Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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formulation for the Newtonian theory of space,

time, gravitation, and hydrodynamics.

The possibility to split the connection  into a flat

part which is independent of matter and a tensorial

part depending on matter and given by the vector

field g



= s





, 

(with  from eqn [11]), arises only

from supplementing the local laws [7] by the global,

resp. asymptotic, conditions (1)–(2), [14], [17]

stated above. The introduction of inertial coordi-

nates is then convenient, but not necessary. In

noninertial, rigid frames of referenc e, 









gives

rise to inertial forces.

It should be possible to define spatial asymptotic

flatness in the CFF, but that has not been done.

Remarks on Newtonian Cosmology

In cosmology, the conditions (2) and [17] of the

last subsection are not appropriate. Instead one

keeps the laws [7] and adds to the m eqn [9],so

that with respect to nonrotating coordinates the

laws [8] with w = 0andeqn [10] remain valid.

Then, there are no longer inertial coordinate

systems, and the potential U is not a 4-scalar.

For a slightly different approach, see Ru¨ede and

Straumann (1997).

For the purpose of this article, the term

‘‘cosmological model’’ w ill be applied to those

solutions of the laws [7] and [9] which satisfy >0

and which have a symmetry group which acts

transitively on the set of world lines representing

the motion of the fluid. This strong symmetry

assumption determ ines the time-evolution even in

the ‘‘Newtonian’’ case  = 0 in spite of the absence

of an evolution equation for the gravitational

field g.

Newtonian Limits of Families

of GR Solutions

The disc ussion in the sections ‘‘The Car tan–

Fried richs form alism’’ an d ‘‘N ewton’s theory in

spacetime form’’ suggests the following:

Definition 1 Let a family F() = (t



(), ...)of

basic fields parametrized by , obeying the laws [7]

of the CFF, be given for 0  <a. We assume the

underlying mani folds M() to be open submanifolds

of a fixed manifold M such that M( 

)  M(

)if



<

and



M() = M. Then we write

lim

 !0

FðÞ¼Fð0Þ½19

if the fields of F() and their first derivatives

converge pointwise to those of F(0).

F(0) is then said to be a CF limit of the sequence of

(-rescaled) solutions F() of GRT. If the fields of a

-family of GR solutions (>0) and their first

derivatives converge for  ! 0 locally uniformly,

then the limit fields satisfy eqns [7].IfF(0) has the

additional property [9], the limit is locally Newtonian.

On the basis of the section ‘‘The Car tan–

Fried richs formalism ’’ one may con jecture that if

eqn [19] holds and the F() for >0 are spatially

asymptotically flat, F(0) will represent an asympto-

tically flat Newtonian spacetime. Examples such as

Example 1 below are in agreement with this

conjecture, but a general proof is not known.

Example 1 The interior solution for a static,

spherically symmetric fluid ball of constant energy

density (Schwarzschild 1916) is given by

þ r

ðd#

þ sin

# d’



ð3a

 aÞ

 ¼ const:>0; p ¼ c

a  a

 a

U ¼

 a

; aðrÞ¼ 1 

8

2





1=2

¼aðr

Inserting into these expressions the parameter

 = c

2

and treating  and r

as -independent

constants results in a -family with 0  <

((8=3)kr

)

1

. The limit solution represents a

Newtonian fluid ball of constant mass density .

The Schwarzschild vacuum fields belonging to these

fluid balls also have the appropriate Newtonian

limits. The resulting complete spacetimes are asymp-

totically flat. A dimensionless small parameter

which could be used instead of  to measure the

deviation of the GR solut ion from its Newtonian

limit is the ratio of Schwarzschil d radius and the

geometric radius:

2kM

8

kr

Example 2 A Friedmann–Lemaitre cosmological

model of GR containing dust and radiation is given by

¼ R

ðtÞ



d

1 ð1=4ÞðE=c

Þ



ðÞ

 c

where R(t) obeys



8



¼ E

Newtonian Limit of General Relativity 507

M is a mass constant,  = M=R

is the mass densi ty

of ‘‘dust,’’ S is an entropy constant,  = S=R

the

energy density a nd p = (1=3) the pressure of

radiation; and E is a constant of dimension

(speed)

. The world lines of the fluid elements

are given by 

= const. (Lagrangian comoving

coordinates).

