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ˆ
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

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. 68
( 78 .)



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