SYMMETRIES AND FUNDAMENTAL FORCES. Marc Henneaux Sciences Appliquées ULB 15 février 2007

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1 SYMMETRIES AND FUNDAMENTAL FORCES Marc Henneaux Sciences Appliquées ULB 15 février 2007

2 There exist four fundamental forces in terms of which on can describe all the interactions among matter constitutents: - the gravitational force; - the electromagnetic force; - the strong nuclear force; - the weak nuclear force. 2

3 We understand rather well the electromagnetic and the nuclear (strong and weak) forces. However the remaining force, gravity, remains a puzzle in spite of the remarkable works by giants such as Newton, Einstein 3

4 Symmetry as one of the «thematic melodies» (C.N. Yang) of 20th century physics Principles of symmetry have invaded theoretical physics and underline the description of all fundamental forces. This is one of the important lessons of 20th century physics. 4

5 Need for a new type of symmetry Finite-dimensional symmetries underlie the nongravitational interactions. There are many indications that a deeper understanding of gravity requires infinite-dimensional symmetries, about which very little is understood (frontiers research). 5

6 Purpose of colloquium Explain this statement; Give an idea of what the new structures look like; Give a sense of the beauty of physics. 6

7 Outline What is symmetry? The language of symmetry: group theory Symmetry and physics Mathematical exploration: Coxeter groups and Kac-Moody algebras Gravity and E 10 Conclusions 7

8 Beauty in physics «Einstein was quite convinced that beauty was a guiding principle in the search for important results in theoretical physics.» H. Bondi 8

9 «The search for beauty in physics was a theme that ran throughout Dirac s work and indeed through much of the history of modern physics» S. Weinberg 9

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11 What is «symmetry»? A symmetry principle is a statement that something looks the same from different points of view; or, equivalently, is invariant under some set of transformations. Example: right-left (bilateral) symmetry 11

12 Reflections Reflection in a line s 2 = 1 in a plane 12

13 Left-right symmetry = invariance under a reflection 13

14 Group theory is the mathematical language for describing symmetries A group is a set of elements (transformations) which can be composed to produce a new element of the group (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 such that a a -1 = e = a -1 a 14

15 Michael Atiyah Born 1929, London Origin: Lebanese & Scottish Fields Medal 1966 in Moscow Abel Prize 2004 (d après Buekenhout) 15

16 Languages start off with a primitive vocabulary and rules but, in response to more complex needs, they develop and become capable of expressing more refined ideas. It is a long road from the language of the cave-man to the plays of Shakespeare. Mathematics has developed in a similar way in response to the changing needs of science. If language is the distinguishing feature of homo sapiens then mathematics is the distinguishing feature of homo scientificus! 16

17 The symmetry of a figure is completely captured by the group of transformations that leaves the figure invariant («symmetry group»). 17

18 Symmetry group for bilateral symmetry Reflection in a line (hyperplane) s 2 = 1 Group is {e,s} = Z 2 where e is the identity transformation 18

19 Other examples Finite groups (reflections and rotations) 19

20 Other examples (continued) Infinite (discrete) groups Note: rotations and translations can be written as products of reflections 20

21 At the frontiers of mathematics Previous finite or infinite groups are subgroups of isometries of Euclidean types and are well understood. (Discrete) infinite groups of more general types is an active research area in mathematics. 21

22 Other examples (continued) Infinite (continuous) group of symmetry, called O(3) Depends on 3 parameters: 2 to characterize the axis and one for the angle of rotation Finite dimension (3) 22

23 Other examples (continued) - Rotation groups SO(n) in higher dimensions - Unitary groups U(n) and special unitary groups SU(n) - Symplectic groups - Exceptional groups E 8, E 7, E 6, F 4, G 2 - SU(3) X SU(2) X U(1) All of finite dimensions 23

24 Unexplored territory The study of infinite-dimensional groups is again a subject at the frontiers of current mathematical research this subject appears to be what is needed for a complete understandng of gravity. 24