Taking E, M, S constant and  = c

2

as a parameter

provides a -family of GR models with Newtonian

limit. In the limit, t is the Newtonian time, and the

spatial metric R



d

describes an expanding

Euclidian space R

(if E  0) or an open ball of

radius 2R(t)init(ifE > 0). In the coordinates (

, t)

the connection does not have the ‘‘Newtonian’’

components [8a], instead its nonvanishing compo-

nents are 

= (

R=R)

. In local inertial coordinates

= R

centered on the particle with 

= 0(which

could be any particle because of the homogeneity of

the model), the spatial metric is dx

,andthe

connection components are Newtonian, with

U = (2=3)kx

and U = 4k . In the limit, the

radiation no longer influences the expansion; one gets

the Newtonian dust models (eqn [9] is satisfied). The

connection is, of course, not asymptotically flat. The

curvature tensor R









= (4=3) kt





exhibits

homogeneity and isotropy. The Gaussian sectional

curvature of the 3-space at time t is K = E=R

.As

a dimensionless smallness parameter one can take

E=c

. In the ‘‘open’’ models, with E  0, the

coordinates 

cover the whole 3-manifold of fluid

particles, while in the ‘‘closed’’ case, E < 0, one

particle, the antipode of 

= 0 on the 3-sphere, is not

covered. That particle is missing in the Newtonian

limit model. In the Newtonian case the expanding

Euclidian space R

can be replaced by a torus; in the

GR cases this is possible only for E = 0.

Many examples of GR fam ilies with Newtonian

limits are known (see, e.g., Ehlers (1997) and

references therein). An example of a -family

which has an almost Newtonian limit which does

not satisfy eqn [9] is pr ovided by NUT spacetimes

(see Ehlers 1997), interpreted as due t o a

gravitomagnetic monopole (Lynden-Bell and

Nouri-Zonez 1998).

Applications and Problems

Can one construct, for a given Newtonian solution

N,a-family of GR solutions which converges to

N? Some answers are known and listed below.

U Heilig (1995) has shown: given a solution t o

the Euler–Poisson equations representing a station-

ary, rigidly rotating, self-gravitating fluid body

with its surrounding gravitational field, there exists

a -family of corresponding solutions to the

Einstein–Euler system having the given solutions

as its limit.

The proof is based on the fact that one can

reformulate eqns [1], [2] in terms of harmonic

coordinates and new dependent gravitational vari-

ables instead of g



such that the new equations

given in Lottermoser (1992) are analytic in  and

reduce, for  = 0, to the Euler–Poisson system. In the

stationary case these equations are elliptic for   0.

Using appropriate function spaces, Heilig shows, via

the implicit function theorem, that a solution for

 = 0 can be extended to small, positive values of .

Since L Lichtenstein has constructed solutions as

assumed in the theorem, the existence of GR

solutions follows.

The gravitational part of the system of equations

referred to above is hyperbolic for >0, but

becomes elliptic for  = 0, whereas the fluid equa-

tions remain hyperbolic. In spite of this difficulty

Rendall (199 4) has shown that -families of time-

dependent, asymptotically flat solutions to the

Einstein–Vlasov system representing gravitating

systems of collisionless particles have Poisson–

Vlasov limits, and that any Poisson–Vlasov solution

can be so obtained.

Lottermoser (1992) succeeded in proving the exis-

tence of -families of solutions to the Einstein constraint

equations which have Newtonian initial data as limits.

Nothing seems to be known about solutions evolving

from such data. Lottermoser has given an interesting

discussion concerning possible extension of his work

which apparently has gone unnoticed.

Rendall (1992) has defined and analyzed post-

Newtonian expansions to Einstein’s equations and

their solvability, assuming -families whose t



, s



are a few times differentiable in  =

ﬃﬃﬃ



at  = 0. He

found that for low orders the equations have

asymptotically flat solutions, but that at order 

divergences occur for general Newtonian seed

solutions. Modifications of the method to overcome

these difficulties have been considered by Rendall

and others; the problem is open.

In cosmology, one uses homogeneous back-

ground models and studies their perturbations.

The latter are frequently based on Newtonian

equations. This can perhaps be justified as follows.

According to Example 2 the fields of Friedmann–

Lemaitre models differ fr om their Newtonian limits

by arbitrarily small amounts uniformly in

spacetime regions where the terms involving  are

small, that is,

 RðtÞ;

ﬃﬃﬃﬃﬃﬃ

jEj

jxjRðtÞ

508 Newtonian Limit of General Relativity

Additional conditions will be needed to ensure that

Newtonian perturbations approximate relativistic

ones and that gravitational wave perturbations can

be neglected.

See also: Cosmology: Mathematical Aspects; Einstein

Equations: Exact Solutions; General Relativity: Overview;

Gravitational Lensing; Shock Wave Refinement of the

Friedman–Robertson–Walker Metric.

Further Reading

Cartan E (1923) Sur les varie´te´s a` connexion affine et la the´orie de

la relativite´ge´ne´ralis

ee. Annales Scientifiques de l’Ecole

Normale Supe´rieure XL: 325–412.

Cartan E (1924) Sur les varie´te´s a` connexion affine et la the´orie de

la relativite´ge´ne´ralis

ee. Annales Scientifiques de l’Ecole

Normale Supe´rieure XLI: 1–25.

Ehlers J (1981) U

ber den Newtonschen Grenzwert der Ein-

steinschen Gravitationstheorie. In: Nitsch J et al. (eds.)

Grundlagenprobleme der modernen Physik, pp. 65–84.

Mannheim: Bibliografisches Insitut Wissenschaftsverlag.

