Table Of ContentNoncommutative Geometry as the Key to Unlock the Secrets of Space-Time
Ali H.Chamseddine
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0 Physics Department, American University of Beirut, Lebanon
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2 email: [email protected]
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h Abstract
t I give a summary of the progress made on using the elegant construction of Alain
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p Connes noncommutaive geometry to explore the nature of space-time at very high
e
energies. In particular I show that by making very few natural and weak assump-
h
tions about the structure of the noncommutative space, one can deduce the struc-
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ture of all fundamental interactions at low energies.
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1. Introduction
This article is dedicated to Alain Connes on the occasion of his 60th birthday.
I havecome to know Alain well during my first visit to IHES in 1996. I wasimme-
diately overwhelmed with his brilliance and the overflow of his ideas, and within
a short time started to collaborate with him on the interface of noncommutative
geometry, his invention, and the ideas of unification in theoretical physics. This
collaborationhasbeenveryfruitful,andwehavecometoappreciatethemysterious
links between geometry and physics. Many problems remain, but I am optimistic
thatthe challengeoffinding a quantumtheoryofgravityusingthe geometrictools
thatAlaindeveloped,iswithinreach. Atthepersonallevel,IdiscoveredthatAlain
is a very warm person, full of life, and has fantastic sense of humor. I am proud of
his friendship.
What I will present here, is a summary of a forthcoming long article in col-
laboration with Alain, which hopefully will appear in the near future, where a self
contained exposition of the methods of noncommutative geometry applied to par-
ticlephysicsareexplainedinalanguageaccessibletophysicists[1]. Agoodpartof
this forthcoming article will elaborate and build on the results that were first ob-
tainedwiththecrucialinputofthecollaborationwithMarcolli[2]. Inaddition,the
introduction presentin a recentpaper [3] can be used to help introduce the reader
to the general philosophy of our program. Our aim is to provide enough material
to help students and young researchers who wish to learn about this promising
direction of research.
The laws of physics at low energies are well encoded by the action functional
which is the sum of the Einstein-Hilbert action and that of the standard model.
Thesetwopartshavedifferentproperties,thefirstbeingdependentonthegeometry
oftheunderlyingmanifold(M,g)whereg isthemetric,whiletheotherisgoverned
byinternalsymmetriesofagaugegroupGwhichcanbewelldescribedusingthelan-
guageofvectorbundles. Theunderlyingsymmetriesarealsodifferent. Generalrel-
ativity is governedby diffeomorphism invariance (outer automorphisms of (M,g))
whilegaugesymmetriesarebasedonlocalgaugeinvariance(innerautomorphisms).
Thus the natural group of invariance is the semi direct product
G=U ⋊Diff(M)
where
U =C (M,U(1) SU(2) SU(3)).
∞
× ×
It is possible to trace back the failure of finding a unified theory of all interactions
includingquantumgravitytothedifferencebetweenthesetwokindsofsymmetries.
In addition, there are many unanswered questions within the established formula-
tionofthestandardmodel. Forexample,thefollowingquestionshavenocompelling
answer: Whythegaugegroupisspecificallygivenby U(1) SU(2) SU(3)? Why
× ×
the fermions occupy the particular representations that they do? Why there are
three families and why there are 16 fundamental fermions per family? What is
thetheoreticaloriginoftheHiggsmechanismandspontaneousbreakdownofgauge
symmetries? What is the Higgs mass and how to explain all the fermionic masses?
These are only few of the questions that have to be answered by the ultimate uni-
fied theory of all interactions. We shall attempt to answer some of these questions
taking as a starting point the following observations. At energies well below the
2
Planck scale
1 1
M = =2.43 1018 Gev
P
8πG ≡ κ ×
r
gravity can be safely considered as a classical theory. But as energies approach
the Planck scale one expects the quantum nature of space-time to reveal itself,
and for the Einstein-Hilbert action to become an approximationof some deformed
theory. In addition the other three forces must be unified with gravity in such a
way that all interactions will correspond to one underlying symmetry. One thus
would expect that the nature of space-time, and thus of geometry, would change
at Planckianenergies, in such a way that at lower energies, one recoversthe above
pictureofdiffeomorphismandinternalgaugesymmetries. Itisnotrealistictoguess
the exact properties of space-time at Planckian energies and to make directly an
extrapolationof17ordersofmagnitudefromourpresentenergies. Wearetherefore
led to take an indirect approach where we search for the hidden structure in the
functionalofgravitycoupledto the standardmodelat presentenergies. To do this
we shall make a basic conjecture which we will take as a starting point:
Conjecture1. Atsomeenergylevel,space-timeistheproductofacontinuous
four-dimensional manifold times a discrete space F.
