This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 AA1 = I A1 A = I 14 I (360°) A 2 (720°) A 4 15 (1) Is A 4 = I ? (2) Is A 2 = I ? 16 A ‧ A1 = I B ‧ B1 = I 17 ABA BAB ABA = BAB Artin 18 ABBA = I 19 AA1 = A1 A = BB1 = B1 B = I ABBA = I ABA = BAB Algebra of Dirac’s game 20 ABBA = I B1 A1 ( ABBA )AB = B1 A1 I AB = 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 A1 = B1 = i satisfy all 3 rules: AA1 = A1 A = BB1 = B1 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 1 1 + z 2 1 + 3z 2 + z 4 30 Two knots with different Alexander Polynomials are inequivalent.are inequivalent....
View
Full Document
 Winter '08
 JARVIS
 Math, 1, A′, Knot theory, Dirac’s Game, Knot polynomial

Click to edit the document details