�,respe ts the unit, 1� = 1, and is su h that a�� = a. The set of n � n matri es with omplex oeÆ ients,Mn(C ), is an example of su h an algebra, and more generally, the set of endomor-phisms or operators on a given Hilbert spa e H. The multipli ation is matrix multipli ationor more generally omposition of operators, the involution is Hermitean onjugation or moregenerally the adjoint of operators.A representation � of an abstra t algebra A on a Hilbert spa e H is a way to write A on retely as operators as in the last example, � : A ! End(H). In the group ase, therepresentation had to reprodu e the multipli ation law. Now it has to reprodu e, the linearstru ture: �(�a + b) = ��(a) + �(b); �(0) = 0; the multipli ation: �(ab) = �(a)�(b); �(1) = 1;and the involution: �(a�) = �(a)�: Therefore the tensor produ t of two representations �1 and �2of A on Hilbert spa es H1 3 1 and H2 3 2 is not a representation: ((�1�2)(�a)) ( 1 2) =(�1(�a) 1) (�2(�a) 2) = �2(�1 �2)(a) ( 1 2).The group of unitaries U(A) := fu 2 A; uu� = u�u = 1g is a subset of the algebra A.Every algebra representation indu es a unitary representation of its group of unitaries. On theother hand, only few unitary representations of the group of unitaries extend to an algebrarepresentation. These representations des ribe elementary parti les. Composite parti les aredes ribed by tensor produ ts, whi h are not algebra representations.An anti-linear operator J on a Hilbert spa e H 3 ; ~ is a map from H into itself satisfyingJ(� + ~ ) = ��J( ) + J( ~ ): An anti-linear operator J is anti-unitary if it is invertible andpreserves the s alar produ t, (J ; J ~ ) = ( ~ ; ). For example, on H = C n 3 we an de�nean anti-unitary operator J in the following way. The image of the olumn ve tor under J isobtained by taking the omplex onjugate of and then multiplying it with a unitary n � nmatrix U , J = U � or J = U Æ omplex onjugation. In fa t, on a �nite dimensional Hilbertspa e, every anti-unitary operator is of this form.67

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