The aim then is to find supporting evidence for this conjecture. Once this is
done the next step would be to find the true geometry at Planckian energies, for
which this product in turn is a limit.
This is the minimal extension where no new extra dimensions are assumed.
The task now is to determine with minimal input the properties of the discrete
space F, and construct the associated physical theory. Remarkably, we will show
thatthisinformationwillallowustodeterminethe hiddenstructureofspace-time,
and answer some, but not all (so far) of the questions posed above.
2. A Brief Summary of Alain Connes NCG
The basic idea is based on physics. The modern way of measuring distances
is spectral. The unit of distance is taken as the wavelength of atomic spectra. To
adapt this geometrically the notion of real variable which one takes as a function
f on a set X where f : X R has to be replaced. This is now taken to be a
→
self adjoint operator in a Hilbert space as in quantum mechanics. The space X is
described by the algebra of coordinates which is represented as operators in a
A
fixedHilbertspace . Thegeometryofthenoncommutativespaceisdeterminedin
H
terms of the spectral data ( , , ,J ,γ) . A real, even spectral triple is defined
A H D
by [4], [5]
is an associative algebra with unit 1 and involution .
• A ∗
is a complex Hilbert space carrying a faithful representation π of the
• H
algebra.
D is a self-adjoint operator on with the resolvent (D λ)−1,λ R of
• H − ∈
D compact.
J is an anti–unitary operator on , which is a real structure (charge
• H
conjugation.)
γ is a unitary operator on , the chirality.
• H
We require the following axioms to hold:
J2 =ε , (ε=1 in zero dimensions and ε= 1 in 4 dimensions).
• −
3
[a,bo] = 0 for all a,b , where bo = Jb J 1. This is the zeroth order
∗ −
• ∈ A
condition. This is needed to define the right action on elements of :
H
ζb=boζ, and is a statement that left action and right action commute.
DJ = εJD, Jγ = ε γJ, Dγ = γD where ε,ε,ε 1,1 . The
′ ′′ ′ ′′
• − ∈ {− }
reality conditions resemble the conditions governing the existence of Ma-
jorana (real) fermions.
[[D,a],bo]=0 for all a,b . This is the first order condition.
• ∈A
γ2 = 1 and [γ,a]=0 for all a . These properties allow the decompo-
• ∈A
sition = .
L R
H H ⊕H
is endowed with bimodule structure aζb=aboζ.
• H A
The notion of dimension is governed by growth of eigenvalues, and may
•
be fractal or complex.
has a well defined unitary group
A
= u ; uu∗ =u∗u=1 .
U { ∈A }
The natural adjoint action of on is given by ζ uζu = uJuJ ζ ζ .
∗ ∗
U H → ∀ ∈ H
Then
ζ,Dζ
h i
is not invariant under the above transformation:
(uJuJ∗)D(uJuJ∗)∗ =D+u[D,u∗]+J(u[D,u∗])J∗.
However,the action ζ,D ζ is invariant, where
A
h i
D =D+A+εJAJ 1, A= ai D,bi
A ′ −
i
X (cid:2) (cid:3)
and A = A is self-adjoint. This is similar to the appearance of the interaction
∗
term for the photon with the electrons
iψγµ∂ ψ iψγµ(∂ +ieA )ψ
µ µ µ
→
to maintain invariance under the variations ψ eieα(x)ψ.
→
Thepropertieslistedaboveoftheanti-linearisometryJ : arecharacter-
istic of a real structure of KO-dimension n Z/8 on the speHctr→al tHriple ( , ,D).
