ψB (j0 , ..., j3 ) exp(’ijµ xµ ),

ψB (x) =: B = 1, 2, 3, 4 (111)

j0 ,...,j3 ∈Z

the eigenvalue equation ‚ ψ = »ψ reads

/

« ˆ «

«

’j0 ’ ij3 ’ij1 ’ j2 ˆ

0 0 ψ1 ψ1

’j0 + ij3 · ¬ ψ2 · ¬ψ ·

’ij1 + j2 ˆ ˆ

¬ 0 0

¬ ˆ · = » ¬ ˆ2 · . (112)

ψ3 ψ3

’j0 + ij3 ij1 + j2 0 0

ij1 ’ j2 ’j0 ’ ij3 ˆ ˆ

0 0 ψ4 ψ4

2

Its characteristic equation is »2 ’ (j0 + j1 + j2 + j3 ) = 0 and for ¬xed jµ ,

2 2 2 2

each eigenvalue » = ± j0 + j1 + j2 + j3 has multiplicity two. Therefore asymp-

2 2 2 2

totically for large Λ there are 4B4 Λ4 eigenvalues (counted with their multiplicity)

314 T. Sch¨cker

u

whose absolute values are smaller than Λ. B4 = π 2 /2 denotes the volume of the

unit ball in R4 . En passant, we check . Let us arrange the absolute values of the

eigenvalues in an increasing sequence and number them by naturals n, taking

due account of their multiplicities. For large n, we have

n 1/4

|»n | ≈ . (113)

2π 2

The exponent is indeed the inverse dimension. To check the heat kernel expan-

sion, we compute the right-hand side of equation (110):

Λc f0

dV = (2π)4 4

= 2π 2 Λ4 ,

SCC = 4π 2 Λ (114)

8πG

M

which agrees with the asymptotic count of eigenvalues, 4B4 Λ4 . This example was

the ¬‚at torus. Curvature will modify the spectrum and this modi¬cation can be

used to measure the curvature = gravitational ¬eld, exactly as the Zeemann or

Stark e¬ect measures the electro“magnetic ¬eld by observing how it modi¬es

the spectral lines of an atom.

In the spectral action, we ¬nd the Einstein“Hilbert action, which is linear

in curvature. In addition, the spectral action contains terms quadratic in the

curvature. These terms can safely be neglected in weak gravitational ¬elds like

in our solar system. In homogeneous, isotropic cosmologies, these terms are a

surface term and do not modify Einstein™s equation. Nevertheless the quadratic

terms render the (Euclidean) Chamseddine“Connes action positive. Therefore

this action has minima. For instance, the 4-sphere with a radius of the order of

√

the Planck length G is a minimum, a ˜ground state™. This minimum breaks the

di¬eomorphism group spontaneously [23] down to the isometry group SO(5).

The little group is the isometry group, consisting of those lifted automorphisms

that commute with the Dirac operator ‚ . Let us anticipate that the spontaneous

/

symmetry breaking via the Higgs mechanism will be a mirage of this gravita-

tional break down. Physically this ground state seems to regularize the initial

cosmological singularity with its ultra strong gravitational ¬eld in the same way

in which quantum mechanics regularizes the Coulomb singularity of the hydro-

gen atom.

We close this subsection with a technical remark. We noticed that the matrix

’1 µ

e a in equation (103) is symmetric. A general, not necessarily symmetric ma-

trix e’1 µ a can be obtained from a general Lorentz transformation Λ ∈ M SO(4):

ˆ

e’1 µ a Λa b = e’1 µ b ,

ˆ (115)

which is nothing but the polar decomposition of the matrix e’1 . These trans-

ˆ

formations are the gauge transformations of general relativity in Cartan™s for-

mulation. They are invisible in Einstein™s formulation because of the complete

(symmetric) gauge ¬xing coming from the initial coordinate system xµ .

˜˜

5.2 Almost Commutative Geometry

We are eager to see the spectral action in a noncommutative example. Technically

the simplest noncommutative examples are almost commutative. To construct

Forces from Connes™ Geometry 315

the latter, we need a natural property of spectral triples, commutative or not:

The tensor product of two even spectral triples is an even spectral triple. If both

are commutative, i.e. describing two manifolds, then their tensor product simply

describes the direct product of the two manifolds.

Let (Ai , Hi , Di , Ji , χi ), i = 1, 2 be two even, real spectral triples of even

dimensions d1 and d2 . Their tensor product is the triple (At , Ht , Dt , Jt , χt ) of

dimension d1 + d2 de¬ned by

At = A1 — A2 , Ht = H1 — H2 ,

Dt = D1 — 12 + χ1 — D2 ,

Jt = J1 — J2 , χt = χ1 — χ2 .

The other obvious choice for the Dirac operator, D1 — χ2 + 11 — D2 , is unitar-

ily equivalent to the ¬rst one. By de¬nition, an almost commutative geometry

is a tensor product of two spectral triples, the ¬rst triple is a 4-dimensional

spacetime, the calibrating example,

C ∞ (M ), L2 (S), ‚ , C, γ5 ,

/ (116)

and the second is 0-dimensional. In accordance with , a 0-dimensional spectral

triple has a ¬nite dimensional algebra and a ¬nite dimensional Hilbert space.

We will label the second triple by the subscript ·f (for ¬nite) rather than by ·2 .

The origin of the word almost commutative is clear: we have a tensor product of

an in¬nite dimensional commutative algebra with a ¬nite dimensional, possibly

noncommutative algebra.