Ehlers J (1997) Examples of Newtonian limits of relativistic

spacetimes. Classical and Quantum Gravity 14: A119–A126.

Friedrichs KO (1927) Eine invariante Formulierung des New-

tonschen Gravitationsgesetzes und des Grenzu¨ berganges vom

Einsteinschen zum Newtonschen Gesetz. Mathematische

Annalen 98: 566–575.

Heilig U (1995) On the existence of rotating stars in general

relativity. Communications in Mathematical Physics 166:

457–493.

Ku¨ nzle HP (1972) Galilei and Lorentz structures on space-time:

comparison of the corresponding geometry and physics.

Annales de l’Institut Henri Poincare´ XVII: 337–362.

Lottermoser M (1992) A convergent post-Newtonian approxima-

tion for the constraint equations in general relativity. Annales

de l’Institut Henri Poincare´ 57: 279–317.

Lynden-Bell D and Nouri-Zonoz M (1998) Classical monopoles:

Newton, NUT space, gravomagnetic lensing, and atomic

spectra. Reviews of Modern Physics 70: 427–445.

Rendall AD (1992) On the definition of post-Newtonian

approximations. Proceedings of the Royal Society of London

Series A 438: 341–360.

Rendall AD (1994) The Newtonian limit for asymptotically flat

solutions of the Vlasov–Einstein system. Communications in

Mathematical Physics 163: 89–112.

Ru¨ ede Ch and Straumann N (1997) On Newton–Cartan

cosmology. Helvetica Physica Acta 318–335.

Noncommutative Geometry and the Standard Model

T Schu¨ cker, Universite

de Marseille, Marseille,

France

Introduction

The aim of this contribution is to explain how

Connes derives the standard model of electromag-

netic, weak, and strong forces from noncommuta-

tive geometry. The reader is supposed to be aware of

two other derivations in fundamental physics: the

derivation of the Balmer–Rydberg formula for the

spectrum of the hydrogen atom from quantum

mechanics and Einstein’s derivation of gravity from

Riemannian geometry.

At the end of the nineteenth century, new physics

was discovered in atoms, namely their discrete

spectra. Balmer and Rydberg succeeded to put

order into the fast-growing set of experimental

results with the help of a phenomenological ansatz

for the frequencies  of the spectral rays of, for

example, the hydrogen atom,

 ¼ gðn

 n

Þ; n

2 N; q 2 Z; g 2 R ½1

The integer variables n

and n

reflect the

discreteness of the spectrum. On the other hand,

the discr ete parame ter q and the continuous

parameter g were fitted by experiment: q = 2

and g = 3.289  10

Hz, the famous Rydberg

constant. Later quantum me chanics was discov-

ered and allowed to derive the Balmer–Rydberg

ansatz and to constrain its parameters:

q ¼ 2 and g ¼

4h

ð4

½2

in beautiful agreement with the anterior experi-

mental fit.

The Standard Model

We propose to introduce the standard model (see

Standard Model of Particle Physics) in analogy with

the Balmer–Rydberg formula (Table 1).

Table 1 An analogy between atomic and particle physics elements

Atomic physics Particle physics

New physics Discrete spectra Forces mediated by

gauge bosons

Ansatz  = g(n

 n

) Yang–Mills–Higgs

models

Experimental

fit

q = 2, g = 3.289  10

Hz Standard model

Underlying

theory

Quantum mechanics Noncommutative

geometry

Noncommutative Geometry and the Standard Model 509

The Yang–Mills–Higgs Ansatz

The variables of this Lagrangian ansatz are spin-1

particles A, spin-(1/2) particles decomposed into left-

and right-handed components = (

) and spin-

0 particles ’. There are four discrete parameters, a

compact real Lie group G, the ‘‘gauge group,’’ and

three unitary representations on complex Hilbert

spaces H

, H

,andH

. The spin-1 particles come in a

multiplet living in the complexified of the Lie algebra

of G, A 2 Lie(G )

. The left- and right-handed spinors

come in multiplets living in the Hilbert spaces,

, respectively. The (Higgs) scalar is

another multiplet, ’ 2H

. The Yang–Mills–Higgs

Lagrangian, together with its Feynman diagrams, is

spelled out in Table 2.

There are several continuous parameters: the

gauge coupling g 2 R

, the Higgs self-couplings

,  2 R

, and several Yukawa couplings g

2 C.

Let us choose G = U(1) 3 e

i

. Its irreducible unitary

representations are all one-dimensional, H= C 3

characterized by the charge q 2 Z: (e

i

) = e

iq

Then with q

= q

and H

= {0}, we get Maxwell’s

theory with the photon (or gauge boson or 4-potential)

A coupled to the Dirac theory of a massless spinor of

electric charge q

whose (relativistic) wave function is

. The gauge coupling is given by g = e=

ﬃﬃﬃﬃ



.Gauge

invariance of the Yang–Mills–Higgs Lagrangian

implies, via Noether’s theorem, electric charge con-

servation in this case (see Symmetries and Conserva-

tion Laws).