∈ A H
The numbers ε,ε,ε 1,1 are a function of n mod 8 given by
′ ′′
∈{− }
n 0 1 2 3 4 5 6 7
ε 1 1 -1 -1 -1 -1 1 1
ε 1 -1 1 1 1 -1 1 1
′
ε 1 -1 1 -1
′′
We take the algebra to be given by a tensor product which geometrically
A
correspondstoaproductspace. Thespectralgeometryof isgivenbytheproduct
A
rule =C (M) where the algebra is finite dimensional, and
∞ F F
A ⊗A A
=L2(M,S) , D =D 1+γ D ,
F M 5 F
H ⊗H ⊗ ⊗
where L2(M,S) is the Hilbert space of L2 spinors, and D is the Dirac operator
M
of the Levi-Civita spin connection on M, D =γµ(∂ +ω ). The Hilbert space
M µ µ
istakentoinclude the physicalfermions. The chiralityoperatorisγ =γ γ
F 5 F
H ⊗
and the reality operator is J =C J , where C is the charge conjugation matrix.
F
⊗
Inordertoavoidthefermiondoublingproblemwherethefermionsζ,ζc,ζ ,ζc ,
∗ ∗
ζ , should not be all independent, it was shown that the finite dimensional
∈ H
4
space must be taken to be of K-theoretic dimension 6 [6], [7], where in this case
(ε,ε,ε”)=(1,1, 1)(so as to impose the condition Jζ =ζ).Thismakesthetotal
′
−
K-theoretic dimension of the noncommutative space to be 10 and would allow to
impose the reality (Majorana) condition and the Weyl condition simultaneously
in the Minkowskian continued form, a situation very familiar in ten-dimensional
supersymmetry. Inthe Euclideanversion,the use ofthe J inthe fermionic action,
would give for the chiral fermions in the path integral, a Pfaffian instead of deter-
minant [6], and will thus cut the fermionic degrees of freedom by a factor of 2. In
other words, in order to have the fermionic sector free of the fermionic doubling
problem we must make the choice
J2 =1, J D =D J , J γ = γ J .
F F F F F F F − F F
Inwhatfollowswewillrestrictourattentiontodeterminationofthefinitealgebra,
and will omit the subscript F.
3. Classification of Finite Noncommutative Spaces
There are two main constraints on the algebra from the axioms of noncom-
mutative geometry. We first look for involutive algebras of operators in such
A H
that,
[a,b0]=0, a,b ,
∀ ∈A
where for any operator a in , a0 = Ja J 1. This is called the order zero
∗ −
H
condition. Weshallassumethattherepresentationsof andJ in areirreducible.
A H
The classification of the irreducible triplets ( , ,J) leads to the following
A H
theorem [8], [9]:
Theorem 2. The center Z( C) is C or C C.
A ⊕
If the center Z( C) is C then we can state the following theorem:
A
Theorem 3. Let be a Hilbert space of dimension n. Then an irreducible
solution with Z( C)=HC exists iff n=k2 is a square. It is given by AC =Mk(C)
A
acting by left multiplication on itself and anti-linear involution
J(x)=x , x M (C).
∗ k
∀ ∈
For C = Mk(C) we have =Mk(C), Mk(R) or Ma(H) for even k = 2a,
where HAis the field of quaternAions [10]. These correspond respectively to the
unitary, orthogonaland symplectic case.
If the center Z( C) is C C then we can state the theorem:
A ⊕
Theorem 4. Let be a Hilbert space of dimension n. Then an irreducible
solution with Z( C) =HC C exists iff n = 2k2 is twice a square. It is given
by AC = Mk(C)A Mk(C)⊕acting by left multiplication on itself and anti-linear
⊕
involution
J(x,y)=(y ,x ), x,y M (C).
