¯

’h2

h1 det w 0

= ±u0 ’ 12 . (200)

¯

h2 h1 0 det w

¯

The Higgs is characterized by one complex doublet, (h1 , h2 )T . Again it will be

convenient to pass to the homogeneous Higgs variable,

D = LDL’1 = D + H + JHJ ’1

˜ ˜

«

ˆ

0 ¦0 0

ˆ—

¬¦ 0·

00

= ¦ + J¦J ’1 = ¯ (201)

ˆ

0 0 0— ¦

¯

ˆ

0 0¦ 0

with

«

’•2 Md

•1 M u ¯

— 13 0

¬ ·

•2 Mu •1 Md

¯

¦=¬ · = ρL (φ)M

ˆ (202)

’•2 Me

¯

0

•1 Me

¯

and

’•2

•1 ¯ det w 0

= ±u0

φ= . (203)

•2 •1

¯ 0 det w

¯

In order to satisfy the ¬rst order condition, the representation of M3 (C) c

had to commute with the Dirac operator. Therefore the Higgs is a colour singlet

Forces from Connes™ Geometry 335

and the gluons will remain massless. The ¬rst two of the six intriguing properties

of the standard model listed in Sect. 3.3 have a geometric raison d™ˆtre, the ¬rst

e

order condition. In turn, they imply the third property: we have just shown that

the Higgs • = (•1 , •2 )T is a colour singlet. At the same time the ¬fth property

follows from the fourth: the Higgs of the standard model is an isospin doublet

because of the parity violating couplings of the quaternions H. Furthermore, this

Higgs has hypercharge y• = ’ 1 and the last ¬ne tuning of the sixth property (57)

2

also derives from Connes™ algorithm: the Higgs has a component with vanishing

electric charge, the physical Higgs, and the photon will remain massless.

In conclusion, in Connes version of the standard model there is only one

intriguing input property, the fourth: explicit parity violation in the algebra

representation HL • HR , the ¬ve others are mathematical consequences.

Spectral Action. Computing the spectral action SCC = f (Dt /Λ2 ) in the stan-

2

dard model is not more di¬cult than in the minimax example, only the matrices

are a little bigger,

«

ˆ

‚ L γ5 ¦

/ 0 0

¬ γ ¦— ‚ ·

ˆ /R 0 0

˜ t L’1 = ¬ 5 ·

Dt = Lt D t ¬ ¯ ·. (204)

’1 ˆ

0 0 C ‚LC

/ γ5 ¦

¯—

C ‚ C ’1

ˆ

0 0 γ5 ¦ / R

ˆ ˆ

The trace of the powers of ¦ are computed from the identities ¦ = ρL (φ)M and

φ— φ = φφ— = (|•1 |2 + |•2 |2 )12 = |•|2 12 by using that ρL as a representation

respects multiplication and involution.

The spectral action produces the complete action of the standard model

coupled to gravity with the following relations for coupling constants:

2 2 9

g3 = g2 = N ». (205)

Our choice of central charges, q = (1, 0)T , entails a further relation, g1 = 3 g2 , i.e.

2 2

˜ 5

sin2 θw = 3/8. However only products of the Abelian gauge coupling g1 and the

hypercharges yj appear in the Lagrangian. By rescaling the central charges, we

can rescale the hypercharges and consequently the Abelian coupling g1 . It seems

quite moral that noncommutative geometry has nothing to say about Abelian

gauge couplings.