Yang–Millsmodelsaretherefore simply nonabelian

generalizations of electromagnetism where the abelian

gauge group U(1) is replaced by any compact real Lie

group. We insist on a compact group because all

irreducible unitary representations of compact groups

are finite dimensional. Finally, the Higgs scalar is

added to give masses to spinors and gauge bosons via

spontaneous symmetry breaking (see Symmetry

Breaking in Field Theory).

We use compact groups and unitary representations

as (discrete) parameters. One motivation is Noether’s

theorem and conserved quantities. The other comes

from Wigner’s theorem: the irreducible unitary

representations of the Poincare´ group are classified

by mass and spin. Its orthonormal basis vectors

are classified by energy–momentum and by the

z-component of angular momentum. This theorem

leads to the widely accepted definition of a particle as

an orthonormal basis vector in a Hilbert space

H carrying a unitary representation  of a group G.

A precious property of the Yang–Mills–Higgs

ansatz is its perturbative renormalizability necessary

for fine-structure calculations like the anomalous

magnetic moment of the muon.

The Experimental Fit

Physicists have spent some 30 years and some 10

Swiss

Francs to distill the fit (Particle Data Group 2004):

G ¼ SUð2ÞUð1ÞSUð 3Þ=ðZ

 Z

Þ½3

ð2;

; 3Þð2; 

; 1Þ



½4

ð1;

; 3Þð1; 

; 3Þð1; 1; 1Þ



½5

¼ð2; 

; 1Þ½6

Here (n

, y, n

) denotes the tensor product of an

-dimensional representation of SU(2), ‘‘(weak) iso-

spin,’’ an n

-dimensional representation of SU(3),

‘‘color,’’ and the one-dimensional representation of

Table 2 The Yang–Mills–Higgs Lagrangian and its Feynman

diagrams

L[A; ;’] =

tr(@









 @







)

þg tr(@







; A



])

þg

tr([A



; A



][A



; A



])



þig



(







)(A



)



’



’

gf(





)’)



’ þ @



’







)’g

(





)’)







)’

þ’



’’



’





’



’

þg



’ þ



’



510 Noncommutative Geometry and the Standard Model

U(1) with ‘‘hyper charge’’ y. For historical reasons, the

hypercharge is an integer multiple of 1/6. This is

irrelevant: in the abelian case, only the product of the

hypercharge with its gauge coupling is measurable, and

we do not need multivalued representations, which are

characterized by noninteger, rational hypercharges. In

the direct sum, we recognize the three generations of

fermions,thequarks,‘‘up,down,charm,strange,top,

bottom,’’ are SU(3) triplets, the leptons, ‘‘electron,

, ’’ and their neutrinos, are color singlets. The basis

of the fermion representation space is



;



;







;







;







; c

; t

; e

;

;

; s

; b

;

The parentheses indicate isospin doublets.

The eight gauge bosons associated with su(3) are

called gluons. Warning: the U(1) is not the one of electric

charge; it is called hypercharge, the electric charge is a

linear combination of hypercharge and weak isospin.

This mixing is necessary to give electric charges to the W

bosons. The W

and W



are pure isospin states, while

the Z

and the photon are (orthogonal) mixtures of the

third isospin generator and hypercharge.

As the group G contains three simple factors,

there are three gauge couplings,

¼ 0:6518  0:0003

¼ 0:3574  0:0001

¼ 1:218  0:01

½7

The Higgs couplings are usually expressed in terms

of the W and Higgs masses:

v ¼ 80:419  0:056 GeV ½8

’

¼ 2

ﬃﬃﬃ



v > 98 GeV ½9

with the vacuum expectation value v := (1=2)=

ﬃﬃﬃ



Because of the high degree of reducibility of the spin-

(1/2) representations there are 27 complex Yukawa

couplings. They constitute the fermionic mass matrix

which contains the fermion masses and mixings:

¼ 0:510998902  0:000000021 MeV

¼ 3  2 MeV; m

¼ 6  3 MeV



¼ 0:105658357  0:000000005 GeV

¼ 1:25  0:1 GeV; m

¼ 0:125  0:05 GeV



¼ 1:77703  0:00003 GeV

¼ 174:3  5:1 GeV; m

¼ 4:2  0:2 GeV

For simplicity, we have taken massless neutrinos.

Then mixing only occurs for quarks and is given by

a unitary matrix, the Cabibbo–Kobayashi–Maskawa

matrix

:¼

½10

whose matrix elements in terms of absolute values are:

0:9750 0:0008 0:223 0:004 0:0040:002

0:222 0:003 0:9742 0:0008 0:040 0:003

0:009 0:005 0:039 0:004 0:9992 0:0003

½11

Mathematically, the Cabibbo–Kobayashi–Maskawa

matrix comes from a polar decomposition of the

mass matrix. The physical meaning of the quark

mixings is the following: when a sufficiently

energetic W

decays into a u quark, this u quark

is produced together with a



d quark with prob-

ability jV

,an



s quark with probability jV

and a



b quark with probability jV

The phenomenological success of the standard

model is phenomenal: with only a handful of

parameters, it reproduces correctly some millions

of experimental numbers: cross sections, lifetimes,

branching ratios.

