1 The subtle relationship between Physics and Mathematics

2 I. Physics of a neutron. After 1926, the mathema- tics of QM shows that a Fermion rotated 360° does not come back to itself. It acquires a phase of -1.

3  Werner et al. PRL 35(1975)1053

4 II. Dirac’s Game

5 (1)After a rotation of 720°, could the strings be disentangled without moving the block? (2)After a rotation of 360°, could the strings be disentangled without moving the block?

6 The answers to (1) and (2), (yes or no) cannot depend on the original positions of the strings.

7 720° 360°

8 720° 360°

9 720° 360°

10 720° 360°

11 Algebraic representations of braids (and knots).

12 720° 360°

13 AA -1 = I A -1 A = I

14 I (360°) A 2 (720°) A 4

15 (1) Is A 4 = I ? (2) Is A 2 = I ?

16 A ‧ A -1 = IB ‧ B -1 = I

17 ABA BAB ABA = BAB Artin

18 ABBA = I

19 AA -1 = A -1 A = BB -1 = B -1 B = I ABBA = I ABA = BAB Algebra of Dirac’s game

20 ABBA = I B -1 A -1 (ABBA)AB = B -1 A -1 IAB = I B AAB = I

21 ABA = BAB ABA ABA = BAB BAB A 2 = B 2 ABBA = I → A 4 = I Hence answer to (1): Yes

22 The algebra of the last 3 slides shows how to do the disentangling.

23 A = B = i A -1 = B -1 = -i satisfy all 3 rules: AA -1 = A -1 A = BB -1 = B -1 B = I ABBA = I BA = BAB

24 But A 2 = -1 ≠ I Hence answer to (2): No

25 III. Mathematics of Knots

26 Planar projections of prime knots and links with six or fewer crossings.

27 Knots are related to Braids

28 Fundamental Problem of Knot Theory: How to classify all knots

29 Alexander Polynomial z z 2 + z 4

30 Two knots with different Alexander Polynomials are inequivalent.

31 But Alexander Polynomial is not discriminating enough.

32 Both knots have Alexander Polynomial = 1 (from C. Adams: The Knot Book)

33 Jones Polynomial (1987) Homfly Polynomial Kauffman Polynomial etc.

34 Statistical Mechanics (Many Body Problem) 1967: Yang Baxter Equation

35 ABA = BAB (12)(23)(12)=(23)(12)(23) A(u)B(u+v)A(v) = B(v)A(u+v)B(u)

36 IV. Topology The different positions of the block form a “space”, called SO 3.

37 We need a geometric representation of this “space”.

38 For example, consider the following six positions:

39

40 60°

41 120°

42 180°

43 240°

44 300°

45 Each of these six positions (i.e. each rotation) will be represented by a point:

46 ･･ 0°

47 ････ 60°

48 ････････････････････････ 120°

49 ････････････････････････････････････････････ 180°

50 ････････････････････････････････････････････ 180°= −180°

51 ･･････････････････････････････････････ 240°= −120°

52 ･･････････････････ 300°= −60°

53 ･･････････････････ 360°= 0°

54 Each rotation is represented by the end of a vector, the length of which measures the angle of rotation. The direction indicates the direction of rotation.

55 Thus we have a geometrical representation of SO 3 as a solid sphere: diametrically opposite points like A and A’ or B and B’ are considered identical.

56  A′ A B B’

57 Now consider a loop in SO 3 Space: Rotation 0°→360°

58  ････････････････････････ ･･････････････････ ････････････････････ ････････････････････ ････････････････････ ･･････････････････ A′ O A

59  ････････････････････････ ･･････････････････ ････････････････････ ････････････････････ ････････････････････ ･･････････････････ A′ O A....

60 The path starts and ends in the same position. Thus it is a closed loop in SO 3.

61   A′ O A ....

62   A′ O A 

63 No distortion of the loop can shrink it into one point, because there is always at least one jump.

64 Now consider a series of rotations: 0 degree to 720 degrees. i.e. trace the loop twice:

65  ････････････････････････ ･･････････････････ ････････････････････ ････････････････････ ････････････････････ ･･････････････････ A′ O A......

66  A′ O A    B’ B x y Let the A ’ OB segment of the double loop be detoured to A ’ XYB.

67  ････････････････････････ ････････････････････ ････････････････････ ･･････････････････ A′ O A...... x y B B’

68  ････････････････････････ ････････････････････ ････････････････････ ･･････････････････ A′ O A    x y B B’ Now we further detour path A ’ to B.

69  ････････････････････ ････････････････････ ･･････････････････ A′ O A     B B’

70  ････････････････････ ････････････････････ ･･････････････････ A′ O A    B But A’ to B = A to B’ B’

71  ････････････････････ ････････････････････ ･･････････････････ A′ O A   B B’ 

72 Further distortion of the double loop.

73  A′ O A   B B’ 

74 Thus we have shown how the double loop can be continuously distorted, and shrunk into a point.

75 Thus SO 3 is said to be “doubly connected” a topological property

76 After the development of quantum mechanics gradually physicists realize the importance of topology in physics.

77 Dirac ’ s Game Knots and Braids Continuous Groups General Relativity Fiber Bundles

78 V. The subtle relationship between Physics and Mathematics

79 In Maxwell’s ( ) original papers & books, the vector potential A played a central role.

80 Later Heaviside ( ) and Hertz ( ) eliminated A and created a dogma:

81 Electromagnetic effects reside in E and H. If E=H=0, then no electromagnetic effects.

82 In 1959, Aharonov and Bohm proposed the Aharonov-Bohm experiment:

Aharonov-Bohm A B

84

85 Tonomura et al. 1985

86 L

87 L L

88 This experiment showed ① A cannot be completely eliminated. ② EM has topological structure, called fiber bundles in math.

89 Thus the foundation of EM is related to deep concepts in fundamental math.

90 Physics Math

91 Deep as the relationship is between mathematics and physics, it would be wrong, however, to think that the two disciplines overlap that much. They do not: They have their separate aims and tastes.

92 They have distinctly different value judgements, and they have different traditions. At the fundamental conceptual level they amazingly share some concepts, but even there, the life force of each discipline runs along its own veins. (92c)