and potentials are added opportunistically. This problem is avoided naturally

by considering the tensor product of the noncommutative torus with a ¬nite

spectral triple, but I am sure that the axioms of noncommutative geometry can

be rediscovered by playing long enough with model building.

In quantum mechanics and in general relativity, time and space play radically

di¬erent roles. Spatial position is an observable in quantum mechanics, time is

not. In general relativity, spacial position loses all meaning and only proper time

can be measured. Distances are then measured by a particular observer as (his

proper) time of ¬‚ight of photons going back and forth multiplied by the speed

of light, which is supposed to be universal. This de¬nition of distances is oper-

ational thanks to the high precision of present day atomic clocks, for example

in the GPS. The ˜Riemannian™ de¬nition of the meter, the forty millionth part

of a complete geodesic on earth, had to be abandoned in favour of a quantum

mechanical de¬nition of the second via the spectrum of an atom. Connes™ de¬-

nition of geometry via the spectrum of the Dirac operator is the precise counter

part of today™s experimental situation. Note that the meter stick is an extended

(rigid ?) object. On the other hand an atomic clock is a pointlike object and

experiment tells us that the atom is sensitive to the potentials at the location

of the clock, the potentials of all forces, gravitational, electro“magnetic, ... The

special role of time remains to be understood in noncommutative geometry [56]

as well as the notion of spectral triples with Lorentzian signature and their 1+3

split [57].

Let us come back to our initial claim: Connes derives the standard model of

electro“magnetic, weak, and strong forces from noncommutative geometry and,

at the same time, uni¬es them with gravity. If we say that the Balmer“Rydberg

formula is derived from quantum mechanics, then this claim has three levels:

Explain the nature of the variables: The choice of the discrete variables

nj , contains already a “ at the time revolutionary “ piece of physics, energy

quantization. Where does it come from?

Explain the ansatz: Why should one take the power law (11)?

Explain the experimental ¬t: The ansatz comes with discrete parameters,

the ˜bills™ qj , and continuous parameters, the ˜coins™ gj , which are determined

by an experimental ¬t. Where do the ¬tted values, ˜the winner™, come from?

How about deriving gravity from Riemannian geometry? Riemannian geom-

etry has only one possible variable, the metric g. The minimax principle dictates

the Lagrangian ansatz:

[Λc ’ q

1

S[g] = 16πG R ] dV. (208)

M

Experiment rules on the parameters: q = 1, G = 6.670 · 10’11 m3 s’2 kg, New-

ton™s constant, and Λc ∼ 0. Riemannian geometry remains silent on the third

340 T. Sch¨cker

u

Table 3. Deriving some YMH forces from gravity

Einstein

Riemannian

- gravity

geometry

£

Connes

?

Connes

noncommutative

- gravity + Yang“Mills“Higgs

geometry

level. Nevertheless, there is general agreement, gravity derives from Riemannian

geometry.

Noncommutative geometry has only one possible variable, the Dirac operator,

which in the commutative case coincides with the metric. Its ¬‚uctuations explain

the variables of the additional forces, gauge and Higgs bosons. The minimax

principle dictates the Lagrangian ansatz: the spectral action. It reproduces the

Einstein“Hilbert action and the ansatz of Yang, Mills and Higgs, see Table 3.

On the third level, noncommutative geometry is not silent, it produces lots of

constraints, all compatible with the experimental ¬t. And their exploration is

not ¬nished yet.

I hope to have convinced one or the other reader that noncommutative geom-

etry contains elegant solutions of long standing problems in fundamental physics

and that it proposes concrete strategies to tackle the remaining ones. I would like

to conclude our outlook with a sentence by Planck who tells us how important

the opinion of our young, unbiased colleagues is. Planck said, a new theory is

accepted, not because the others are convinced, because they die.

Acknowledgements

It is a pleasure to thank Eike Bick and Frank Ste¬en for the organization of a

splendid School. I thank the participants for their unbiased criticism and Kurusch

Ebrahimi-Fard, Volker Schatz, and Frank Ste¬en for a careful reading of the

manuscript.

Appendix

A.1 Groups

Groups are an extremely powerful tool in physics. Most symmetry transforma-

tions form a group. Invariance under continuous transformation groups entails

conserved quantities, like energy, angular momentum or electric charge.

Forces from Connes™ Geometry 341

A group G is a set equipped with an associative, not necessarily commutative

(or ˜Abelian™) multiplication law that has a neutral element 1. Every group

element g is supposed to have an inverse g ’1 .

