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  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. 360° 720°

  8. 360° 720°

  9. 360° 720°

  10. 360° 720°

  11. Algebraic representations of braids (and knots).

  12. 360° 720°

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

  14. I (360°) A2 (720°)A4

  15. (1) Is A4 = I ? • (2) Is A2 = I ?

  16. A‧A-1 = I B‧B-1 = I

  17. ABA BAB ABA = BAB Artin

  18. ABBA = I

  19. AA-1 = A-1A = BB-1 = B-1B = IABBA = IABA = BABAlgebra of Dirac’s game

  20. ABBA = IB-1A-1(ABBA)AB = B-1A-1IAB = IBAAB = I

  21. ABA = BABABA • ABA = BAB • BAB A2 = B2ABBA = I → A4 = IHence 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-1A = BB-1 = B-1B = I • ABBA = I • BA = BAB

  24. But A2 = -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 1 1 + z2 1 + 3z2 + z4

  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 SO3.

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

  38. For example, consider the following six positions:

  39. 60°

  40. 120°

  41. 180°

  42. 240°

  43. 300°

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

  45.  0°

  46. ・  60°

  47. ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・  120°

  48. ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・  180°

  49. 180°= −180° ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ 