SYMMETRIES AND FUNDAMENTALFORCES

Marc Henneaux – Sciences Appliquées ULB –15 février 2007

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There exist four fundamental forces interms of which on can describe all theinteractions among matter constitutents:- the gravitational force;- the electromagnetic force;- the strong nuclear force;- the weak nuclear force.

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We understand rather well theelectromagnetic and the nuclear (strongand weak) forces.

However the remaining force, gravity,remains a puzzle in spite of theremarkable works by giants such asNewton, Einstein …

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Symmetry as one of the « thematic melodies »(C.N. Yang) of 20th century physics

Principles of symmetry have invadedtheoretical physics and underline thedescription of all fundamental forces.

This is one of the important lessons of20th century physics.

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Need for a new type of symmetry

Finite-dimensional symmetries underlie the non-gravitational interactions.

There are many indications that a deeper understandingof gravity requires infinite-dimensional symmetries,about which very little is understood (frontiersresearch).

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Purpose of colloquium

Explain this statement; Give an idea of what the new structures

look like; Give a sense of the beauty of physics.

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Outline

What is symmetry? The language of symmetry: group theory Symmetry and physics Mathematical exploration: Coxeter groups

and Kac-Moody algebras Gravity and E10

Conclusions

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Beauty in physics

« … Einstein was quiteconvinced that beauty was aguiding principle in the searchfor important results intheoretical physics. »

H. Bondi

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« The search for beauty inphysics was a theme that ranthroughout Dirac’s work andindeed through much of thehistory of modern physics »

S. Weinberg

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What is « symmetry »?

A symmetry principleis a statement thatsomething looks thesame from differentpoints of view; or,equivalently, isinvariant under someset oftransformations.

Example: right-left(bilateral) symmetry

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s2 = 1

Reflections

Reflection in a line

in a plane

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Left-right symmetry = invariance under areflection

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Group theory is the mathematicallanguage for describing symmetries

A group is a set of elements (transformations) whichcan be composed to produce a new element of thegroup (called the product), with the following properties- Associativity (a • b) • c = a • (b• c)- Existence of a neutral element e such that e • a = a =a • e-- Existence of an inverse a-1 for each element a suchthat a • a-1 = e = a-1 • a

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Michael AtiyahBorn 1929, LondonOrigin: Lebanese & Scottish

Fields Medal 1966 in Moscow Abel Prize 2004

(d’après Buekenhout)

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Languages start off with a primitive vocabulary and rules but,in response to more complex needs, they develop and becomecapable of expressing more refined ideas. It is a long roadfrom the language of the cave-man to the plays ofShakespeare. Mathematics has developed in a similar way inresponse to the changing needs of science.

… If language is the distinguishing feature of homo sapiensthen mathematics is the distinguishing feature of homoscientificus!

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The symmetry of a figure is completelycaptured by the group of transformationsthat leaves the figure invariant(« symmetry group »).

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s2 = 1

Symmetry group forbilateral symmetry

Group is {e,s} = Z2 where e is the identity transformation

Reflection in a line (hyperplane)

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Other examples

Finite groups (reflectionsand rotations)

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Other examples (continued)

Infinite (discrete) groups

Note: rotations and translations canbe written as products ofreflections

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At the frontiers ofmathematics

Previous finite or infinite groups aresubgroups of isometries of Euclideantypes and are well understood.

(Discrete) infinite groups of more generaltypes is an active research area inmathematics.

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Other examples (continued)

Infinite (continuous)group of symmetry,called O(3)

Depends on 3parameters: 2 tocharacterize the axisand one for theangle of rotation

Finite dimension (3)

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Other examples (continued)

- Rotation groups SO(n) in higher dimensions- Unitary groups U(n) and special unitary groups SU(n)- Symplectic groups- Exceptional groups E8, E7, E6, F4, G2

- SU(3) X SU(2) X U(1)

All of finite dimensions

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Unexplored territory

The study of infinite-dimensional groupsis again a subject at the frontiers ofcurrent mathematical research … thissubject appears to be what is needed fora complete understandng of gravity.

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Symmetry and physics

Crystallography Einstein and special relativity (Lorentz

group – Poincaré - Minkoswki) Noether theorem – symmetries and

conserved quantities Quantum mechanics

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With the development of quantum mechanics,symmetry gradually became the main thematicmelody (1927-1970)

atomic, molecular physics

nuclear physics

elementary particle physics

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Periodic table and SO(3)(d’après Yang)

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Dimensions ofrepresentations of SO3are

1, 3, 5 etc.

