«

N N

(wc ) := ρ q1,j1

(wcjM )qM,jM ,

(wcj1 ) , ..., (167)

j1 =1 jM =1

N N

(wcjM +N )qM +N ,jM +N 1nN Jρ(...) J ’1

(wcjM +1 )qM +1,jM +1 1n1 , ...,

jM +1 =1 jM +N =1

with the (M + N ) — N matrix of charges qkj . The extension satis¬es indeed

p( (wc )) = 1 ∈ Int(A) for all wc ∈ U nc (A).

Having adjoined the harmful, continuous central unitaries, we may now stream

line our notations and write the group of inner automorphisms as

«

N

Int(A) = — SU (nk ) /“ [wσ ] = [(wσ1 , ..., wσN )] mod γ, (168)

k=1

where “ is the discrete group

N

“ = — Znk (z1 1n1 , ..., zN 1nN ),

k=1

zk = exp[’mk 2πi/nk ], mk = 0, ..., nk ’ 1 (169)

and the quotient is factor by factor. This way to write inner automorphisms

is convenient for complex matrices, but not available for real and quaternionic

matrices. Equation (161) remains the general characterization of inner automor-

phisms.

The lift L(wσ ) = (L —¦ i’1 )(wσ ), wσ = w mod U nc (A), is multi-valued with,

ˆ

N

depending on the representation, up to |“ | = j=1 nj values. More precisely the

multi-valuedness of L is indexed by the elements of the kernel of the projection

p restricted to the image L(Int(A)). Depending on the choice of the charge

matrix q, the central extension may reduce this multi-valuedness. Extending

harmless central unitaries is useless for any reduction. With the multi-valued

group homomorphism

(hσ , hc ) : U n (A) ’’ Int(A) — U nc (A)

’’ ((wσj , wcj )) = ((wj (det wj )’1/nj , (det wj )1/nj )),(170)

(wj )

we can write the two lifts L and together in closed form L : U n (A) ’ AutH (A):

Forces from Connes™ Geometry 327

L(w) = L(hσ (w)) (hc (w))

«

N N

= ρ q1,j1

˜

(det wjM )qM,jM ,

˜

(det wj1 ) , ...,

j1 =1 jM =1

N N

(det wjN +M )qN +M ,jN +M

(det wjM +1 )qM +1,jM +1 , ..., wN

˜ ˜

w1

jM +1 =1 jN +M =1

— Jρ(...)J ’1 . (171)

We have set

« « « ’1

n1

0M —N

q := q ’

..

˜ . (172)

.

1N —N nN

Due to the phase ambiguities in the roots of the determinants, the extended

lift L is multi-valued in general. It is single-valued if the matrix q has integer

˜

0

, then q = 0 and L(w) = L(w). On the other hand, q = 0

ˆ

entries, e.g. q = ˜

1N

gives L(w) = L(i’1 (hσ (w))), not always well de¬ned as already noted. Unlike the

ˆ

extension (157), and unlike the map i, the extended lift L is not necessarily even.

We do impose this symmetry L(’w) = L(w), which translates into conditions

on the charges, conditions that depend on the details of the representation ρ.

Let us note that the lift L is simply a representation up to a phase and

as such it is not the most general lift. We could have added harmless central

unitaries if any present, and, if the representation ρ is reducible, we could have

chosen di¬erent charge matrices in di¬erent irreducible components. If you are

not happy with central extensions, then this is a sign of good taste. Indeed

commutative algebras like the calibrating example have no inner automorphisms

and a huge centre. Truly noncommutative algebras have few outer automorphism

and a small centre. We believe that almost commutative geometries with their

central extensions are only low energy approximations of a truly noncommutative

geometry where central extensions are not an issue.

6.2 Output

From the input data of a ¬nite spectral triple, the central charges and the three

moments of the spectral function, noncommutative geometry produces a coupled

to gravity. Its entire Higgs sector is computed from the input data, Fig. 6. The

Higgs representation derives from the ¬‚uctuating metric and the Higgs potential

from the spectral action.

To see how the Higgs representation derives in general from the ¬‚uctuating

Dirac operator D, we must write it as ˜¬‚at™ Dirac operator D plus internal 1-

˜

form H like we have done in equation (127) for the minimax example without

328 T. Sch¨cker

u

fj

f

f,q f

S

Fig. 6. Connes™ slot machine

extension. Take the extended lift L(w) = ρ(w)Jρ(w)J ’1 with the unitary

N N

q1j1

˜

(det wjM )qM jM ,

˜

w= (det wj1 ) , ..., (173)

j1 =1 jM =1

N N

qM +1,jM +1

˜

(det wjN +M )qN +M ,jN +M .