Experiment tells us that the weak and strong couplings are unequal, equation

(49) at energies corresponding to the Z mass, g2 = 0.6518±0.0003, g3 = 1.218±

0.01. Experiment also tells us that the coupling constants are not constant, but

that they evolve with energy. This evolution can be understood theoretically

in terms of renormalization: one can get rid of short distance divergencies in

perturbative quantum ¬eld theory by allowing energy depending gauge, Higgs,

and Yukawa couplings where the theoretical evolution depends on the particle

content of the model. In the standard model, g2 and g3 come together with

increasing energy, see Fig. 8. They would become equal at astronomical energies,

Λ = 1017 GeV, if one believed that between presently explored energies, 102 GeV,

336 T. Sch¨cker

u

g3

1.4

1.2

1

g2

0.8

0.6

0.4

0.2

E

9

mZ 10 GeV

Fig. 8. Running coupling constants

and the ˜uni¬cation scale™ Λ, no new particles exist. This hypothesis has become

popular under the name ˜big desert™ since Grand Uni¬ed Theories. It was believed

that new gauge bosons, ˜lepto-quarks™ with masses of order Λ existed. The lepto-

quarks together with the W ± , the Z, the photon and the gluons generate the

simple group SU (5), with only one gauge coupling, g5 := g3 = g2 = 5 g1 at Λ. In

2 2 2 2

3

the minimal SU (5) model, these lepto-quarks would mediate proton decay with

a half life that today is excluded experimentally.

If we believe in the big desert, we can imagine that “ while almost commuta-

tive at present energies “ our geometry becomes truly noncommutative at time

scales of /Λ ∼ 10’41 s. Since in such a geometry smaller time intervals cannot

be resolved, we expect the coupling constants to become energy independent

at the corresponding energy scale Λ. We remark that the ¬rst motivation for

noncommutative geometry in spacetime goes back to Heisenberg and was pre-

cisely the regularization of short distance divergencies in quantum ¬eld theory,

see e.g. [45]. The big desert is an opportunistic hypothesis and remains so in

the context of noncommutative geometry. But in this context, it has at least the

merit of being consistent with three other physical ideas:

Planck time: There is an old hand waving argument combining of phase space

with the Schwarzschild horizon to ¬nd an uncertainty relation in space-

time with a scale Λ smaller than the Planck energy ( c5 /G)1/2 ∼ 1019

GeV: To measure a position with a precision ∆x we need, following Heisen-

berg, at least a momentum /∆x or, by special relativity, an energy c/∆x.

According to general relativity, such an energy creates an horizon of size

G c’3 /∆x. If this horizon exceeds ∆x all information on the position is

lost. We can only resolve positions with ∆x larger than the Planck length,

∆x > ( G/c3 )1/2 ∼ 10’35 m. Or we can only resolve time with ∆t larger

than the Planck time, ∆t > ( G/c5 )1/2 ∼ 10’43 s. This is compatible with

the above time uncertainty of /Λ ∼ 10’41 s.

Forces from Connes™ Geometry 337

Stability: We want the Higgs self coupling » to remain positive [46] during its

perturbative evolution for all energies up to Λ. A negative Higgs self coupling

would mean that no ground state exists, the Higgs potential is unstable.

This requirement is met for the self coupling given by the constraint (205)

at energy Λ, see Fig. 8.

Triviality: We want the Higgs self coupling » to remain perturbatively small [46]

during its evolution for all energies up to Λ because its evolution is computed

from a perturbative expansion. This requirement as well is met for the self

coupling given by the constraint (205), see Fig. 8. If the top mass was larger

than 231 GeV or if there were N = 8 or more generations this criterion

would fail.

Since the big desert gives a minimal and consistent picture we are curious to know

its numerical implication. If we accept the constraint (205) with g2 = 0.5170 at

the energy Λ = 0.968 1017 GeV and evolve it down to lower energies using

the perturbative renormalization ¬‚ow of the standard model, see Fig. 8, we

retrieve the experimental nonAbelian gauge couplings g2 and g3 at the Z mass

by construction of Λ. For the Higgs coupling, we obtain

» = 0.06050 ± 0.0037 at E = mZ . (206)

The indicated error comes from the experimental error in the top mass, mt =

174.3 ± 5.1 GeV, which a¬ects the evolution of the Higgs coupling. From the

Higgs coupling at low energies we compute the Higgs mass,

√

√ »

mW = 171.6 ± 5 GeV.

mH = 4 2 (207)

g2

For details of this calculation see [47].