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These numbers are related to thestructure of the periodic table.

2 = 2 x (1)8 = 2 x (1+3)18=2 x (1+3+5)

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Symmetry (and group theory) has alsoinvaded elementary particle physics.

Classification of elementary particles.

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Symmetry DictatesInteraction

Symmetry andfundamental forces

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Electromagnestism(Weyl, Yang, Mills)

Quantum mechanics is invariant undermultiplication of the wave function by anarbitrary constant phase.

Invariance under multiplication by anarbitrary, spacetime dependent phase,requires electromagnetism.

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(Gauge) Invariance under local SU(3) XSU(2) X U(1) « explains » all nongravitational forces.

Invariance under arbitrarydiffeomorphisms « explains » Einsteingravity.

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However, Einstein gravity (general relativity), inspite of its beauty and remarkable successes,cannot be the final formulation of thegravitational force: it appears to beincompatible with quantum mechanics.

We do not know yet the ultimate theory ofgravity.

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Candidate for a more fundamentalformulation of gravity: string « theory » =« M-theory »

But we do not know what is string, or M-theory!

In particular:

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What are the underlyingsymmetries of M-theory?

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Appropriate language

There are indications that infinite Coxetergroups and infinite-dimensional Kac-Moody groups might capture (some of)the symmetries of a more fundamentalformulation of gravity.

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Coxeter Groups

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Dihedral groups

I2(3), order 6 I2(4), order 8 I2(5), order 10

etc …

I2(6), order 12

12

3

(s1)2=1,

(s2)2=1,

(s1s2)p = 1(fundamental domain in red)

{3}

12

3

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{4}

12

3

4

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{5}

12

3

4

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{6}

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Coxeter GroupsThe previous groups are examples of Coxeter groups: these are (bydefinition) generated by a finite set of reflections si obeying therelations:

(si)2 = 1;(sisj)mij = 1

with mij = mji positive integers (=1 for i = j and >1 for different i,j’s)

Notation: (s r)p = 1angles between reflection axes: π/p

no line if p = 2

p not written when it is equal to 3

(2 lines if p = 4, 3 lines if p = 6)

ps r

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Crystallographic dihedral groups

p = 3, 4, 6A2

B2 – C2

G2

643N

1286|G|

G2B2/C2A2

Hexagonal lattice

Square lattice

|G| = group order

N = number of reflections

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THE FIVE PLATONIC SOLIDS

Dodecahedron {5,3}Icosahedron {3,5}

Cube {4,3}Octahedron {3,4}

Tetrahedron {3,3}

http://home.teleport.com/~tpgettys/platonic.shtml

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Symmetries of Platonic Solids

15120Icosahedronanddodecahedron

948Cube andoctahedron

624Tetrahedron

N|G|

A3

B3/C3

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H3

G is in all cases a Coxeter group{s1, s2, s3}; (si)2 = 1; (sisj)mij = 1; mij = 2,3,4,5 (i different from j)

H3 is not crystallographic

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List of Finite Reflection Groups(= Finite Coxeter Groups)

6014400H4

15120H3

612G2

2427 32F4

120214 35 52

7E8

63210 34 57

E7

3627 34 5E6

n(n-1)2n-1 n!Dn

n22n n!Bn/Cn

n(n+1)/2(n+1)!An

N|G|

Coxeter graphs of finite Coxeter groups(source: J.E. Humphreys, Reflection Groups andCoxeter Groups, Cambridge University Press 1990)

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Affine Reflection GroupsIn previous cases, the hyperplanes ofreflection contain the origin and thusleave the unit sphere invariant(« spherical case »)

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One can relax this condition andconsider reflections aboutarbitrary hyperplanes inEuclidean space (« affine case »).

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Regular tilings of the plane

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Classification of affineCoxeter groups

Coxeter graphs of affine Coxeter groups(source: J.E. Humphreys, Reflection Groups andCoxeter Groups, Cambridge University Press 1990)

Remarks

• Affine Coxeter Groups areinfinite

• Fundamental region is anEuclidean simplex

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Hyperbolic Reflection Groups

One can also consider reflection groups inhyperbolic space.

These groups are also infinite.

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Tilings of the hyperbolic plane

http://www.hadron.org/~hatch/HyperbolicTesselations/

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Circle-limits (M.C. Escher)

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Classification

Hyperbolic simplex reflection groups exist only in hyperbolic spacesof dimension < 10. In the maximum dimension 9, the groups are generatedby 10 reflections. There are three possibilities, all of which are relevant toM-theory . (See e.g. Humphreys, Reflection Groups and Coxeter Groups, for the complete list.). Understanding the properties of these groups is a challenge.

