1 / 92

The subtle relationship between Physics and Mathematics

The subtle relationship between Physics and Mathematics. 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. . Werner et al. PRL 3 5(1975)1053. II. Dirac’s Game.

Download Presentation

The subtle relationship between Physics and Mathematics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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° ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ ・ 

More Related