Abstract
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.
Highlights
The importance of topology for various applications in physics is constantly increasing, in particular, in quantum information theory [1,2]
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
Using a simple paper strip model revealing the knot structure of quantum phases in Hilbert space including inner twists, a haptic model of entanglement and gauge symmetries is proposed, which may be valid for physics education
Summary
The importance of topology for various applications in physics is constantly increasing, in particular, in quantum information theory [1,2]. Symmetry 2019, 11, 1399 to homotopically equivalent knots describing the quantum phase are the key ingredient of our model, revealing the close relation between minimal interaction and entanglement. We want to show that a more detailed understanding of the geometry of the qubit (and its generalization to higher spin states) in S3 is important in order to reveal the knot structure of bosonic and fermionic quantum states, and to find some appropriate models for gauge interaction and entanglement. The knot structure in the three-dimensional bulk S3 is mapped to a node structure on the boundary, which is nothing but the Bloch sphere S2 as shown in Figures 3 and 4 for the case j = 1/2. The antipode of this node is called ’direction of the spin’, describing the direction of maximal amplitude.
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