∗ ∗ k
∀ ∈
With each of the Mk(C) in C we can have the three possibilities Mk(C),
M (R), or M (H), where k = 2aA. At this point we make the hypothesis that we
k a
are in the “symplectic–unitary” case, thus restricting the algebra to the form
A
=M (H) M (C), k =2a.
a k
A ⊕
5
The dimension of the Hilbert space is n = 2k2, however, because of the reality
condition, these correspond to k2 fundamental fermions , where k = 2a is an even
integer. The first possible value for k is 2 corresponding to a Hilbert space of four
fermions and an algebra = H M (C). This is ruled out because it does not
2
A ⊕
allow to impose grading on the algebra. It is also ruled out by the existence of
quarks. The next possible value for k is 4 thus predicting the number of fermions
to be 16.
In the Z( C)= C case, one can show that it is not possible to have the finite
A
space to be of K-theoretic dimension 6 consistentwith the relation Jγ = γJ [8].
−
We therefore can proceed directly to the second case.
One then takes the grading γ of so that the K-theoretic dimension of the
H
finite space is 6 and this is consistent with the condition Jγ = γJ. It is given by
−
γ(ζ,η)=(γζ, γη).
−
Thisgradingbreaksthealgebra =M (H) M (C),whichisnontriviallygraded
2 4
only for the M (H) component,Ato its even p⊕art:
2
ev = H H M (C) .
R L 4
A ⊕ ⊕
The Dirac operator must connect the two pieces non-trivially, and therefore
must satisfy
[D,Z( )]= 0 .
A 6 { }
The physical meaning of this constraint, is to allow some of the fermions to ac-
quire Majorana masses, realizing the see-saw mechanism, and thus connecting the
fermions to their conjugates.
ev
We haveto lookforsubalgebras ,the evenpartofthe algebra , for
F
A ⊂A A
which [[D,a],b0]=0, a,b . We can state the theorem:
F
∀ ∈A
Theorem 5. Up to automorphisms of Aev, there exists a unique involutive
ev
subalgebraA A ofmaximaldimensionadmittingoff-diagonalDiracoperators
F
⊂
= λ λ q, λ m λ C, q H, m M (C)
F 3
A ⊕ ⊕ ⊕ | ∈ ∈ ∈
H H M (C).
(cid:8)(cid:0) (cid:1) 4 (cid:9)
⊂ ⊕ ⊕
It is isomorphic to C H M (C).
3
⊕ ⊕
4. Tensor Notation
Itishelpfultowritetheresultsobtainedaboutthestandardmodelusingtensor
notation. The Dirac action must take the form
Ψ DNΨ
∗M M N
ψ
where ΨM = ψAA′ and we have denoted ψA′ =ψAc, the conjugate spinor. We
(cid:18) (cid:19)
start with the algebra
=M (C) M (C)
4 4
A ⊕
and denote the spinors by ψ = ψ , A = αI, α = 1, ,4, I = 1, ,4, and
A αI
thus DB =DβJ. The Dirac operator takes the form ··· ···
A αI
′
DB DB
D = A A′ ,
DAB′ DAB′ !
6
where A = αI , α = 1, ,4, I = 1, ,4, and DB′ = DB, DB = DB′
′ ′ ′ ′ ′ ··· ′ ′ ′ ··· ′ A′ A A′ A
and overbar denotes complex conjugation.
Elements of the algebra are matrices aN of the special form:
A M
XβδJ 0
a= α0 I δβ′YJ′ ,
(cid:18) α′ I′ (cid:19)
where Xβ is an element of the first M (C) and YJ′ is an element of the second
α 4 I′
M (C). The reality operator J is defined by
4
0 δβ′δJ′
J = α I complex conjugation.
δβ δJ 0 ×
(cid:18) α′ I′ (cid:19)
In this representation we deduce that ao takes the form
δβYJ 0
ao =Ja∗J−1 = α0I Xαβ′′δIJ′′′ !,
e
where denotes transposition. It is trivial to verifey that [a,bo]=0.
The order one condition is
e [[D,a],bo]=0
If we write
δβWJ 0
bo = α0 I Zβ′δJ′ ,
α′ I′ !
then
B B′
[[D,X],W] ((DY XD)Z W (DY XD))
[[D,a],bo]= A B − − B′ − A
((DX −YD)W −Z(DX −YD))A′ [[D,Y],Z]A′ !
=0.