We denote by Zn the cyclic group of n elements. You can either think of

Zn as the set {0, 1, ..., n ’ 1} with multiplication law being addition modulo

n and neutral element 0. Or equivalently, you can take the set {1, exp(2πi/n),

exp(4πi/n), ..., exp((n’1)2πi/n)} with multiplication and neutral element 1. Zn

is an Abelian subgroup of the permutation group on n objects.

Other immediate examples are matrix groups: The general linear groups

GL(n, C) and GL(n, R) are the sets of complex (real), invertible n — n matrices.

The multiplication law is matrix multiplication and the neutral element is the

n — n unit matrix 1n . There are many important subgroups of the general linear

groups: SL(n, ·), · = R or C, consist only of matrices with unit determinant.

S stands for special and will always indicate that we add the condition of unit

determinant. The orthogonal group O(n) is the group of real n — n matrices g

satisfying gg T = 1n . The special orthogonal group SO(n) describes the rotations

in the Euclidean space Rn . The Lorentz group O(1, 3) is the set of real 4 — 4

matrices g satisfying g·g T = ·, with · =diag{1, ’1, ’1, ’1}. The unitary group

U (n) is the set of complex n — n matrices g satisfying gg — = 1n . The unitary

symplectic group U Sp(n) is the group of complex 2n — 2n matrices g satisfying

gg — = 12n and gIg T = I with

«

01

··· 0

’1 0

¬ ·

¬ ·

I := ¬ ·.

. .

.. (A.1)

¬ ·

. .

.

. .

0 1

···

0

’1 0

The center Z(G) of a group G consists of those elements in G that commute

with all elements in G, Z(G) = {z ∈ G, zg = gz for all g ∈ G}. For example,

Z(U (n)) = U (1) exp(iθ) 1n , Z(SU (n)) = Zn exp(2πik/n) 1n .

2

All matrix groups are subsets of R2n and therefore we can talk about com-

pactness of these groups. Recall that a subset of RN is compact if and only

if it is closed and bounded. For instance, U (1) is a circle in R2 and therefore

compact. The Lorentz group on the other hand is unbounded because of the

boosts.

The matrix groups are Lie groups which means that they contain in¬nitesimal

elements X close to the neutral element: exp X = 1 + X + O(X 2 ) ∈ G. For

instance,

«

0 0

’ 0 0,

X= small, (A.2)

0 00

342 T. Sch¨cker

u

describes an in¬nitesimal rotation around the z-axis by an in¬nitesimal angle .

Indeed

«

cos sin 0

exp X = ’ sin 0 ∈ SO(3), 0 ¤ < 2π,

cos (A.3)

0 0 1

is a rotation around the z-axis by an arbitrary angle . The in¬nitesimal trans-

formations X of a Lie group G form its Lie algebra g. It is closed under the

commutator [X, Y ] = XY ’ Y X. For the above matrix groups the Lie algebras

are denoted by lower case letters. For example, the Lie algebra of the special uni-

tary group SU (n) is written as su(n). It is the set of complex n — n matrices X

satisfying X + X — = 0 and tr X = 0. Indeed, 1n = (1n + X + ...)(1n + X + ...)— =

1n +X +X — +O(X 2 ) and 1 = det exp X = exp tr X. Attention, although de¬ned

in terms of complex matrices, su(n) is a real vector space. Indeed, if a matrix

X is anti-Hermitean, X + X — = 0, then in general, its complex scalar multiple

iX is no longer anti-Hermitean.

However, in real vector spaces, eigenvectors do not always exist and we will

have to complexify the real vector space g: Take a basis of g. Then g consists of

linear combinations of these basis vectors with real coe¬cients. The complexi-

¬cation gC of g consits of linear combinations with complex coe¬cients.

The translation group of Rn is Rn itself. The multiplication law now is vector

addition and the neutral element is the zero vector. As the vector addition is

commutative, the translation group is Abelian.

The di¬eomorphism group Di¬(M ) of an open subset M of Rn (or of a mani-

fold) is the set of di¬erentiable maps σ from M into itself that are invertible (for

the composition —¦) and such that its inverse is di¬erentiable. (Attention, the last

condition is not automatic, as you see by taking M = R x and σ(x) = x3 .)

By virtue of the chain rule we can take the composition as multiplication law.

The neutral element is the identity map on M , σ = 1M with 1M (x) = x for all

x ∈ M.

A.2 Group Representations

We said that SO(3) is the rotation group. This needs a little explanation. A

rotation is given by an axis, that is a unit eigenvector with unit eigenvalue,

and an angle. Two rotations can be carried out one after the other, we say

˜composed™. Note that the order is important, we say that the 3-dimensional

rotation group is nonAbelian. If we say that the rotations form a group, we

mean that the composition of two rotations is a third rotation. However, it is

not easy to compute the multiplication law, i.e., compute the axis and angle of

the third rotation as a function of the axes and angles of the two initial rotations.

