0 0 0 U1

U2 ∈ U (2), U1 ∈ U (1). (124)

The covariance property is ful¬lled, iU ρ(a, b) = ρ(iU2 a, b) and the projection,

p(U ) = ±(det U2 )’1/2 U2 , has kernel Z2 . The lift,

318 T. Sch¨cker

u

«

±u 0 0 0

¬0 0·

1 0

L(±u) = ρ(±u, 1)Jρ(±u, 1)J ’1 = , (125)

±¯

0 0 u 0

0 0 0 1

is double-valued. The spin group is the image of the lift, L(Aut(A)) = SU (2),

a proper subgroup of the receptacle AutH (A) = U (2) — U (1). The ¬‚uctuated

Dirac operator is

«

±uM

0 0 0

¬ (±uM)— 0·

0 0

D := L(±u)DL(±u)’1 =

˜ . (126)

±uM

0 0 0

(±uM)—

0 0 0

An absolutely remarkable property of the ¬‚uctuated Dirac operator in internal

space is that it can be written as the ¬‚at Dirac operator plus a 1-form:

D = D + ρ(±u, 1) [D, ρ(±u’1 , 1)] + J ρ(±u, 1) [D, ρ(±u’1 , 1)] J ’1 . (127)

˜

The anti-Hermitean 1-form

«

0 h 00

¬ h— 0 0·

0

(’i)ρ(±u, 1) [D, ρ(±u’1 , 1)] = (’i) ,

0 0 00

0 0 00

h := ±uM ’ M (128)

is the internal connection. The ¬‚uctuated Dirac operator is the covariant one with

respect to this connection. Of course, this connection is ¬‚at, its ¬eld strength =

curvature 2-form vanishes, a constraint that is relaxed by the equivalence princi-

ple. The result can be stated without going into the details of the reconstruction

of 2-forms from the spectral triple: h becomes a general complex doublet, not

necessarily of the form ±uM ’ M.

Now we are ready to tensorize the spectral triple of spacetime with the inter-

nal one and compute the spectral action. The algebra At = C ∞ (M )—A describes

a two-sheeted universe. Let us call again its sheets ˜left™ and ˜right™. The Hilbert

space Ht = L2 (S) — H describes the neutrino and the electron as genuine ¬elds,

that is spacetime dependent. The Dirac operator Dt = ˜ — 16 + γ5 — D is the

˜ ˜

‚

/

¬‚at, free, massive Dirac operator and it is impatient to ¬‚uctuate.

The automorphism group close to the identity,

SU (2)/Z2 ] — Di¬(M )

M

Aut(At ) = [Di¬(M ) ((σL , σ±u ), σR ), (129)

now contains two independent coordinate transformations σL and σR on each

sheet and a gauged, that is spacetime dependent, internal transformation σ±u .

The gauge transformations are inner, they act by conjugation i±u . The receptacle

group is

(Spin(4) — U (2) — U (1)).

M

AutHt (At ) = Di¬(M ) (130)

Forces from Connes™ Geometry 319

It only contains one coordinate transformation, a point on the left sheet trav-

els together with its right shadow. Indeed the covariance property forbids to

lift an automorphism with σL = σR . Since the mass term multiplies left- and

right-handed electron ¬elds, the covariance property saves the locality of ¬eld

theory, which postulates that only ¬elds at the same spacetime point can be

multiplied. We have seen examples where the receptacle has more elements than

the automorphism group, e.g. six-dimensional spacetime or the present internal

space. Now we have an example of automorphisms that do not ¬t into the re-

ceptacle. In any case the spin group is the image of the combined, now 4-valued

lift Lt (σ, σ±u ),

(Spin(4) — SU (2)).

M

Lt (Aut(At )) = Di¬(M ) (131)

The ¬‚uctuating Dirac operator is

«

‚L

/ γ5 • 0 0

¬ γ •— ·

‚R

/ 0 0

Dt = Lt (σ, σ±u )Dt Lt (σ, σ±u )’1

˜ = 5 ,(132)

C ‚ L C ’1

0 0 / γ5 •¯

γ5 •— C ‚ R C ’1

0 0 ¯ /

with

e’1 = ‚ L = ie’1 µ a γ a [‚µ + s(ω(e)µ ) + Aµ ],

JJT, / (133)

Aµ = ’ ± u ‚µ (±u’1 ), ‚ R = ie’1 µ a γ a [‚µ + s(ω(e)µ )],

/ (134)

• = ±uM. (135)

Note that the sign ambiguity in ±u drops out from the su(2)-valued 1-form A =

Aµ dxµ on spacetime. This is not the case for the ambiguity in the ˜Higgs™ doublet

• yet, but this ambiguity does drop out from the spectral action. The variable •

is the homogeneous variable corresponding to the a¬ne variable h = • ’ M in

the connection 1-form on internal space. The ¬‚uctuating Dirac operator Dt is still

¬‚at. This constraint has now three parts, e’1 = J (σ)J (σ)T , A = ’ud(u’1 ),

and • = ±uM. According to the equivalence principle, we will take e to be

any symmetric, invertible matrix depending di¬erentiably on spacetime, A to

be any su(2)-valued 1-form on spacetime and • any complex doublet depending

di¬erentiably on spacetime. This de¬nes the new kinematics. The dynamics of

the spinors = matter is given by the ¬‚uctuating Dirac operator Dt , which is

covariant with respect to i.e. minimally coupled to gravity, the gauge bosons

and the Higgs boson. This dynamics is equivalently given by the Dirac action

(ψ, Dt ψ) and this action delivers the awkward Yukawa couplings for free. The

Higgs boson • enjoys two geometric interpretations, ¬rst as connection in the

discrete direction. The second derives from Connes™ distance formula: 1/|•(x)|

is the “ now x-dependent “ distance between the two sheets. The calculation

behind the second interpretation makes explicit use of the Kaluza“Klein nature

of almost commutative geometries [35].