25 Symmetry and physics Crystallography Einstein and special relativity (Lorentz group Poincaré - Minkoswki) Noether theorem symmetries and conserved quantities Quantum mechanics 25

26 With the development of quantum mechanics, symmetry gradually became the main thematic melody ( ) atomic, molecular physics nuclear physics elementary particle physics 26

27 Periodic table and SO(3) (d après Yang) 27

28 Dimensions of representations of SO 3 are 1, 3, 5 etc. 28

29 These numbers are related to the structure of the periodic table. 2 = 2 x (1) 8 = 2 x (1+3) 18=2 x (1+3+5) 29

30 Symmetry (and group theory) has also invaded elementary particle physics. Classification of elementary particles. 30

31 Symmetry and fundamental forces Symmetry Dictates Interaction 31

32 Electromagnestism (Weyl, Yang, Mills) Quantum mechanics is invariant under multiplication of the wave function by an arbitrary constant phase. Invariance under multiplication by an arbitrary, spacetime dependent phase, requires electromagnetism. 32

33 (Gauge) Invariance under local SU(3) X SU(2) X U(1) «explains» all non gravitational forces. Invariance under arbitrary diffeomorphisms «explains» Einstein gravity. 33

34 However, Einstein gravity (general relativity), in spite of its beauty and remarkable successes, cannot be the final formulation of the gravitational force: it appears to be incompatible with quantum mechanics. We do not know yet the ultimate theory of gravity. 34

35 Candidate for a more fundamental formulation of gravity: string «theory» = «M-theory» But we do not know what is string, or M- theory! In particular: 35

36 What are the underlying symmetries of M-theory? 36

37 Appropriate language There are indications that infinite Coxeter groups and infinite-dimensional Kac- Moody groups might capture (some of) the symmetries of a more fundamental formulation of gravity. 37

38 Coxeter Groups 38

39 Dihedral groups {3} 3 {4} 4 {5} I 2 (3), order 6 I 2 (4), order 8 I 2 (5), order {6} etc (s 1 ) 2 =1, (s 2 ) 2 =1, (s 1 s 2 ) p = 1 I 2 (6), order 12 (fundamental domain in red) 39

40 Coxeter Groups The previous groups are examples of Coxeter groups: these are (by definition) generated by a finite set of reflections s i obeying the relations: (s i ) 2 = 1; (s i s j ) m ij = 1 with m ij = m ji positive integers (=1 for i = j and >1 for different i,j s) Notation: (s r) p = 1 s p r angles 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) 40

41 Crystallographic dihedral groups Hexagonal lattice A 2 p = 3, 4, 6 B 2 C 2 G 2 A 2 B 2 /C 2 G 2 Square lattice G N G = group order N = number of reflections 41

42 THE FIVE PLATONIC SOLIDS Tetrahedron {3,3} Octahedron {3,4} Cube {4,3} Icosahedron {3,5} Dodecahedron {5,3} 42

43 Symmetries of Platonic Solids G is in all cases a Coxeter group {s 1, s 2, s 3 }; (s i ) 2 = 1; (s i s j ) m ij = 1; m ij = 2,3,4,5 (i different from j) G N Tetrahedron A Cube and octahedron 48 9 B 3 /C 3 Icosahedron and dodecahedron H H 3 is not crystallographic 43

44 List of Finite Reflection Groups (= Finite Coxeter Groups) A n B n / G (n+1)! 2 n n! N n(n+1)/2 n 2 Coxeter graphs of finite Coxeter groups (source: J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press 1990) C n D n E E 7 E 8 2 n-1 n! n(n-1) F G H H

45 Affine Reflection Groups In previous cases, the hyperplanes of reflection contain the origin and thus leave the unit sphere invariant («spherical case») 45

46 One can relax this condition and consider reflections about arbitrary hyperplanes in Euclidean space («affine case»). 46

47 Regular tilings of the plane 47

48 Classification of affine Coxeter groups Remarks Affine Coxeter Groups are infinite Fundamental region is an Euclidean simplex Coxeter graphs of affine Coxeter groups (source: J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press 1990) 48

