Mathematics and Physics_ASC_Aug2009

Mathematics and Physics_ASC_Aug2009 - 1 The subtle...

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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. Diracs 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 = I B 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 Diracs game 20 ABBA = I B-1 A-1 ( ABBA )AB = B-1 A-1 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 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 1 1 + z 2 1 + 3z 2 + z 4 30 Two knots with different Alexander Polynomials are inequivalent.are inequivalent....
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Mathematics and Physics_ASC_Aug2009 - 1 The subtle...

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