This tensor product is, in fact, already familiar to you from the quantum

mechanics of spin, whose Hilbert space is the in¬nite dimensional Hilbert space

of square integrable functions on con¬guration space tensorized with the 2-

dimensional Hilbert space C2 on which acts the noncommutative algebra of spin

observables. It is the algebra H of quaternions, 2 — 2 complex matrices of the

x ’¯ y

x, y ∈ C. A basis of H is given by {12 , iσ1 , iσ2 , iσ3 }, the identity

form

yx ¯

matrix and the three Pauli matrices (82) times i. The group of unitaries of H is

SU (2), the spin cover of the rotation group, the group of automorphisms of H

is SU (2)/Z2 , the rotation group.

A commutative 0-dimensional or ¬nite spectral triple is just a collection of

points, for examples see [31]. The simplest example is the two-point space,

A f = CL • CR Hf = C4 ,

(aL , aR ),

« «

aL 0 0 0 0 m 0 0

¬0 0· ¬m 0·

aR 0 ¯ 0 0

Df = , m ∈ C,

ρf (aL , aR ) = ,

0 0 aR

¯ 0 0 0 0 m

¯

0 0 0 aR

¯ 0 0 m 0

«

’1 000

¬0 1 0 0·

0 12

—¦ c c, χf = .

Jf = (117)

0 ’1 0

12 0 0

0 001

316 T. Sch¨cker

u

The algebra has two points = pure states, δL and δR , δL (aL , aR ) = aL . By

Connes™ formula (86), the distance between the two points is 1/|m|. On the

other hand Dt = ‚ — 14 + γ5 — Df is precisely the free massive Euclidean Dirac

/

operator. It describes one Dirac spinor of mass |m| together with its antiparticle.

The tensor product of the calibrating example and the two point space is the

two-sheeted universe, two identical spacetimes at constant distance. It was the

¬rst example in noncommutative geometry to exhibit spontaneous symmetry

breaking [32,33].

One of the major advantages of the algebraic description of space in terms

of a spectral triple, commutative or not, is that continuous and discrete spaces

are included in the same picture. We can view almost commutative geometries

as Kaluza“Klein models [34] whose ¬fth dimension is discrete. Therefore we will

also call the ¬nite spectral triple ˜internal space™. In noncommutative geome-

try, 1-forms are naturally de¬ned on discrete spaces where they play the role of

connections. In almost commutative geometry, these discrete, internal connec-

tions will turn out to be the Higgs scalars responsible for spontaneous symmetry

breaking.

Almost commutative geometry is an ideal playground for the physicist with

low culture in mathematics that I am. Indeed Connes™ reconstruction theorem

immediately reduces the in¬nite dimensional, commutative part to Riemannian

geometry and we are left with the internal space, which is accessible to anybody

mastering matrix multiplication. In particular, we can easily make precise the

last three axioms of spectral triples: orientability, Poincar´ duality and regularity.

e

In the ¬nite dimensional case “ let us drop the ·f from now on “ orientability

means that the chirality can be written as a ¬nite sum,

ρ(aj )Jρ(˜j )J ’1 , aj , aj ∈ A.

χ= a ˜ (118)

j

The Poincar´ duality says that the intersection form

e

©ij := tr χ ρ(pi ) Jρ(pj )J ’1 (119)

must be non-degenerate, where the pj are a set of minimal projectors of A.

Finally, there is the regularity condition. In the calibrating example, it ensures

that the algebra elements, the functions on spacetime M , are not only continuous

but di¬erentiable. This condition is of course empty for ¬nite spectral triples.

Let us come back to our ¬nite, commutative example. The two-point space

is orientable, χ = ρ(’1, 1)Jρ(’1, 1)J ’1 . It also satis¬es Poincar´ duality, there

e

are two minimal projectors, p1 = (1, 0), p2 = (0, 1), and the intersection form is

0 ’1

©= .

’1 2

Forces from Connes™ Geometry 317

5.3 The Minimax Example

It is time for a noncommutative internal space, a mild variation of the two point

space:

«

a0 0 0

¬0 ¯ 0 0·

b

A = H • C (a, b), H = C6 , ρ(a, b) = , (120)

0 0 b12 0

00 0 b

«

M

0 0 0

—

˜ = ¬M 0·

0 0 0

D M= m ∈ C,

¯ , , (121)

M

0 0 0 m

¯—

0M

0 0

«

’12 0 0 0

¬0 0·

0 13 1 0

—¦ c c, χ= .

J= (122)

0 ’12

13 0 0 0

0 0 0 1

The unit is (12 , 1) and the involution is (a, b)— = (a— , ¯ where a— is the Her-

b),

mitean conjugate of the quaternion a. The Hilbert space now contains one

massless, left-handed Weyl spinor and one Dirac spinor of mass |m| and M

is the fermionic mass matrix. We denote the canonical basis of C6 symboli-

cally by (ν, e)L , eR , (ν c , ec )L , ec . The spectral triple still describes two points,

R

δL (a, b) = 1 tr a and δR (a, b) = b separated by a distance 1/|m|. There are

2

still two minimal projectors, p1 = (12 , 0), p2 = (0, 1) and the intersection form

0 ’2

©= is invertible.

’2 2

Our next task is to lift the automorphisms to the Hilbert space and ¬‚uctuate

the ˜¬‚at™ metric D. All automorphisms of the quaternions are inner, the complex

˜

numbers considered as 2-dimensional real algebra only have one non-trivial au-

tomorphism, the complex conjugation. It is disconnected from the identity and

we may neglect it. Then

σ±u (a, b) = (uau’1 , b).

Aut(A) = SU (2)/Z2 σ±u , (123)

The receptacle group, subgroup of U (6) is readily calculated,

«

U2 0 0 0

¬0 0·

U1 0

AutH (A) = U (2) — U (1) U = ,

¯2

0 0 U 0