ρL (a) := , ρR (b) := 0 b1

a — 1N

0 ¯N

0 0 b1

«

1N — c 0 0

12 — 1N — c 0

ρc (b, c) := 0 0 .

1N — c

ρc (b, c) := ,

¯ 2 — 1N

L R

0 b1 ¯N

0 0 b1

The apparent asymmetry between particles and antiparticles “ the former are

subject to weak, the latter to strong interactions “ will disappear after applica-

tion of the lift L with

0 115N

—¦ complex conjugation.

J= (179)

115N 0

For the sake of completeness, we record the chirality as matrix

«

’18N 0 0 0

¬0 0·

17N 0

χ= . (180)

’18N

0 0 0

0 0 0 17N

The internal Dirac operator

«

M

0 0 0

—

˜ ¬M 0·

0 0

D= ¯ (181)

M

0 0 0

0 M—

¯

0 0

Forces from Connes™ Geometry 331

is made of the fermionic mass matrix of the standard model,

«

10 00

¬ 0 0 — M u — 13 + 0 1 — M d — 1 3 0 ·

¬ · , (182)

M=

0

— Me

0

1

with

« «

mu 0 0 md 0 0

Mu := 0 0 , Md := CKM 0 ms 0 ,

mc (183)

0 0 mt 0 0 mb

«

me 0 0

Me := 0 mµ 0 . (184)

0 0 m„

From the booklet we know that all indicated fermion masses are di¬erent from

each other and that the Cabibbo“Kobayashi“Maskawa matrix CKM is non-de-

generate in the sense that no quark is simultaneously mass and weak current

eigenstate.

We must acknowledge the fact “ and this is far from trivial “ that the ¬nite

spectral triple of the standard model satis¬es all of Connes™ axioms:

• It is orientable, χ = ρ(’12 , 1, 13 )Jρ(’12 , 1, 13 )J ’1 .

• Poincar´ duality holds. The standard model has three minimal projectors,

e

« «

100

p1 = (12 , 0, 0), p2 = (0, 1, 0), p3 = 0, 0, 0 0 0 (185)

000

and the intersection form

«

01 1

© = ’2N 1 ’1 ’1 , (186)

1 ’1 0

is non-degenerate. We note that Majorana masses are forbidden because of the

axiom Dχ = ’χD. On the other hand if we wanted to give Dirac masses to

˜ ˜

all three neutrinos we would have to add three right-handed neutrinos to the

standard model. Then the intersection form,

«

01 1

© = ’2N 1 ’2 ’1 , (187)

1 ’1 0

would become degenerate and Poincar´ duality would fail.

e

• The ¬rst order axiom is satis¬ed precisely because of the ¬rst two of the

six ad hoc properties of the standard model recalled in Sect. 3.3, colour couples

vectorially and commutes with the fermionic mass matrix, [D, ρ(12 , 1, c)] = 0. As

332 T. Sch¨cker

u

an immediate consequence the Higgs scalar = internal 1-form will be a colour

singlet and the gluons will remain massless, the third ad hoc property of the

standard model in its conventional formulation.

• There seems to be some arbitrariness in the choice of the representation under

C b. In fact this is not true, any choice di¬erent from the one in equations

(179,179) is either incompatible with the axioms of spectral triples or it leads to

charged massless particles incompatible with the Lorentz force or to a symmetry

breaking with equal top and bottom masses. Therefore, the only ¬‚exibility in

the fermionic charges is from the choice of the central charges [40].

Central Charges. The standard model has the following groups,

SU (2) — U (1) — U (3)

U (A) = u = (u0 , v, w), (188)

Z2 — U (1) — U (1)

U c (A) = uc = (uc0 , vc , wc 13 ),

—

U n (A) = SU (2) U (3) (u0 , w),

Z2 —

nc

U (A) = U (1) (uc0 , wc 13 ),

—

Int(A) = [SU (2) SU (3)]/“ uσ = (uσ0 , wσ ),

Z2 — Z3

“= γ = (exp[’m0 2πi/2], exp[’m2 2πi/3]),

with m0 = 0, 1 and m2 = 0, 1, 2. Let us compute the receptacle of the lifted

automorphisms,

AutH (A) (189)

= [U (2)L —U (3)c —U (N )qL —U (N ) L —U (N )uR —U (N )dR ]/[U (1)—U (1)]

—U (N )eR .

The subscripts indicate on which multiplet the U (N )s act. The kernel of the

projection down to the automorphism group Aut(A) is

ker p = [U (1)—U (1)—U (N )qL —U (N ) L —U (N )uR —U (N )dR ]/[U (1)—U (1)]

—U (N )eR , (190)

and its restrictions to the images of the lifts are

ker p © L(Int(A)) = Z2 — Z3 , ker p © L(U n (A)) = Z2 — U (1). (191)

The kernel of i is Z2 — U (1) in sharp contrast to the kernel of L, which is trivial.