Noncommutative Geometry

Noncommutative geometry is an analytic geometry

generalizing three other geometries that also had

important impact on our understanding of forces

and time. Let us start by briefly recalling the three

forerunners (Table 3). Euclidean geometry underlies

Newton’s mechanics as a geometry in the space of

positions. Forces are described by vectors living in

the same space and the Euclidean scalar product is

needed to define work and potential energy. Time

is not part of geometry – it is absolute. This point

of view is abandoned in special relativity unifying

space and time into Minkowskian geometry. This

new point of view allows to derive the magnetic

Table 3 Four nested analytic geometries

Geometry Force Time

Euclidean E =

F  d

x Absolute

Minkowskian

E, 

)

B, 

= 

1

2

Universal

Riemannian Coriolis $ gravity Proper, 

Noncommutative Gravity ) YMH,  =

  10

40

Noncommutative Geometry and the Standard Model 511

field from the electric field as a pseudoforce

associated with a Lorentz boost. Although time

becomes relative, one can still imagine a grid of

synchronized clocks, that is, a universal time. The

next generalization is ‘‘Riemannian geome-

try = curved spacetime.’’ Here gravity can be

viewed as the pseudoforce associated with a

uniformly accelerated coordinate transformation.

At the same time, universal time loses all meaning

and we must content ourselves with proper time.

With today’s precision in time measurement, this

complication of life becomes a bare necessity, for

example, the global positioning system (GPS).

Our last generalization is ‘‘noncommutative

geometry = curved space(time) with an uncertainty

principle.’’ As in quantum mechanics, this uncertainty

principle is introduced via noncommutativity.

Quantum Mechanics

Consider the classical harmonic oscillator. Its phase

space is R

with points labeled by position x and

momentum p. A classical observable is a differenti-

able function on phase space such as the total energy

=(2m) þ kx

. Observables can be added and multi-

plied, and they form the algebra C

), which is

associative and commutative. To pass to quantum

mechanics, this algebra is rendered noncommutative

by means of a noncommutation relation for the

generators x and p:[x, p] = ih1. Let us call A the

resulting algebra ‘‘of quantum observables.’’ It is still

associative, and has an involution 



(the adjoint or

Hermitian conjugation) and a unit 1.

Of course, there is no space anymore of which A is

the algebra of functions. Nevertheless, we talk about

such a ‘‘quantum phase space’’ as a space that has no

points or a space with an uncertainty relation. Indeed,

the noncommutation relation implies Heisenberg’s

uncertainty relation xp  h=2andtellsusthat

points in phase space lose all meaning; we can only

resolve cells in phase space of volume h=2, see Figure 1.

To define the uncertainty a for an observable a 2A,

we need a faithful representation of the algebra on a

Hilbert space, that is, an injective homomorphism 

from A into the algebra of operators on H. For the

harmonic oscillator, this Hilbert space is H= L

(R).

Its elements are the wave functions (x), square-

integrable functions on configuration space. Finally,

the dynamics is defined by the Hamiltonian, a self-

adjoint observable H = H



2A via Schro¨dinger’s

equation (ih@=@t  (H)) (t, x) = 0. Here time is an

external parameter; in particular, time is not an

observable. This is different in the special-relativistic

setting, where Schro¨ dinger’s equation is replaced by

Dirac’s equation 6@ = 0. Now the wave function is

the four-component spinor consisting of left- and right-

handed, particle and antiparticle wave functions.

Unlike the Hamiltonian, the Dirac operator does not

lie in A, but it is still an operator on H.InEuclidean

spacetime, the Dirac operator is also self-adjoint,



= 6@.

Spectral Triples

Noncommutative geometry (Connes 1994, 1995)

does to a compact Riemannian spin manifold M

what quantum mechanics does to phase space. A

noncommutative geometry is defined by the three

purely algebraic items (A, H, 6@), called a spectral

triple. A is a real, associative, and possibly non-

commutative involution algebra with unit, faithfully

represented on a complex Hilbert space H,and6@ is

a self-adjoint operator on H. As the spectral triple,

also the axioms linking its three items are motivated

by relativistic quantum mechanics.

When A= C

(M), the functions on a Riemannian

spin manifold M, represented on spinors ,and6@ is

the gravitational Dirac operator, one has a spectral

triple. The converse is also true when A is a

suitable commutative algebra (Connes 1996), but

the axioms make sense even when A is not

commutative. As for quantum phase space, Connes

defines a noncommutative geometry by a spectral

triple whose algebra is allowed to be noncommu-

tative and he shows how important properties like

dimensions, distances, differentiation, integration,

general coordinate transformations, and direct

products generalize to the noncommutative setting.

