˜

= 0. (1)

dp2

After a general coordinate transformation, x» = σ » (˜), Newton™s equation reads

x

d2 x» dxµ dxν

»

+ “ µν (g) = 0. (2)

dp2 dp dp

Pseudo forces have appeared. They are coded in the Levi“Civita connection

‚ ‚ ‚

gκµ ’

“ » µν (g) = 1 g »κ gκν + gµν , (3)

2 ‚xµ ‚xν ‚xκ

where gµν is obtained by ˜¬‚uctuating™ the ¬‚at metric ·µν = diag(1, ’1, ’1, ’1, )

˜˜ ˜

with the Jacobian of the coordinate transformation σ:

gµν (x) = J (x)’1˜ µ ·µν J (x)’1˜ ν , J (˜)µ µ := ‚σ µ (˜)/‚ xµ .

µ ν

˜˜

x˜ x (4)

˜˜

For the coordinates of the rotating disk, the pseudo forces are precisely the cen-

trifugal and Coriolis forces. Einstein takes uniformly accelerated coordinates,

ct = ct, z = z + 1 cg2 (ct)2 with g = 9.81 m/s2 . Then the geodesic equation (2)

˜ ˜

˜2

reduces to d2 z/dt2 = ’g. So far this gravity is still a pseudo force which means

that the curvature of its Levi“Civita connection vanishes. This constraint is re-

laxed by the equivalence principle: pseudo forces and true gravitational forces

are coded together in a not necessarily ¬‚at connection “ , that derives from a

potential, the not necessarily ¬‚at metric g. The kinematical variable to describe

gravity is therefore the Riemannian metric. By construction the dynamics of

matter, the geodesic equation, is now covariant under general coordinate trans-

formations.

2.2 Second Stroke: Dynamics

Now that we know the kinematics of gravity let us see how Einstein obtains its

dynamics, i.e. di¬erential equations for the metric tensor gµν . Of course Einstein

wants these equations to be covariant under general coordinate transformations

and he wants the energy-momentum tensor Tµν to be the source of gravity. From

288 T. Sch¨cker

u

Riemannian geometry he knew that there is no covariant, ¬rst order di¬erential

operator for the metric. But there are second order ones:

Theorem: The most general tensor of degree 2 that can be constructed from

the metric tensor gµν (x) with at most two partial derivatives is

±, β, Λ ∈ R..

±Rµν + βRgµν + Λgµν , (5)

Here are our conventions for the curvature tensors:

Riemann tensor : R» µνκ = ‚ν “ » µκ ’ ‚κ “ » µν + “ · µκ “ » ν· ’ “ · µν “ » κ· , (6)

Rµκ = R» µ»κ ,

Ricci tensor : (7)

µν

curvature scalar : R = Rµν g . (8)

The miracle is that the tensor (5) is symmetric just as the energy-momentum

tensor. However, the latter is covariantly conserved, Dµ Tµν = 0, while the former

one is conserved if and only if β = ’ 1 ±. Consequently, Einstein puts his equation

2

Rµν ’ 1 Rgµν ’ Λc gµν = 8πG

Tµν . (9)

c4

2

He chooses a vanishing cosmological constant, Λc = 0. Then for small static mass

density T00 , his equation reproduces Newton™s universal law of gravity with G

the Newton constant. However for not so small masses there are corrections

to Newton™s law like precession of perihelia. Also Einstein™s theory applies to

massless matter and produces the curvature of light. Einstein™s equation has an

agreeable formal property, it derives via the Euler“Lagrange variational principle

from an action, the famous Einstein“Hilbert action:

’1 2Λc

R dV ’

SEH [g] = dV, (10)

16πG 16πG

M M

with the invariant volume element dV := | det g·· |1/2 d4 x.

General relativity has a precise geometric origin: the left-hand side of Ein-

stein™s equation is a sum of some 80 000 terms in ¬rst and second partial deriva-

tives of gµν and its matrix inverse g µν . All of these terms are completely ¬xed by

the requirement of covariance under general coordinate transformations. General

relativity is veri¬ed experimentally to an extraordinary accuracy, even more, it

has become a cornerstone of today™s technology. Indeed length measurements

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 source, in contrast to Maxwell™s equation that does allow point charges

as source. From this point of view it is not satisfying to consider point-like

matter.

• The gravitational force is coded in the connection “ . Nevertheless we have

accepted its potential, the metric g, as kinematical variable.

Forces from Connes™ Geometry 289

• The equivalence principle states that locally, i.e. on the trajectory of a point-

like particle, one cannot distinguish gravity from a pseudo force. In other

words, there is always a coordinate system, ˜the freely falling lift™, in which

gravity is absent. This is not true for electro“magnetism and we would like

to derive this force (as well as the weak and strong forces) as a pseudo force

coming from a geometric transformation.

• So far general relativity has resisted all attempts to reconcile it with quantum

mechanics.

3 Slot Machines and the Standard Model

Today we have a very precise phenomenological description of electro“magnetic,

weak, and strong forces. This description, the standard model, works on a pertur-

bative quantum level and, as classical gravity, it derives from an action principle.

Let us introduce this action by analogy with the Balmer“Rydberg formula.