ˆ

ˆ

The isospin SU (2)L and the colour SU (3)c are the image of the lift L. If q = 0,

the image of consists of one U (1) wc = exp[iθ] contained in the ¬ve ¬‚avour

U (N )s. Its embedding depends on q:

L(12 , 1, wc 13 ) = (wc ) (192)

= diag (uqL 12 — 1N — 13 , u L 12 — 1N , uuR 1N — 13 , udR 1N — 13 , ueR 1N ;

uqL 12 — 1N — 13 , u L 12 — 1N , uuR 1N — 13 , udR 1N — 13 , ueR 1N )

¯ ¯ ¯ ¯ ¯

Forces from Connes™ Geometry 333

with uj = exp[iyj θ] and

= ’q1 , ydR = ’q1 + q2 , yeR = ’2q1 .(193)

yqL = q2 , y yuR = q1 + q2 ,

L

Independently of the embedding, we have indeed derived the three fermionic con-

ditions of the hypercharge ¬ne tuning (57). In other words, in noncommutative

geometry the massless electro“weak gauge boson necessarily couples vectorially.

Our goal is now to ¬nd the minimal extension that renders the extended

lift symmetric, L(’u0 , ’w) = L(u0 , w), and that renders L(12 , w) single-valued.

The ¬rst requirement means { q1 = 1 and q2 = 0 } modulo 2, with

˜ ˜

q1

˜ q1 0

’

1

= . (194)

3

q2

˜ q2 1

The second requirement means that q has integer coe¬cients.

˜

0

The ¬rst extension which comes to mind has q = 0, q =

˜ . With

’1/3

respect to the interpretation (168) of the inner automorphisms, one might object

that this is not an extension at all. With respect to the generic characterization

(161), it certainly is a non-trivial extension. Anyhow it fails both tests. The most

general extension that passes both tests has the form

2z1 + 1 6z1 + 3

z1 , z2 ∈ Z.

q=

˜ , q= , (195)

2z2 6z2 + 1

Consequently, y L = ’q1 cannot vanish, the neutrino comes out electrically neu-

tral in compliance with the Lorentz force. As common practise, we normalize the

hypercharges to y L = ’1/2 and compute the last remaining hypercharge yqL ,

1

+ z2

q2

=6

yqL = . (196)

2q1 1 + 2z1

We can change the sign of yqL by permuting u with dc and d with uc . Therefore

it is su¬cient to take z1 = 0, 1, 2, ... The minimal such extension, z1 = z2 = 0,

recovers nature™s choice yqL = 1 . Its lift,

6

L(u0 , w) = ρ(u0 , det w, w)Jρ(u0 , det w, w)J ’1 , (197)

is the anomaly free fermionic representation of the standard model considered

as SU (2) — U (3) . The double-valuedness of L comes from the discrete group

Z2 of central quaternionic unitaries (±12 , 13 ) ∈ Z2 ‚ “ ‚ U nc (A). On

the other hand, O™Raifeartaigh™s [5] Z2 in the group of the standard model (45),

±(12 , 13 ) ∈ Z2 ‚ U nc (A), is not a subgroup of “ . It re¬‚ects the symmetry

of L.

Fluctuating Metric. The stage is set now for ¬‚uctuating the metric by means

of the extended lift. This algorithm answers en passant a long standing question

in Yang“Mills theories: To gauge or not to gauge? Given a fermionic Lagrangian,

334 T. Sch¨cker

u

e.g. the one of the standard model, our ¬rst re¬‚ex is to compute its symmetry

group. In noncommutative geometry, this group is simply the internal receptacle

(190). The painful question in Yang“Mills theory is what subgroup of this sym-

metry group should be gauged? For us, this question is answered by the choices

of the spectral triple and of the spin lift. Indeed the image of the extended lift

is the gauge group. The ¬‚uctuating metric promotes its generators to gauge

bosons, the W ± , the Z, the photon and the gluons. At the same time, the Higgs

representation is derived, equation (174):

«

ˆ

0 H 0 0

ˆ—

¬H 0·

0 0

H = ρ(u0 , det w, w)[D, ρ(u0 , det w, w)’1 ] =

˜ (198)

0 0 0 0

0 0 0 0

with

«

¯

’h2 Md

h1 M u

— 13 0

¬ ·

¯

h2 Mu h1 Md

H=¬ ·

ˆ (199)

¯

’h2 Me

0 ¯

h1 Me