As a bonus, the algebraic axioms of a spectral

triple, commutative or not, include discrete, that is,

zero-dimensional spaces that now are naturally

equipped with a differential calculus. These spaces

have finite-dimensional algebras and Hilbert

spaces, meaning that their algebras are just matrix

algebras.

An ‘‘almost commutative geometry’’ is defined as a

direct product of a four-dimensional commutative

geometry, ‘‘ordinary spacetime,’’ by a zero-dimensional

noncommutative geometry, the ‘‘internal space.’’ If the

•

h/2

Figure 1 The first example of noncommutative geometry.

512 Noncommutative Geometry and the Standard Model

latter is also commutative, for example, the ordinary

two-point space, then the direct product describes a

two-sheeted universe or a Kaluza–Klein space whose

fifth dimension is discrete, (Madone 1995). In general,

the axioms of spectral triples imply that the Dirac

operator of the internal space is precisely the fermionic

mass matrix.

As a generic example, here is the internal spectral

triple underlying the standard model with one

generation of quarks and leptons. The algebra

A= H  C  M

that is, 2  2 matrices of the form

a ¼

x 





; x; y 2 C

complex numbers b and complex 3  3matricesc.

The Hilbert space is 30-dimensional, where we

count particles and antiparticles ( 

) separately:

H= H

H

= C

 C

.The

representation is block-diagonal, with the four

blocks



ðaÞ : ¼

a  1

0 a



ðbÞ :¼



½12



ðb; cÞ :¼

 c 0







ðb; cÞ :¼

c 00

0 c 0



½13

The internal Dirac operator (= fermionic mass

matrix) contains two quark masses m

, m

and one

lepton mass m

, and no mixing:

D¼

0 M 00



000





M¼

0 m



 1



½14

These matrices look rather ad hoc; they are not.

They define an irreducible spectral triple and, for a

given algebra, there is only a finite number of such

triples.

The Spectral Action

Chamseddine and Connes (1997) generalize general

relativity to noncommutative spacetimes in two

strokes, kinematics and dynamics. They explicitly

compute this generalization for almost commutative

geometries.

Kinematics In noncommutative geometry, gen-

eral coordinate transformations are algebra auto-

morphisms lifted to the Hilbert space of spinors.

For almost commutative geometries, these transfor-

mations are precisely general coordinate trans-

formations of ordinary spacetime and gauge

transformations. Now remember how Einstein uses

the equivalence principle to produce ‘‘gravity =

curvature’’ starting from the flat metric, which in

Connes’ language is the ordinary flat Dirac opera-

tor. When applied to an almost commutative

geometry (Connes 1996), the equivalence principle

produces again a curved metric via the ordinary

coordinate transformations on M, while the gauge

transformations applied to the fermionic mass

matrix produce a new field, the Higgs scalar ’. For

the example above, this field is precisely the isospin

doublet, color singlet with hypercharge 1=2 of eqn

[6]. Gauge transformations also apply to the

ordinary Dirac operator, thereby producing the

gauge fields A.

Dynamics The group of generalized coordinate

transformations allowed us to construct the con-

figuration space. In the almost commutative case it

consists of Riemannian metrics, gauge fields, and

Higgs scalars. We now want a dynamics on this

configuration space. Of course, we want this

dynamics to be invariant under the group of

generalized coordinate transformations. Note that

the spectrum of the Dirac operator is invariant

under this group and Chamseddine and Connes

(1997) define the spectral action as a regularized

partition function of these eigenvalues.

On almost commutative geometries, the spectral

action is equal to the Einstein–Hilbert action plus

the Yang–Mills–Higgs ansatz (Figure 2). In other

words, almost commutative geometry explains the

forces mediated by gauge bosons and Higgs scalars

as pseudoforces accompanying the gravitational

force in the same way that Minkowskian geometry

(i.e., special relativity) explains the magnetic force as

a pseudoforce accompanying the electric force.

Noncommutative Geometry and the Standard Model 513

There are constraints on the discrete and contin-

uous parameters in the Yang–Mills–Higgs ansatz

deriving from the spectral action Figure 3.

In particular, if we consider only irreducible spectral

triples and among them only those which produce

nondegenerate fermion masses compatible with renor-

malization, then we only get the standard model with

one generation of quarks and leptons, with a massless

neutrino and with an arbitrary number of colors, and a

few submodels thereof. More than one generation and

neutrino masses are possible but imply reducible

triples. However, in at least one generation, the

neutrino must remain purely left and massless.

For the standard model with N generations

and N

colors, we have the constraints

= g

= (9=N) on the continuous parameters. If

we put N = N

= 3 and if we believe in the popular

‘‘big desert’’ then these constraints yield a ‘‘unifica-

tion scale’’ =10

GeV at which the uncertainty

relation in spacetime should become manifest,

 =h=, and a Higgs mass of m

’

= 171.6 

5 GeV for m

= 174.3  5.1 GeV (see Figure 4).

It is clear that almost commutative geometries

only scratch the surface of a gold mine. May we

hope that a genuinely noncommutative geometry

will solve our present problems with quantum field

theory and quantum gravity?