49 Hyperbolic Reflection Groups One can also consider reflection groups in hyperbolic space. These groups are also infinite. 49

50 Tilings of the hyperbolic plane 50

51 Circle-limits (M.C. Escher) 51

52 Classification Hyperbolic simplex reflection groups exist only in hyperbolic spaces of dimension < 10. In the maximum dimension 9, the groups are generated by 10 reflections. There are three possibilities, all of which are relevant to M-theory. (See e.g. Humphreys, Reflection Groups and Coxeter Groups, for the complete list.). Understanding the properties of these groups is a challenge. E 10 BE 10 CE 10 DE 10 52

53 Infinite-dimensional Symmetry Groups 53

54 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 in physics so far are finite-dimensional (depend on a finite number of continuous parameters). A great mathematical achievement has been the complete classification of all finite-dimensional, simple Lie groups (Lie algebras are the vector spaces of «infinitesimal transformations»). 54

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56 The connection between crystallographic finite Coxeter groups and finite-dimensional simple Lie algebras is that the Coxeter groups are the «Weyl groups» of the Lie algebras. Example: unitary symmetry and permutation group. The Coxeter group A n is isomorphic to the permutation group S n+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 Cartan subalgebra»). The Weyl = Coxeter group A n is what is left of the original unitary symmetry once U has been diagonalized since the diagonal form of U is determined up to a permutation of the n+1 eigenvalues. In particular: Z 2 and SO(3) 56

57 Infinite Coxeter groups The same connection holds for infinite Coxeter groups; but in that case the corresponding Lie group is infinite-dimensional and of the Kac-Moody type. Infinite-dimensional Lie groups (i.e., infinite-dimensional symmetries) are playing an increasingly important role in physics. In the gravitational case, the relevant Kac-Moody groups are of hyperbolic or Lorentzian type (beyond the affine case). These groups are unfortunately still poorly understood. 57

58 Infinite Coxeter groups of hyperbolic type emerge when one investigates the dynamics of gravity in extreme situations. For M-theory, it is E 10 that is relevant. 58

59 Billiard description The dynamics of gravitational theories can be mapped on billiard dynamics in some region of hyperbolic space. Furthermore, the billiard region is the fundamental region of a hyperbolic Coxeter group (the reflections against the walls being the fundamental reflections generating the group). 59

60 Examples Pure gravity in 4 spacetime Dimensions. The billiard is a triangle with angles π/2, π/3 and 0, corresponding to the Coxeter group (2,3, infinity). The triangle is the fundamental region of the group PGL(2,Z). 60

61 M-theory and E 10 Truncation to 11-dimensional supergravity Billiard is fundamental Weyl chamber of E 10 Heterotic string: BE 10 Bosonic string DE 10 Is E 10 the symmetry algebra (or a subalgebra of the symmetry algebra) of M-theory? (perhaps E 10 (Z), E 11, E 11 (Z)) 61

62 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 and mathematics meet. 62

63 Conclusions Gravity remains the most mysterious of all the fundamental interactions. There are indications that infinite-dimensional Lie groups related to hyperbolic structures will be crucial ingredients for a deeper understanding of gravity (characteristic feature of gravity). Progress will require advances on both mathematical and physical fronts. Ideas of symmetries will continue to pervade theoretical physics in the years to come. 63

64 To those who do not know mathematics, it is difficult to get across a real feeling as to the beauty of nature if you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. R. Feynman 64

65 «La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi a gli occhi (io dico l'universo) ma non si può intender se prima non s'impara a intender la lingua e conoscere i caratteri ne' quali è scritto. Egli è scritto in lingua matematica e i caratteri sono triangoli, cerchi, ed altre figure geometriche senza i quali mezi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro laberinto.» «Philosophy is written in this grand book of the Universe which stands continually open to our gaze. But we cannot read it without having first learnt the language and the characters in which it is written. It is written in the language of mathematics and the characters are triangles, circles and other geometrical shapes without the means of which it is humanly impossible to decipher a single word; without these we are wandering in vain through a dark labyrinth.» Galileo Galilei, Il Saggiatore (1623) 65