One of the new features of atomic physics was the appearance of discrete

frequencies and the measurement of atomic spectra became a highly developed

art. It was natural to label the discrete frequencies ν by natural numbers n. To

¬t the spectrum of a given atom, say hydrogen, let us try the ansatz

ν = g1 nq1 + g2 nq2 . (11)

1 2

We view this ansatz as a slot machine. You input two bills, the integers q1 , q2

and two coins, the two real numbers g1 , g2 , and compare the output with the

measured spectrum. (See Fig. 1.) If you are rich enough, you play and replay

on the slot machine until you win. The winner is the Balmer“Rydberg formula,

i.e., q1 = q2 = ’2 and g1 = ’g2 = 3.289 1015 Hz, which is the famous Rydberg

constant R. Then came quantum mechanics. It explained why the spectrum of

the hydrogen atom was discrete in the ¬rst place and derived the exponents and

the Rydberg constant,

e4

me

R= , (12)

4π 3 (4π 0 )2

from a noncommutativity, [x, p] = i 1.

q g

1 1

q g

2 2

Fig. 1. A slot machine for atomic spectra

290 T. Sch¨cker

u

G gn

L g

Y

R

S

Fig. 2. The Yang“Mills“Higgs slot machine

To cut short its long and complicated history we introduce the standard

model as the winner of a particular slot machine. This machine, which has be-

come popular under the names Yang, Mills and Higgs, has four slots for four

bills. Once you have decided which bills you choose and entered them, a certain

number of small slots will open for coins. Their number depends on the choice of

bills. You make your choice of coins, feed them in, and the machine starts work-

ing. It produces as output a Lagrange density. From this density, perturbative

quantum ¬eld theory allows you to compute a complete particle phenomenology:

the particle spectrum with the particles™ quantum numbers, cross sections, life

times, and branching ratios. (See Fig. 2.) You compare the phenomenology to

experiment to ¬nd out whether your input wins or loses.

3.1 Input

The ¬rst bill is a ¬nite dimensional, real, compact Lie group G. The gauge

bosons, spin 1, will live in its adjoint representation whose Hilbert space is the

complexi¬cation of the Lie algebra g (cf. Appendix).

The remaining bills are three unitary representations of G, ρL , ρR , ρS , de-

¬ned on the complex Hilbert spaces, HL , HR , HS . They classify the left- and

right-handed fermions, spin 1 , and the scalars, spin 0. The group G is chosen

2

compact to ensure that the unitary representations are ¬nite dimensional, we

want a ¬nite number of ˜elementary particles™ according to the credo of particle

physics that particles are orthonormal basis vectors of the Hilbert spaces which

carry the representations. More generally, we might also admit multi-valued

representations, ˜spin representations™, which would open the debate on charge

quantization. More on this later.

The coins are numbers, coupling constants, more precisely coe¬cients of

invariant polynomials. We need an invariant scalar product on g. The set of all

these scalar products is a cone and the gauge couplings are particular coordinates

of this cone. If the group is simple, say G = SU (n), then the most general,

invariant scalar product is

—

X, X ∈ su(n).

2

(X, X ) = gn tr [X X ], (13)

2

Forces from Connes™ Geometry 291

If G = U (1), we have

Y, Y ∈ u(1).

1¯

(Y, Y ) = 2Y Y , (14)

g1

We denote by ¯ the complex conjugate and by ·— the Hermitean conjugate. Mind

·

the di¬erent normalizations, they are conventional. The gn are positive numbers,

the gauge couplings. For every simple factor of G there is one gauge coupling.

Then we need the Higgs potential V (•). It is an invariant, fourth order,

stable polynomial on HS •. Invariant means V (ρS (u)•) = V (•) for all u ∈ G.

Stable means bounded from below. For G = U (2) and the Higgs scalar in the

fundamental or de¬ning representation, • ∈ HS = C2 , ρS (u) = u, we have

V (•) = » (•— •)2 ’ 1 µ2 •— •. (15)

2

The coe¬cients of the Higgs potential are the Higgs couplings, » must be positive

for stability. We say that the potential breaks G spontaneously if no minimum

of the potential is a trivial orbit under G. In our example, if µ is positive, the

√

minima of V (•) lie on the 3-sphere |•| = v := 1 µ/ ». v is called vacuum

2

expectation value and U (2) is said to break down spontaneously to its little

group

10

U (1) . (16)

0 ei±

The little group leaves invariant any given point of the minimum, e.g. • = (v, 0)T .

On the other hand, if µ is purely imaginary, then the minimum of the potential

is the origin, no spontaneous symmetry breaking and the little group is all of G.

Finally, we need the Yukawa couplings gY . They are the coe¬cients of the

— —

most general, real, trilinear invariant on HL — HR — (HS • HS ). For every

1-dimensional invariant subspace in the reduction of this tensor representa-

tion, we have one complex Yukawa coupling. For example G = U (2), HL =

C2 , ρL (u)ψL = (det u)qL u ψL , HR = C, ρR (u)ψR = (det u)qR ψR , HS = C2 ,

ρS (u)• = (det u)qS u •. If ’qL + qR + qS = 0 there is no Yukawa coupling,

—

otherwise there is one: (ψL , ψR , •) = Re(gY ψL ψR •).

If the symmetry is broken spontaneously, gauge and Higgs bosons acquire

masses related to gauge and Higgs couplings, fermions acquire masses equal to

the ˜vacuum expectation value™ v times the Yukawa couplings.

As explained in Jan-Willem van Holten™s and Jean Zinn-Justin™s lectures at

this School [11,12], one must require for consistency of the quantum theory that

the fermionic representations be free of Yang“Mills anomalies,