See also: Compact Groups and Their Representations;

Dirac Fields in Gravitation and Nonabelian Gauge

Theory; Effective Field Theories; General Relativity:

Overview; Hopf Algebras and q-Deformation Quantum

Groups; Positive Maps on C



-Algebras; Quantum Hall

Effect; Standard Model of Particle Physics; Symmetries

and Conservation Laws; Symmetry Breaking in Field

Theory; von Neumann Algebras: Introduction, Modular

Theory, and Classification Theory.

Further Reading

Chamseddine A and Connes A (1997) The spectral action

principle. Communications in Mathematical Physics 186:

731 (hep-th/9606001).

Connes A (1994) Noncommutative Geometry. San Diego:

Academic Press.

Connes A (1995) Noncommutative geometry and reality. Journal

of Mathematical Physics 36: 6194.

Connes A (1996) Gravity coupled with matter and the foundation

of noncommutative geometry. Communications in Mathema-

tical Physics 155: 109 (hep-th/9603053).

Gracia-Bondı´a JM, Va´rilly JC, and Figueroa H (2000) Elements

of Noncommutative Geometry. Boston: Birkha¨ user.

Kastler D (2000) Noncommutative geometry and fundamental

physical interactions: the Lagrangian level. Journal of Math-

ematical Physics 41: 3867.

Landi G (1997) An Introduction to Noncommutative Spaces and

Their Geometry, hep-th/9701078. Berlin: Springer.

Riemannian geometry Gravity

Almost commutative

geometry

Einstein

Connes

Gravity

+ Yang–Mills–Higgs

ansatz

+ constraints

Connes

Noncommutative geometry

Figure 2 Deriving the Yang–Mills–Higgs ansatz from gravity.

Yang–Mills–Higgs

Left–right symmetry

GUT

Supersymmetry

NCG

Standard model

Figure 3 Constraints inside the ansatz.

0.2

0.4

0.6

0.8

1.2

1.4

GeV

λ)

1/2

Figure 4 Running coupling constants.

514 Noncommutative Geometry and the Standard Model

Madore J (1995) An Introduction to Noncommutative Differ-

ential Geometry and Its Physical Applications. Cambridge:

Cambridge University Press.

Martı´n CP, Gracia-Bondı´a JM, and Va´rilly JC (1998) The

standard model as a noncommutative geometry: the low

mass regime. Physics Reports 294: 363 (hep-th/9605001).

O’Raifeartaigh L (1986) Group Structure of Gauge Theories.

Cambridge: Cambridge University Press.

Scheck F, Werner W, and Upmeier H (eds.) (2002) Noncommutative

Geometry and the Standard Model of Elementary Particle

Physics, Lecture Notes in Physics, vol. 596. Berlin: Springer.

Schu¨ cker T (2005) Forces from Connes’ geometry. In: Bick E and

Steffen F (eds.) Topology and Geometry in Physics, Lecture

Notes in Physics, vol. 659, hep-th/0111236. Berlin: Springer.

The Particle Data Group, Particle Physics Booklet and http://

pdg.lbl.gov

Noncommutative Geometry from Strings

Chong-Sun Chu, University of Durham, Durham, UK

Noncommutative Geometry from

String Theory

The first use of noncommutative geometry in string

theory appears in the work of Witten on open-string

field theory where the noncommutativity is asso-

ciated with the product of open-string fields.

Noncommutative geometry appears in the recent

development of string theory in the seminal work of

Connes, Douglas, and Schwarz where they con-

structed and identified the compactification of

Matrix theory on a noncommutative torus.

Matrix Theory Compactification and

Noncommutative Geometry

The matrix theory (M-theory) is an 11-dimensional

quantum theory of gravity which is believed to

underlie all superstring theories. Banks, Fischler,

Shenker, and Susskind proposed that the large N

limit of the supersymmetric matrix quantum

mechanics of N D0-branes should describe the

M-theory compactified on a lightlike circle.

Compactification of the M-theory on a torus can be

easily achieved by considering the torus as the quotient

space R

with the quotient conditions

1

¼ X

þ 

2R

; i ¼ 1; ...; d ½1

Here R

are the radii of the torus. The unitary

translation generators U

generate the torus. They

satisfy U

= U

. T-dualizing the D0 brane

system, eqn [1] leads to the dual description as a

(d þ 1)-dimensional supersymmetric gauge theory

on the dual toroidal D-brane. A noncommutative

torus T



is defined by the modified relations

¼ e

i

½2

where 

specify the noncommutativity. Compacti-

fication on a noncommutative torus can be easily

accommodated and leads to noncommutative gauge

theory on the dual D-brane. The parameters 

can

be identified with the components C

ij

of the 3-form

potential in M-theory.

Since M-theory compactified on a circle leads to

IIA string theory, the components C

ij

correspond to

the Neveu–Schwarz (NS) B-field B

in IIA string

theory. The physics of the D0 brane system in the

presence of an NS B-field can also be studied from

the viewpoint of IIA string theory. This led Douglas

and Hull to obtain the same result that a non-

commutative field theory lives on the D-brane.

