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Symmetry

A Haptic Model for the Quantum Phase of Fermions and Bosons in Hilbert Space Based on Knot Theory

Stefan Heusler, Malte S. Ubben

Published Mar 22, 2019• 5 citations
Publisher

Abstract

A generalization of the famous Dirac belt trick opens up the way to a haptic model for quantum phases of fermions and bosons in Hilbert space based on knot theory. We introduce a simple paper strip model as an aid for visualization of the quantum phases before and after Hopf-mapping, which can be extended to arbitrary spin states with almost no mathematical formalism. Knot theory arises naturally, leading to the Jones polynomials derived from Artin’s braid group for fermionic knots and for bosonic links. The paper strip model explicitly illuminates the relation between these knots and links within the S U ( 2 ) -representation of spin-jstates in C 2 j + 1 before Hopf-mapping and the number p = 2 j of nodes in the stellar representation in C P 1 after Hopf mapping.

DOI
10.3390/sym11030426

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A generalization of the famous Dirac belt trick opens up the way to a haptic model for quantum phases of fermions and bosons in Hilbert space based on knot theory. We introduce a simple paper strip model as an aid for visualization of the quantum phases before and after Hopf-mapping, which can be extended to arbitrary spin states with almost no mathematical formalism. Knot theory arises naturally, leading to the Jones polynomials derived from Artin’s braid group for fermionic knots and for bosonic links. The paper strip model explicitly illuminates the relation between these knots and links within the S U ( 2 ) -representation of spin-jstates in C 2 j + 1 before Hopf-mapping and the number p = 2 j of nodes in the stellar representation in C P 1 after Hopf mapping.
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.
Abstract The challenge is to create an efficient quantum algorithm for the bosonic model capable of calculating the Jones polynomials for a knot resulting from interweaving or interlacing n-vertices. This weave is the construction of braid group representations from nineteen-vertex model. We present eigenbases and eigenvalues for lattice generators and their usefulness for the direct computation of Jones polynomials. The calculation shows that the Temperley-Lieb operators can be used for any braid word. Therefore, we propose a quantum sequence using these singular operators as quantum gates operating on the state of n qubits. We show that quantum calculations give the Jones polynomial for achiral knots and links.
Knot theory provides a powerful tool for understanding topological matters in biology, chemistry, and physics. Here knot theory is introduced to describe topological phases in a quantum spin system. Exactly solvable models with long-range interactions are investigated, and Majorana modes of the quantum spin system are mapped into different knots and links. The topological properties of ground states of the spin system are visualized and characterized using crossing and linking numbers, which capture the geometric topologies of knots and links. The interactivity of energy bands is highlighted. In gapped phases, eigenstate curves are tangled and braided around each other, forming links. In gapless phases, the tangled eigenstate curves may form knots. Our findings provide an alternative understanding of phases in the quantum spin system, and provide insights into one-dimension topological phases of matter.
Knot theory provides a powerful tool for the understanding of topological matters in biology, chemistry, and physics. Here knot theory is introduced to describe topological phases in the quantum spin system. Exactly solvable models with long-range interactions are investigated, and Majorana modes of the quantum spin system are mapped into different knots and links. The topological properties of ground states of the spin system are visualized and characterized using crossing and linking numbers, which capture the geometric topologies of knots and links. The interactivity of energy bands is highlighted. In gapped phases, eigenstate curves are tangled and braided around each other forming links. In gapless phases, the tangled eigenstate curves may form knots. Our findings provide an alternative understanding of the phases in the quantum spin system, and provide insights into one-dimension topological phases of matter.
Kauffman and Lomonaco (New J Phys 4:73.1–73.18, 2002. arXiv:quant-ph/0401090 , New J Phys 6:134.1–134.40, 2004) explored the idea of understanding quantum entanglement (the non-local correlation of certain properties of particles) topologically by viewing unitary entangling operators as braiding operators. In Alagic et al. (Yang–Baxter operators need quantum entanglement to distinguish knots, 2015. arXiv:1507.05979v1 ), it is shown that entanglement is a necessary condition for forming non-trivial invariants of knots from braid closures via solutions to the Yang–Baxter equation. We show that the arguments used by Alagic et al. (2015) generalize to essentially the same results for quantum invariant state summation models of knots. In one case (the unoriented swap case) we give an example of a Yang–Baxter operator, and associated quantum invariant, that can detect the Hopf link. Again this is analogous to the results of Alagic et al. (2015). We also give a class of R matrices that are entangling and are weak invariants of classical knots and links yet strong invariants of virtual knots and links. We also give an example of an SU(2) representation of the three-strand braid group that models the Jones polynomial for closures of three-strand braids. This invariant is a quantum model for the Jones polynomial restricted to three-strand braids, and it does not involve quantum entanglement. These relationships between topological braiding and quantum entanglement can be used as a framework for future work in understanding the properties of entangling gates in topological quantum computing. The paper ends with a discussion of the Aravind hypothesis about the direct relationship of knots and quantum entanglement and the ER=EPRER = EPRER=EPR E R = E P R hypothesis about the relationship of quantum entanglement with the connectivity of space. We describe how, given a background space and a quantum tensor network, to construct a new topological space that welds the network and the background space together. This construction embodies the principle that quantum entanglement and topological connectivity are intimately related.
We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of “knot invariants,” among which the Jones polynomial plays a prominent role, since it can be associated with observables in topological quantum field theory. Although the problem of computing the Jones polynomial is intractable in the framework of classical complexity theory, it has been recently recognized that a quantum computer is capable of approximating it in an efficient way. The quantum algorithms discussed here represent a breakthrough for quantum computation, since approximating the Jones polynomial is actually a “universal problem,” namely, the hardest problem that a quantum computer can efficiently handle.