6.4 Beyond the Standard Model

A social reason, that made the Yang“Mills“Higgs machine popular, is that it is an

inexhaustible source of employment. Even after the standard model, physicists

continue to play on the machine and try out extensions of the standard model by

adding new particles, ˜let the desert bloom™. These particles can be gauge bosons

coupling only to right-handed fermions in order to restore left-right symmetry.

The added particles can be lepto-quarks for grand uni¬cation or supersymmetric

particles. These models are carefully tuned not to upset the phenomenological

success of the standard model. This means in practice to choose Higgs represen-

tations and potentials that give masses to the added particles, large enough to

make them undetectable in present day experiments, but not too large so that ex-

perimentalists can propose bigger machines to test these models. Independently

there are always short lived deviations from the standard model predictions in

new experiments. They never miss to trigger new, short lived models with new

particles to ¬t the ˜anomalies™. For instance, the literature contains hundreds of

superstring inspired s, each of them with hundreds of parameters, coins, waiting

for the standard model to fail.

338 T. Sch¨cker

u

Of course, we are trying the same game in Connes™ do“it“yourself kit. So far,

we have not been able to ¬nd one single consistent extension of the standard

model [41“43,48]. The reason is clear, we have no handle on the Higgs repre-

sentation and potential, which are on the output side, and, in general, we meet

two problems: light physical scalars and degenerate fermion masses in irreducible

multiplets. The extended standard model with arbitrary numbers of quark gen-

erations, Nq ≥ 0, of lepton generations, N ≥ 1, and of colours Nc , somehow

manages to avoid both problems and we are trying to prove that it is unique as

such. The minimax model has Nq = 0, N = 1, Nc = 0. The standard model has

Nq = N =: N and Nc = 3 to avoid Yang“Mills anomalies [12]. It also has N = 3

generations. So far, the only realistic extension of the standard model that we

know of in noncommutative geometry, is the addition of right-handed neutrinos

and of Dirac masses in one or two generations. These might be necessary to

account for observed neutrino oscillations [13].

7 Outlook and Conclusion

Noncommutative geometry reconciles Riemannian geometry and uncertainty and

we expect it to reconcile general relativity with quantum ¬eld theory. We also

expect it to improve our still incomplete understanding of quantum ¬eld theory.

On the perturbative level such an improvement is happening right now: Connes,

Moscovici, and Kreimer discovered a subtle link between a noncommutative

generalization of the index theorem and perturbative quantum ¬eld theory. This

link is a Hopf algebra relevant to both theories [49].

In general, Hopf algebras play the same role in noncommutative geometry

as Lie groups play in Riemannian geometry and we expect new examples of

noncommutative geometry from its merging with the theory of Hopf algebras.

Reference [50] contains a simple example where quantum group techniques can

be applied to noncommutative particle models.

The running of coupling constants from perturbative quantum ¬eld theory

must be taken into account in order to perform the high precision test of the

standard model at present day energies. We have invoked an extrapolation of

this running to astronomical energies to make the constraint g2 = g3 from the

spectral action compatible with experiment. This extrapolation is still based on

quantum loops in ¬‚at Minkowski space. While acceptable at energies below the

scale Λ where gravity and the noncommutativity of space seem negligible, this

approximation is unsatisfactory from a conceptual point of view and one would

like to see quantum ¬elds constructed on a noncommutative space. At the end of

the nineties ¬rst examples of quantum ¬elds on the (¬‚at) noncommutative torus

or its non-compact version, the Moyal plane, were published [51]. These examples

came straight from the spectral action. The noncommutative torus is motivated

from quantum mechanical phase space and was the ¬rst example of a noncom-

mutative spectral triple [52]. Bellissard [53] has shown that the noncommutative

torus is relevant in solid state physics: one can understand the quantum Hall

e¬ect by taking the Brillouin zone to be noncommutative. Only recently other

Forces from Connes™ Geometry 339

examples of noncommutative spaces like noncommutative spheres where uncov-

ered [54]. Since 1999, quantum ¬elds on the noncommutative torus are being

studied extensively including the ¬elds of the standard model [55]. So far, its