As in pure gravity, the dynamics of the new kinematics derives from the

Chamseddine“Connes action,

320 T. Sch¨cker

u

SCC [e, A, •] = tr f (Dt /Λ2 )

2

’ + a(5 R2 ’ 8 Ricci2 ’ 7 Riemann2 )

2Λc 1

= [ 16πG R

16πG

M

—

+ 1 (Dµ •)— Dµ •

µν

1

2 tr Fµν F 2

2g2

+ O(Λ’2 ),

»|•| ’ 2 µ |•| + 12 |•|2 R ] dV

4 12 2 1

(136)

where the coupling constants are

6f0 2 π ’2 f4

Λc = Λ , G= Λ , a= ,

960π 2

f2 2f2

6π 2 π2 2f2 2

2

, µ2 =

g2 = , »= Λ. (137)

f4 2f4 f4

Note the presence of the conformal coupling of the scalar to the curvature scalar,

+ 12 |•|2 R. From the ¬‚uctuation of the Dirac operator, we have derived the scalar

1

representation, a complex doublet •. Geometrically, it is a connection on the ¬-

nite space and as such uni¬ed with the Yang“Mills bosons, which are connections

on spacetime. As a consequence, the Higgs self coupling » is related to the gauge

2

coupling g2 in the spectral action, g2 = 12 ». Furthermore the spectral action

contains a negative mass square term for the Higgs ’ 1 µ2 |•|2 implying a non-

2

trivial ground state or vacuum expectation value |•| = v = µ(4»)’1/2 in ¬‚at

spacetime. Reshifting to the inhomogeneous scalar variable h = • ’ v, which

vanishes in the ground state, modi¬es the cosmological constant by V (v) and

1

Newton™s constant from the term 12 v 2 R:

3π ’2

Λ c = 6 3 f0 ’ f2

Λ2 , G= Λ. (138)

f2 f4 2f2

Now the cosmological constant can have either sign, in particular it can be zero.

This is welcome because experimentally the cosmological constant is very close

to zero, Λc < 10’119 /G. On the other hand, in spacetimes of large curvature,

like for example the ground state, the positive conformal coupling of the scalar to

the curvature dominates the negative mass square term ’ 1 µ2 |•|2 . Therefore the

2

vacuum expectation value of the Higgs vanishes, the gauge symmetry is unbroken

and all particles are massless. It is only after the big bang, when spacetime loses

its strong curvature that the gauge symmetry breaks down spontaneously and

particles acquire masses.

The computation of the spectral action is long, let us set some waypoints.

The square of the ¬‚uctuating Dirac operator is Dt = ’∆ + E, where ∆ is the

2

covariant Laplacian, in coordinates:

‚

1 — 1H + 1 ωabµ γ ab — 1H + 14 — [ρ(Aµ ) + Jρ(Aµ )J ’1 ] δ ν ν

∆ = g µ˜

ν

µ4 ˜

4

‚x

’“ ν ν µ 14 — 1H

˜

‚

14 — 1H + 1 ωabν γ ab — 1H + 14 — [ρ(Aν ) + Jρ(Aν )J ’1 ] , (139)

— 4

‚xν

Forces from Connes™ Geometry 321

and where E, for endomorphism, is a zero order operator, that is a matrix of

size 4 dim H whose entries are functions constructed from the bosonic ¬elds and

their ¬rst and second derivatives,

[γ µ γ ν — 1H ] Rµν

1

E= (140)

2

«

14 — ••— ’iγ5 γ µ — Dµ • 0 0

¬ ’iγ5 γ µ — (Dµ •)— ·

14 — • — • 0 0

+¬ ·.

’iγ5 γ — Dµ •

14 — ••— µ

0 0

’iγ5 γ µ — (Dµ •)— 14 — • — •

0 0

R is the total curvature, a 2-form with values in the (Lorentz • internal) Lie

algebra represented on (spinors — H). It contains the curvature 2-form R =

dω + ω 2 and the ¬eld strength 2-form F = dA + A2 , in components

Rµν = 1 Rabµν γ a γ b — 1H + 14 — [ρ(Fµν ) + Jρ(Fµν )J ’1 ]. (141)

4

The ¬rst term in equation (141) produces the curvature scalar, which we also (!)

denote by R,

e’1 µ c e’1 ν d γ c γ d ab

1 1

= 1 R14 .

4 Rabµν γ γ (142)

2 4

We have also used the possibly dangerous notation γ µ = e’1 µ a γ a . Finally D is

the covariant derivative appropriate for the representation of the scalars. The

above formula for the square of the Dirac operator is also known as Lich´rowicz

e

formula. The Lich´rowicz formula with arbitrary torsion can be found in [36].

e

Let f : R+ ’ R+ be a positive, smooth function with ¬nite moments,

∞ ∞

f0 = 0 uf (u) du, f2 = 0 f (u) du, f4 = f (0), (143)

f6 = ’f (0), f8 = f (0), ... (144)

Asymptotically, for large Λ, the distribution function of the spectrum is given