The equivalent ˜representation™ of the rotation group as 3 — 3 matrices is much

more convenient because the multiplication law is simply matrix multiplication.

There are several ˜representations™ of the 3-dimensional rotation group in terms

of matrices of di¬erent sizes, say N — N . It is sometimes useful to know all these

Forces from Connes™ Geometry 343

representations. The N —N matrices are linear maps, ˜endomorphisms™, of the N -

dimensional vector space RN into itself. Let us denote by End(RN ) the set of all

these matrices. By de¬nition, a representation of the group G on the vector space

RN is a map ρ : G ’ End(RN ) reproducing the multiplication law as matrix

multiplication or in nobler terms as composition of endomorphisms. This means

ρ(g1 g2 ) = ρ(g1 ) ρ(g2 ) and ρ(1) = 1N . The representation is called faithful if the

map ρ is injective. By the minimax principle we are interested in the faithful

representations of lowest dimension. Although not always unique, physicists call

them fundamental representations. The fundamental representation of the 3-

dimensional rotation group is de¬ned on the vector space R3 . Two N -dimensional

representations ρ1 and ρ2 of a group G are equivalent if there is an invertible

N — N matrix C such that ρ2 (g) = Cρ1 (g)C ’1 for all g ∈ G. C is interpreted

as describing a change of basis in RN . A representation is called irreducible if

its vector space has no proper invariant subspace, i.e. a subspace W ‚ RN , with

W = RN , {0} and ρ(g)W ‚ W for all g ∈ G.

Representations can be de¬ned in the same manner on complex vector spaces,

C . Then every ρ(g) is a complex, invertible matrix. It is often useful, e.g. in

N

quantum mechanics, to represent a group on a Hilbert space, we put a scalar

product on the vector space, e.g. the standard scalar product on CN v, w,

—

(v, w) := v w. A unitary representation is a representation whose matrices

ρ(g) all respect the scalar product, which means that they are all unitary. In

quantum mechanics, unitary representations are important because they pre-

serve probability. For example, take the adjoint representation of SU (n) g.

Its Hilbert space is the complexi¬cation of its Lie algebra su(n)C X, Y with

scalar product (X, Y ) := tr (X — Y ). The representation is de¬ned by conjugation,

ρ(g)X := gXg ’1 , and it is unitary, (ρ(g)X, ρ(g)Y ) = (X, Y ). In Yang“Mills the-

ories, the gauge bosons live in the adjoint representation. In the Abelian case,

G = U (1), this representation is 1-dimensional, there is one gauge boson, the

photon, A ∈ u(1)C = C. The photon has no electric charge, which means that it

transforms trivially, ρ(g)A = A for all g ∈ U (1).

Unitary equivalence of representations is de¬ned by change of orthonormal

bases. Then C is a unitary matrix. A key theorem for particle physics states that

all irreducible unitary representations of any compact group are ¬nite dimen-

sional. If we accept the de¬nition of elementary particles as orthonormal basis

vectors of unitary representations, then we understand why Yang and Mills only

take compact groups. They only want a ¬nite number of elementary particles.

Unitary equivalence expresses the quantum mechanical superposition principle

observed for instance in the K 0 ’ K 0 system. The unitary matrix C is sometimes

¯

referred to as mixing matrix.

Bound states of elementary particles are described by tensor products: the

tensor product of two unitary representations ρ1 and ρ2 of one group de¬ned on

two Hilbert spaces H1 and H2 is the unitary representation ρ1 — ρ2 de¬ned on

H1 — H2 ψ1 — ψ2 by (ρ1 — ρ2 )(g) (ψ1 — ψ2 ) := ρ1 (g) ψ1 — ρ2 (g) ψ2 . In the case

of electro“magnetism, G = U (1) exp(iθ) we know that all irreducible unitary

representations are 1-dimensional, H = C ψ and characterized by the electric

charge q, ρ(exp(iθ))ψ = exp(iqψ)ψ. Under tensorization the electric charges are

344 T. Sch¨cker

u

added. For G = SU (2), the irreducible unitary representations are characterized

by the spin, = 0, 1 , 1, ... The addition of spin from quantum mechanics is

2

precisely tensorization of these representations.

Let ρ be a representation of a Lie group G on a vector space and let g be

the Lie algebra of G. We denote by ρ the Lie algebra representation of the