Explicitely, the first two equations read:
DγKXβ XγDβK WJ WK DγJ Xβ XγDβJ =0
αI γ − α γI K − I αK γ − α γK
Dαγ′(cid:16)IK′YKJ′′ −XαγDγγI′K Z(cid:17)γβ′′ −WIK D(cid:16)αβ′KK′YKJ′′ −XαγDγβK′J′(cid:17)=0.
(cid:16) (cid:17) (cid:16) (cid:17)
We have shown [8], [2], that the only non-zero solution of the second equation is
Dαβ′IK′ =δα1.δ1β.′′δI1δ1K′′k∗νR
which means that there can be only one non-zero single entry in the off-diagonal
′
16 16 matrix DB , and this implies that
× A
DβJ =Dβ δ1δJ +Dβ δiδJδj
αI α(l) I 1 α(q) I j i
YIJ′′ =δI1′′δ1J′′Y11′′ +δIi′′δjJ′′Yij′′
X11.. =Y11′′, X1α. =0, α6=1.,
7
where we have split the index I = 1,i, and I = 1,i. From the property of
′ ′ ′
commutation of the grading operator
1 0
gβ = 2
α 0 1
2
(cid:18) − (cid:19)
[g,a]=0 a M (C),
4
∈
thealgebraM (C)reducesto M (C) M (C) .Wefurtherimposethecondition
of symplectic i4sometry on M (C)2 RM⊕(C2) L
2 R⊕ 2 L
σ 1 (a)σ 1 =a,
2 2 2 2
⊗ ⊗
which reduces it to H H . We will be using the notation
R L
⊕
. .
α=1,2,a where ξ1.,2. ∈HR, ξa ∈HL.
Together with the above condition this implies that
. . . .
Xαβ =δα1δ1β.X11. +δα2δ2β.X11. +δαaδbβXab
and the algebra H H M (C) reduces to
R L 4
⊕ ⊕
C H M (C)
3
⊕ ⊕
because X11.. =Y11′′. Expanding the Dirac action we get
.
ψA∗DABψB+ψ1∗.′1′D1B.′1′ψB+ψA∗DA1′1′ψ1.′1′ +ψA∗′DAB′′ψB′
The spinors can thus be denoted by
ψ =ψ =(ψ ,ψ )
A αI α1 αi
= ψ1.1,ψ2.1,ψa1,ψ1.i,ψ2.i,ψai
(ν ,e ,l ,u ,d ,q ),
(cid:0) R R a Ri Ri ai (cid:1)
≡
where la =(νL,eL) and qai =(uLi,dLi). The component ψ1.′1′ =ψ1c.1 = νRc which
implies that the Dirac action
ψ DBψ +ν ck νRν +c.c
A∗ A B R∗ ∗ R
has only a mixing term for the right-handed neutrinos.
Having determined the structure of the Dirac operator of the discrete space,
we canform the Dirac operatorof the productspace of this discrete space times a
four-dimensional Riemannian manifold:
D =D 1+γ D .
M 5 F
⊗ ⊗
Since D is a 32 32 matrix tensoredwith the 3 3 matrices of generationspace
F
× ×
and with the Clifford algebra, D is 384 384 matrix.
×
To take inner automorphisms into account, we have to evaluate the Dirac op-
erator
D =D+A+JAJ 1,
A −
where
A= a[D,b].
In particular X
AB = aC DDbB bDDB .
A A C D− C D
Note there are no mixing termsXlike D(cid:0)CD′bBD′ because b(cid:1)is block diagonal.
8
Evaluating all components of the full Dirac operator DN , quoting only the
M
result, the full derivation will be given in a forthcoming paper [1], we obtain:
. 1
(D)11.11 =γµ⊗Dµ⊗13, Dµ =∂µ+ 4ωµcd(e)γcd, 13 =generations
(D)1a.11 =γ5⊗k∗ν ⊗ǫabHb kν =3×3 neutrino mixing matrix
.
(D)22.11 =γµ⊗(Dµ+ig1Bµ)⊗13
(D)2a.11 =γ5⊗k∗e⊗Ha
.
