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

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## Abstract

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## 1. Introduction

## 2. The Hopf Mapping ${S}_{3}\to {S}_{2}$ for a Single Qubit

## 3. Knots and Inner Twists for Spin j States

- The knot in the $(4\pi )$-realm ${S}_{3}$ including ${t}^{\prime}$ inner twists (${t}^{\prime}=4l$ for bosons, ${t}^{\prime}=4l\pm 2$ for fermions), modeled with the paper strip in the $(4\pi )$-realm.
- The corresponding Jones polynomial ${J}_{j}(t)$, describing the knot structure of the spin j state on a homotopic loop in the $(4\pi )$-realm ignoring inner twists.
- The quantum phase of the observable in the $(2\pi )$-realm with $T=2j$ twists, obtained by ‘gluing together’ the two parts of the knot on the surface $\partial {B}_{1}\simeq \partial {B}_{2}$ upon identification of the two pieces $(0,2\pi )$ and $(2\pi ,4\pi )$.
- The corresponding stellar representation on $C{P}^{1}$ given by ${\varphi}_{j}(z)={\prod}_{k=0}^{2j}(z-{z}_{k})$ with $2j$ nodes and $z=u/v$.
- Stereographic projection of the stellar representation on the Bloch sphere ${S}_{2}$. For bosons with $j=l$ the result is equivalent to spherical harmonics ${Y}_{lm}$.

## 4. A Haptic Model of Topological Gauge Fields

## 5. A Haptic Model of Quantum Entanglement

## 6. A Haptic Model of Gauge Symmetries

#### 6.1. Minimal Interaction

#### 6.2. Selection Rules

#### 6.3. Feynman Diagrams and Entanglement

## 7. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix Jones Polynomials and Inner Twists

**Figure A1.**Calculation of Jones polynomials for the torus splitting described in the main text (Figure 16).

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**Figure 1.**Quantum tomography: For an ensemble of identical qubits, successive measurements lead to a random pattern with probabilities for ‘black’ ($+1$ eigenvalues) or ‘white’ ($-1$ eigenvalues) in the $x,y,z$ direction, respectively. With ${p}_{k}^{+}$ as probability for ‘black’, and ${p}_{k}^{-}$ as probability for ‘white’, the relation ${n}_{k}=\mathrm{tr}({\sigma}_{k}\rho )={p}_{k}^{+}-{p}_{k}^{-}$ holds. Here, $\rho =|{0}_{n}\rangle \langle {0}_{n}|$ is the $2\times 2$ density matrix of the single qubit.

**Figure 2.**The operation of the quaternions $I,J,K$ on the Dirac belt in the $(4\pi )$-realm. Note that these operations lead to inner twists of the Dirac belt. In particular, for operation $IJK={e}^{2\pi /2}=-1$ leads to two inner twists.

**Figure 3.**Left: Heegaard splitting of ${S}_{3}$, Right: Hopf mapping to the Bloch sphere. In ${S}_{3}$, a homotopic loop ${S}_{1}$ is considered ranging from $(0,4\pi )$, which is mapped to a great circle ${S}_{1}$ traversed $twice$ on the Bloch sphere ${S}_{2}$. The Dirac belt describing the quantum phase on the homotopic loop in ${S}_{3}$ is equivalent to a Möbius strip when the parts $(0,2\pi )$ and $(2\pi ,4\pi )$ separated in ${S}_{3}$ are ‘glued together’, [see also Figure 4]. Superposition of right- and left-twisted Möbius strips leads to a node on the Bloch sphere. The antipode of this node is called ‘direction of the spin’, describing the direction of maximal amplitude.

**Figure 4.**The node on the Bloch sphere [in this case, the node at $\theta =\pi /2,\varphi =0$ arises due to superposition of the quantum phases $1/\sqrt{2}({e}^{i\varphi /2}-{e}^{-i\varphi /2})$]. In the $(4\pi )$-realm, an infinite number of homotopically equivalent quantum phases in ${S}_{1}\to {S}_{3}$ arise which all map to the great circle $(\theta =\pi /2,\varphi \in \{0,4\pi \})$. The superposition on the Bloch sphere translates to a superposition of Dirac belts.

**Figure 5.**Representation of the superposition state $|-\rangle =\frac{1}{\sqrt{2}}(|0\rangle -|1\rangle )$ on the Bloch sphere ($(2\pi )$-realm). Positions × of nodes are antipodes to the direction of $\overrightarrow{n}$. The superposition of the qubits $|0\rangle ,|1\rangle $ changes the position of the direction of the spin, which in turn changes the position × of the node. The relation between nodes on the Bloch sphere and knots in Hilbert space are shown in Figure 4.

**Figure 6.**For spin j-states, $(2j)$ nodes arise on the Bloch sphere which may be described as a complex function ${\prod}_{k=1}^{2j}(z-{z}_{k})$ in the stellar representation. The corresponding knot structure in the $(4\pi )$-realm is described by the Jones polynomial ${J}_{j}(t)$. Additionally, the quantum phase has $4j$ inner twists (bosons), or $4j+2$ inner twists (fermions), respectively, denoted by (+).

**Figure 7.**Left: Double copy of $\nu $ inner twists, describing a boson with spin $2\nu $. Right: By joining the two pieces to a fermionic knot, two additional inner twists arise, leading to a knot with $4\mathbf{j}\pm 2$.