E10

BE10 – CE10

DE10

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Infinite-dimensionalSymmetry Groups

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Crystallographic Coxeter Groups and Kac-Moody Algebras

There is an intimate connection between crystallographic Coxeter groups and Lie groups/Lie algebras.

Lie groups are continuous groups (e.g. SO(3)). The ones usually met inphysics so far are finite-dimensional (depend on a finite number of continuousparameters). A great mathematical achievement has been the completeclassification of all finite-dimensional, simple Lie groups (Lie algebras arethe vector spaces of « infinitesimal transformations »).

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Example: unitary symmetry and permutation group.

The Coxeter group An is isomorphic to the permutation group Sn+1 of n+1 objects.

Consider the group SU(n+1) of (n+1)-dimensional unitary matrices (of unit determinant).

SU(n+1) acts on itself:

U U’= M* U M

(unitary change of basis, adjoint action)

By a change of basis, one can diagonalize U (« U is conjugate to an element in the Cartansubalgebra »). The Weyl = Coxeter group An is what is left of the original unitary symmetryonce U has been diagonalized since the diagonal form of U is determined up to a permutationof the n+1 eigenvalues.

In particular: Z2 and SO(3)

The connection between crystallographic finite Coxeter groups and finite-dimensionalsimple Lie algebras is that the Coxeter groups are the « Weyl groups » of the Lie algebras.

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Infinite Coxeter groups

The same connection holds for infinite Coxeter groups; but in that casethe corresponding Lie group is infinite-dimensional and of the Kac-Moodytype.

Infinite-dimensional Lie groups (i.e., infinite-dimensional symmetries)are playing an increasingly important role in physics. In the gravitationalcase, the relevant Kac-Moody groups are of hyperbolic or Lorentziantype (beyond the affine case).

These groups are unfortunately still poorly understood.

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Infinite Coxeter groups of hyperbolictype emerge when one investigates thedynamics of gravity in extremesituations. For M-theory, it is E10 thatis relevant.

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The dynamics of gravitationaltheories can be mappedon billiard dynamics in someregion of hyperbolic space.

Furthermore, the billiard regionis the fundamental region of ahyperbolic Coxeter group(the reflections against thewalls being the fundamentalreflections generating thegroup).

Billiard description

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Examples

Pure gravity in 4 spacetimeDimensions.

The billiard is a trianglewith angles π/2, π/3 and 0,corresponding to theCoxeter group (2,3, infinity).

The triangle is the fundamentalregion of the group PGL(2,Z).

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M-theory and E10

Truncation to 11-dimensional supergravity

Billiard is fundamental Weyl chamber ofE10

Is E10 the symmetry algebra (or a subalgebra of the symmetryalgebra) of M-theory? (perhaps E10(Z), E11, E11(Z))

Heterotic string: BE10Bosonic string DE10

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We do not know!

We do not know enough about E10.

We do not know enough about M-theory.

Active area of research where physics andmathematics meet.

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Conclusions

• Gravity remains the most mysterious of all the fundamentalinteractions.

• There are indications that infinite-dimensional Lie groups related tohyperbolic structures will be crucial ingredients for a deeperunderstanding of gravity (characteristic feature of gravity).

• Progress will require advances on both mathematical and physicalfronts.

• Ideas of symmetries will continue to pervade theoretical physics inthe years to come.

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To those who do not know mathematics, it is difficult to get acrossa real feeling as to the beauty of nature … if you want to learnabout nature, to appreciate nature, it is necessary to understandthe language that she speaks in.

R. Feynman

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« Philosophy is written in this grand book of the Universe which stands continually open to ourgaze. But we cannot read it without having first learnt the language and the characters in whichit is written. It is written in the language of mathematics and the characters are triangles, circlesand other geometrical shapes without the means of which it is humanly impossible to deciphera single word; without these we are wandering in vain through a dark labyrinth. »

« La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi agli occhi (io dico l'universo) ma non si può intender se prima non s'impara a intender la linguae conoscere i caratteri ne' quali è scritto. Egli è scritto in lingua matematica e i caratteri sonotriangoli, cerchi, ed altre figure geometriche senza i quali mezi è impossibile a intenderneumanamente parola; senza questi è un aggirarsi vanamente per un oscuro laberinto. »

Galileo Galilei, Il Saggiatore (1623)