(D)11 =γ kν ǫ Hb
a1 5⊗ ⊗ ab
.
(D)21 =γ ke H
a1 5⊗ ⊗ a
i i
(D)b1 =γµ D + g B δb g Wα(σα)b 1 , σα =Pauli
a1 ⊗ µ 2 1 µ a− 2 2 µ a ⊗ 3
(cid:18)(cid:18) (cid:19) (cid:19)
. 2i i
(D)11.ji =γµ⊗ Dµ− 3g1Bµ δij − 2g3Vµm(λm)ji ⊗13, λi =Gell-Mann
(cid:18)(cid:18) (cid:19) (cid:19)
(D)a1.ij =γ5⊗k∗u⊗ǫabHbδij
. i i
(D)22.ji =γµ⊗ Dµ+ 3g1Bµ δij − 2g3Vµm(λm)ji ⊗13
(cid:18)(cid:18) (cid:19) (cid:19)
(D)a2.ij =γ5⊗k∗d⊗Haδij
i i i
(D)bj =γµ D g B δbδj g Wα(σα)bδj g Vm(λm)jδb 1
ai ⊗ µ− 6 1 µ a i − 2 2 µ a i − 2 3 µ i a ⊗ 3
(cid:18)(cid:18) (cid:19) (cid:19)
.
(D)1j =γ ku ǫ Hbδj
ai 5⊗ ⊗ ab i
.
(D)2j =γ kd H δj
ai 5⊗ ⊗ a i
.
(D)11.1′1′ =γ5⊗k∗νRσ generate scale MR by σ →MR
.
(D)11.1′1′ =γ5⊗kνRσ
DAB′′ =DBA, DAB′ =DBA′, DAB′ =DBA′
where B ,Wα and Vm are the U(1), SU(2) and SU(3) gauge fields, and H is a
µ µ µ
complex doublet scalar field and σ is a singlet real scalar field. We have assumed
that the unitary algebra ( ) is restricted to ( ) to eliminate a superfluous
U A SU A
9
U(1) gauge field. Pictorially, the matrix DN has the structure:
M
. . . .
11 21 a1 1i 2i ai
(cid:18) vR eR la uiR diR qiL (cid:19)
.
. (D)1.1 0 (D)a.1 0 0 0
11 11 . 11
. 0 (D)2.1 (D)a.1 0 0 0
21 . 2.1 21
b1 (D)11 (D)21 (D)b1 0 0 0
1.j 0b1 0b1 0a1 (D)1..i 0 (D)a.i
. 1j . 1j
2j 0 0 0 0 (D)2.i (D)a.i
bj . 2.j 2j
0 0 0 (D)1i (D)2i (D)ai
bj bj bj
Needless to say the term ψ DNψ contains all the fermionic terms and their
M∗ M N
interactions in the standard model.
5. The Spectral Action Principle
ThereisashiftofpointofviewinNCGsimilartoFouriertransform,wherethe
usual emphasis on the points x M of a geometric space is now replaced by the
∈
spectrum Σ of the operator D. The existence of Riemannian manifolds which are
isospectral but not isometric shows that the following hypothesis is stronger than
the usual diffeomorphism invariance of the action of general relativity
The physical action depends only on the Σ
Thisisthespectralactionprinciple[11]. Thespectrumisageometricinvariant
and replaces diffeomorphism invariance. We now apply this basic principle to the
noncommutativegeometry defined by the spectrum of the standardmodel to show
that the dynamics of all interactions, including gravity is given by the spectral
action
D 1
A
Trace f + JΨ,D Ψ ,
A
Λ 2h i
(cid:18) (cid:19)
where f is a positive function, Λ a cutoff scale needed to make DA dimensionless,
Λ
and Ψ is a Grassmann variable which represents fermions.
In the case of the cut-off function, f only plays a role through its momenta
f ,f ,f where
0 2 4
∞
f = f(v)vk 1dv, for k >0, , f =f(0).
k − 0
Z
0
These will serve as three free parameters in the model. In this case the action
S [D ]isthe numberofeigenvaluesλ ofD countedwith theirmultiplicities such
Λ A A
that λ Λ.
| |≤