**Figure 8.**Due to the interaction $\mathbf{U}(t)=exp-(i\mathbf{H}t/\hslash )$ with $\mathbf{H}=\hslash \omega ({\sigma}_{z}\times {\sigma}_{z})$, the initial state $|+\rangle |+\rangle $ becomes the entangled Bell state $\frac{1}{\sqrt{2}}(|{0}_{a}{1}_{a}\rangle +|{1}_{a}{0}_{a}\rangle $. We consider the homotopic loop perpendicular to the $|{0}_{a}\rangle \leftrightarrow |{1}_{a}\rangle $-direction.

**Figure 9.**Using transitions to homotopically equivalent knots similar to Type-2 Reidemeister moves in the $(4\pi )$-realm, the constant phase can also be viewed as a combination of two qubits $|0\rangle |1\rangle $ or $|1\rangle |0\rangle $. Since both possibilities are indistinguishable, the superposition $|0\rangle |1\rangle +|1\rangle |0\rangle $ emerges, which is an entangled state. After taking the particle trace, a mixed state arises. The latter result is basis-independent.

**Figure 10.**Paper strip model of the interaction of two qubits: The entangled state is described by one common state with one quantum phase. By rotating the phase once and cutting into two pieces, two separate particles with left- and right twists arise. Here, we use the model of the paper strip near the boundary, where the phases $(0,2\pi )$ and $(2\pi ,4\pi )$ are glued together. The general homotopy in the $(4\pi )$-realm is shown in Section 6.3.

**Figure 11.**In the $(4\pi )$-realm, the constant phase in $(0,2\pi )$ can superpose constructively (symmetric wave function) $|{\mathsf{\Psi}}^{+}\rangle $ or destructively (anti symmetric wave function) $|{\mathsf{\Psi}}^{-}\rangle $ with the constant phase in $(2\pi ,4\pi )$. The situation on the Bloch sphere in the $(2\pi )$-realm is shown in Figure 8.

**Figure 12.**Paper strip model of minimal interaction: The insertion of additional twists is compensated by a gauge field. Only after a torus splitting, the gauge field with $-T$ twists is separated from the original particle, which then has $J+T$ twists due to the gauge interaction.

**Figure 13.**The quantum phase of the state $|3s\rangle $ is homotopically equivalent to $\frac{1}{\sqrt{2}}(|2p,+1\rangle |L\rangle +|2p,-1\rangle |R\rangle )$, see also Figure 9. The entangled state decays into a mixed state due to interaction of the photon with the environment.

**Figure 14.**Paper strip model of the quantum phase of the decay $|3s\rangle \to |2p\rangle $ in the $(2\pi )$-realm. If $RR$ is associated with the right-circular polarized photon, the $LL$ described the quantum phase of the electron in the state $|2p,-1\rangle $. With $50\%$ probability, the roles of the pieces $RR$ and $LL$ are interchanged, see Figure 13.

**Figure 15.**Leading-order Feynman diagrams for electron-electron and electron-positron interaction. In view of entanglement, a direct interpretation of Feynman diagrams in space time is impossible. The quantum phase of the entangled state can be modeled in the $(2\pi )$-realm as shown in Figure 10, and in the $(4\pi )$-realm as shown in Figure 16.

**Figure 16.**Haptic model of the quantum phase of an entangled pair of qubits in the $(4\pi )$-realm. Here, we consider any homotopy in the bulk of ${B}_{1}$ and ${B}_{2}$ in the Heegaard splitting, see Figure 3. After Hopf mapping, the pieces $(0,2\pi )$ and $(2\pi ,4\pi )$ from ${B}_{1}$ and ${B}_{2}$ are ‘glued together’, leading to the description in the $(2\pi )$-realm as shown in Figure 10. As discussed in the text, we associate with the two qubits either a pair of electrons, or an electron-positron pair. (

**A**) The constant phase, see also Figure 11. (

**B**) One rotation leads to a homotopically equivalent configuration, which can be seen as a pair of entangled spin $1/2$ particles, see also Figure 10A. Due to the homotopic equivalences shown in Figure 9, we may view this quantum state also a combination of two spin $1/2$ particles with a (virtual) gauge particle. (

**C**) Encounter of the quantum phases, see also Figure 10B. (

**D**) First splitting of the quantum phase in the $(4\pi )$-realm, see also Figure A1 for the corresponding Jones polynomials. This configuration cannot be mapped to the $(2\pi )$-realm. (

**E**) Second splitting of the quantum phase in the $(4\pi )$-realm. The situation in the $(2\pi )$-realm, where both splitting are combined to one torus splitting is shown in Figure 10C,D. (

**f**) Quantum phase of two distinguishable spin $1/2$ particles in the $(4\pi )$-realm. The inner twists in the Dirac belts have opposite sign, i.e., $(++++)$ and $(----)$.

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**MDPI and ACS Style**

Heusler, S.; Ubben, M. A Haptic Model of Entanglement, Gauge Symmetries and Minimal Interaction Based on Knot Theory. *Symmetry* **2019**, *11*, 1399.
https://doi.org/10.3390/sym11111399

**AMA Style**

Heusler S, Ubben M. A Haptic Model of Entanglement, Gauge Symmetries and Minimal Interaction Based on Knot Theory. *Symmetry*. 2019; 11(11):1399.
https://doi.org/10.3390/sym11111399

**Chicago/Turabian Style**

Heusler, Stefan, and Malte Ubben. 2019. "A Haptic Model of Entanglement, Gauge Symmetries and Minimal Interaction Based on Knot Theory" *Symmetry* 11, no. 11: 1399.
https://doi.org/10.3390/sym11111399