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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....
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This note was uploaded on 11/11/2011 for the course MATH 112 taught by Professor Jarvis during the Winter '08 term at BYU.
 Winter '08
 JARVIS
 Math

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