Toroidally compactified IIA string theory has a

T-duality group SO(d, d ; Z). The T-duality symmetry

gets translated into an equivalence relation between

gauge theories on the noncommutative torus: a gauge

theory on the noncommutative torus T



is equivalent

to that on the noncommutative torus T



if their

noncommutativity parameters and metrics are related

by a T-duality transformation. For example,



¼ðA þ BÞðC þ DÞ

1

;



2 SOðd; d; ZÞ½3

It is remarkable that the T-duality acts within the

field theory level, rather than mixing up the field

theory modes with the string winding states and

other stringy excitations. Mathematically, eqn [3] is

precisely the condition for the noncommutative tori



and T



to be Morita equivalent.

Open-String in B-Field

It was soon realized that the D-brane does not

necessarily need to be toroidal in order to be

noncommutative. A direct canonical quantization

of the open-string system shows that a constant

B-field on a D-brane leads to noncommutative

geometry on the D-brane world volume. Consider

an open string moving in a flat space with metric g

and a constant NS B-field. In the presence of a Dp

brane, the components of the B-field not along the

brane can be gauged away; thus, the B-field can

Noncommutative Geometry from Strings 515

have effects only in the longitudinal directions along

the brane. The world-sheet (bosonic) action for this

part is

S ¼

4





 g

 2





½4

where i, j = 0, 1, ..., p is along the brane. It is easy

to see that the boundary condition g



2i



= 0at = 0,  is not compatible with

the standard canonical quantization [x

(, ),

(, 

)] = 0 at the boundary. Taking t he boundary

condition as constraints and performing canonical

quantization, one obtains the commutation

relations

½a

; a

¼mG



mþn

; ½x

; p

¼iG

;

½x

; x

¼i

½5

Here, the open-string mode expansion is

ð; Þ¼x

þ 2

ðp

  2

ðg

1

BÞ

Þ

ﬃﬃﬃﬃﬃﬃﬃ

2

n6¼0

in

 ia

cos n  2

ðg

1

BÞ

sin n



and 

are the symmetric and antisymmetric

parts of the matrix (g þ 2

1ij

g þ 2

g  2





¼ð2

g þ 2

g  2



½6

It follows from [5] that the boundary coordinates

 x

(, 0) obey the commutation relation

½x

; x

¼i

½7

Relation [7] implies that the D-brane world volume,

where the open-string endpoints live, is a noncom-

mutative manifold. One may also start with the

closed-string Green function and let its arguments to

approach the boundary to obtain the open-string

Green function

ðÞx

ð

Þi ¼ 

lnð  



ð  

Þ½8

where ()isthesignof.From[8], one can

again extract the commutator [7]. G

= g

 (2

)

(Bg

1

is called the open-string metric since it controls

the short-distance behavior of open strings. In contrast,

the short-distance behavior for closed strings is con-

trolled by the closed-string metric g

.Onemayalsotreat

the boundary B-term in [4] asaperturbationtothe

open-str ing conforma l field theory and from which one

may extract [8] from the modified operator product

expansion of the open-string vertex operators.

D-branes in the Wess–Zumino–Witten model

provide another example of noncommutative geo-

metry. In this case, the background is not flat since

there is a nonzero H = dB  k

1=2

, where k is the

level. Examining the vertex operator algebra, one

obtains that D-branes are described by nonassocia-

tive deformations of fuzzy spheres with nonassocia-

tivity controlled by 1=k.

String Amplitudes and Effective Action

The effect of the B-field on the open-string ampli-

tudes is simple to determine since only the x

commutation relation is affected nontrivially. For

example, the noncommutative gauge theory can be

obtained from the tree-level string amplitudes read-

ily. For tree a nd one loop, the vertex operator

formalism can be used. Generally, the vertex

operator can be inserted at either the  = 0or = 

boundary, where the string has zero mode parts x

and y

 x

 (2

)

1

, respectively. The

commutation relations are

½x

; x

¼i

; ½x

; y

¼0;

½y

; y

¼i

½9

The difference in the commutation relation for x

and y

implies that the two boundaries of the open

string have opposite commutativity. This fact is not

so important for tree-level calculations since one can

always choose to put all the interactions at, for

example, the  = 0 boundary. Collecting all these

zero mode parts of the vertex operators, one obtains

a phase factor

 e

¼ e

ði=2Þ

I<J

p

½10

where the ex ternal momenta p

are ordered cycli-

cally on the circle, and momentum conservation has

been used. The computation of the osci llator part of

the amplitude is the same as in the B = 0 case, except

that the metric G is employed in the contractions.

As a result, the effect of the B-field on the tree-level

string amplitude is simply to multiply the amplitude

at B = 0 by the phase factor and to replace the

metric by the metric G. A generic term in the tree-

level effective action simply becomes

pþ1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

tr @



@



pþ1

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det G

tr @



@



½11

516 Noncommutative Geometry from Strings