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The subtle relationship between Physics and Mathematics

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## The subtle relationship between Physics and Mathematics

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**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 35(1975)1053**II.**Dirac’s Game**(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?**The answers to (1) and (2), (yes or no) cannot depend on the**original positions of the strings.**360°**720°**360°**720°**360°**720°**360°**720°**360°**720°**AA-1 = I**A-1A = I**I**(360°) A2 (720°)A4**(1) Is A4 = I ?**• (2) Is A2 = I ?**A‧A-1 = I**B‧B-1 = I**ABA**BAB ABA = BAB Artin**AA-1 = A-1A = BB-1 = B-1B = IABBA = IABA = BABAlgebra of**Dirac’s game**ABA = BABABA • ABA = BAB • BAB A2 =**B2ABBA = I → A4 = IHence answer to (1): Yes**The algebra of the last 3 slides shows how to do the**disentangling.**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**But A2 = -1 ≠ I**• Hence answer to (2): No**III.**• Mathematics of Knots**Planar projections of prime knots and links**with six or fewer crossings.**Fundamental Problem of Knot Theory:**How to classify all knots**Alexander Polynomial**1 1 + z2 1 + 3z2 + z4**Two knots with different Alexander Polynomials are**inequivalent.**Both knots have Alexander Polynomial = 1**(from C. Adams: The Knot Book)**Jones Polynomial (1987)**Homfly Polynomial Kauffman Polynomial etc.**Statistical Mechanics**(Many Body Problem) 1967: Yang Baxter Equation**ABA = BAB**• (12)(23)(12)=(23)(12)(23) • A(u)B(u+v)A(v) = B(v)A(u+v)B(u)**IV.**Topology The different positions of the block form a “space”, called SO3.**Each of these six positions (i.e. each rotation) will be**represented by a point:**･** 0°**･**･ 60°**･**･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ 120°**･**･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ 180°**180°= −180°**･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･ ･