˜

w1 (det wjM +1 ) , ..., wN

jM +1 =1 jN +M =1

Then

D = LDL’1

˜

’1

= ρ(w) Jρ(w)J ’1 D ρ(w) Jρ(w)J ’1

˜

= ρ(w) Jρ(w)J ’1 D ρ(w’1 ) Jρ(w’1 )J ’1

˜

= ρ(w)Jρ(w)J (ρ(w’1 )D + [D, ρ(w’1 )])Jρ(w’1 )J ’1

’1 ˜ ˜

= Jρ(w)J DJρ(w )J + ρ(w)[D, ρ(w’1 )]

’1 ˜ ’1 ’1 ˜

= Jρ(w)Dρ(w’1 )J ’1 + ρ(w)[D, ρ(w’1 )]

˜ ˜

= J(ρ(w)[D, ρ(w’1 )] + D)J ’1 + ρ(w)[D, ρ(w’1 )]

˜ ˜ ˜

= D + H + JHJ ’1 ,

˜ (174)

with the internal 1-form, the Higgs scalar, H = ρ(w)[D, ρ(w’1 )]. In the chain

˜

(174) we have used successively the following three axioms of spectral triples,

[ρ(a), Jρ(˜)J ’1 ] = 0, the ¬rst order condition [[D, ρ(a)], Jρ(˜)J ’1 ] = 0 and

˜

a a

[D,

˜ J] = 0. Note that the unitaries, whose representation commutes with the

internal Dirac operator, drop out from the Higgs, it transforms as a singlet

under their subgroup.

The constraints from the axioms of noncommutative geometry are so tight

that only very few s can be derived from noncommutative geometry as pseudo

forces. No left-right symmetric model can [41], no Grand Uni¬ed Theory can [42],

Forces from Connes™ Geometry 329

Yang-Mills-Higgs

left-right symm.

NCG

GUT

standard model supersymm.

Fig. 7. Pseudo forces from noncommutative geometry

for instance the SU (5) model needs 10-dimensional fermion representations,

SO(10) 16-dimensional ones, E6 is not the group of an associative algebra.

Moreover the last two models are left-right symmetric. Much e¬ort has gone

into the construction of a supersymmetric model from noncommutative geome-

try, in vain [43]. The standard model on the other hand ¬ts perfectly into Connes™

picture, Fig. 7.

6.3 The Standard Model

The ¬rst noncommutative formulation of the standard model was published by

Connes & Lott [33] in 1990. Since then it has evolved into its present form [18“

20,28] and triggered quite an amount of literature [44].

Spectral Triple. The internal algebra A is chosen as to reproduce SU (2) —

U (1) — SU (3) as subgroup of U (A),

A = H • C • M3 (C) (a, b, c). (175)

The internal Hilbert space is copied from the Particle Physics Booklet [13],

HL = C2 — CN — C3 • C 2 — CN — C , (176)

HR = C — CN — C3 • C — C N — C3 • C — CN — C . (177)

In each summand, the ¬rst factor denotes weak isospin doublets or singlets, the

second denotes N generations, N = 3, and the third denotes colour triplets or

singlets. Let us choose the following basis of the internal Hilbert space, counting

330 T. Sch¨cker

u

fermions and antifermions (indicated by the superscript ·c for ˜charge conju-

gated™) independently, H = HL • HR • HL • HR = C90 :

c c

u c t νe νµ ν„

, , , , , ;

d sL b e µ „

L L L L L

uR , cR , tR ,

eR , µR , „R ;

dR , sR , bR ,

c c c c c c

u c t νe νµ ν„

, , , , , ;

d sL b e µ „

L L L L L

uc , cc , tc ,

ec , µc , c

R R R „R .

dR , sc ,

c

bc , R R

R R

This is the current eigenstate basis, the representation ρ acting on H by

«

ρL 0 0 0

¬ 0 ρR 0 0·

ρ(a, b, c) := (178)

c

0 0 ρL 0

¯

0 ρc

0 0 ¯R

with

«

b1N — 13 0 0

a — 1N — 13 0 0 ,