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A Haptic Model of Entanglement, Gauge Symmetries and Minimal Interaction Based on Knot Theory

by Stefan Heusler and Malte Ubben *,†
Institut für Didaktik der Physik, Universität Münster, 48149 Münster, Germany
*
Author to whom correspondence should be addressed.
Current address: Wilhelm-Klemm-Str. 10, 48149 Münster, Germany.
Symmetry 2019, 11(11), 1399; https://doi.org/10.3390/sym11111399
Received: 23 October 2019 / Revised: 6 November 2019 / Accepted: 8 November 2019 / Published: 12 November 2019
The Heegaard splitting of S U ( 2 ) is a particularly useful representation for quantum phases of spin j-representation arising in the mapping S 1 S 3, which can be related to ( 2 j , 2 ) torus knots in Hilbert space. We show that transitions to homotopically equivalent knots can be associated with gauge invariance, and that the same mechanism is at the heart of quantum entanglement. In other words, (minimal) interaction causes entanglement. Particle creation is related to cuts in the knot structure. We show that inner twists can be associated with operations with the quaternions ( I , J , K ), which are crucial to understand the Hopf mapping S 3 S 2. We discuss the relationship between observables on the Bloch sphere S 2, and knots with inner twists in Hilbert space. As applications, we discuss selection rules in atomic physics, and the status of virtual particles arising in Feynman diagrams. Using a simple paper strip model revealing the knot structure of quantum phases in Hilbert space including inner twists, a h a p t i c model of entanglement and gauge symmetries is proposed, which may also be valid for physics education. View Full-Text
Keywords: knot theory; spin-statistics; haptic model; entanglement; gauge symmetry knot theory; spin-statistics; haptic model; entanglement; gauge symmetry
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Heusler, S.; Ubben, M. A Haptic Model of Entanglement, Gauge Symmetries and Minimal Interaction Based on Knot Theory. Symmetry 2019, 11, 1399.

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