# The Topological Origin of Quantum Randomness

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. A Simple Haptic Model for Quantum Randomness

## 3. On the Relation between Distinctions and Space-Time

## 4. The $\mathbf{4}\mathit{\pi}$-Realm and the $\mathbf{2}\mathit{\pi}$-Realm: $\mathit{SL}(\mathbf{2},\mathbb{C})$ as Double Cover of the Lorentz Group $\mathit{SO}(\mathbf{3},\mathbf{1})$

## 5. How Particles in Space-Time Emerge from Making Distinctions

## 6. Knot Theoretic Description of the Dirac- and Maxwell Equations

#### 6.1. Weyl and Dirac Spinor Representations

#### 6.2. Maxwell’s Equations

## 7. How to Undo a Distinction: Haptic Model for Entanglement

#### 7.1. Entanglement of Two Qubits

#### 7.2. Explicit Model for Interactions and Entanglement of Two Qubits

#### 7.3. Homotopic Loops

#### 7.4. Entanglement of Three Qubits

#### 7.5. (Generalized) HS-States

#### 7.6. Dicke States

#### 7.7. The Kauffmann Model

## 8. From Quaternions to Octonions: Normed Division Algebras and $\mathit{SU}\left(\mathbf{3}\right)$

#### 8.1. Modeling Color Confinement and $SU\left(3\right)$ with Virtual Dehn Twists

#### 8.2. Normed Division Algebras

## 9. The Triangle Relation between Interactions, Entanglement, and Observables

- First, we think that the mapping from topological configurations to observables must be incorporated in any model for elementary particles. In ordinary quantum physics, this corresponds to the relation between the wave function and an expectation value. In our model, we have shown how virtual Dehn twists (see Figure 6 for the example of pairs creation of two spin $1/2$-particles in the $2\pi $-realm) are related to a pair of ribbons with inner twists ($\pm 4$ inner twists for two Dirac belts, see Figure 7 for the representation in $4\pi $-realm). We conjecture that only those topological configurations, which can be reached by virtual Dehn twists and torus splitting qualify for observables.
- Second, we think that there cannot be a one-to-one correspondence between knot structures and particles. Rather, depending on the perspective chosen, we may say that the wave function naturally incorporates a knot structure, depending on the chosen homotopic loop (see Figure 8 for the simplest examples). Equivalently, one may consider the full wave function itself as a three dimensional manifold. A detailed description of this ansatz can be found in [27]. Our approach is slightly different, as it is based on the perspective of knots emerging on certain homotopic loops.

## 10. Conclusions and Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Explicit Visualization of Killing Vector Fields and the Corresponding Hopf-Mapping to the Bloch Sphere S_{2}

**Figure A1.**Numerical examples for torus knots in ${S}_{3}$ and the corresponding Hopf mapping to the Bloch sphere ${S}_{2}$ for $j=\frac{1}{2},1,\frac{3}{2}$. The geodesic flow on the Bloch sphere incorporates to the knot structure implicitely: A superposition of crossings of the knots in ${S}_{3}$ are related to nodes in ${S}_{2}$, as explicitely shown in Figure 8.

**Figure A2.**Transition from the unknot to two spin 1 fields by a full virtual Dehn twist ($2\pi $-rotation) of the torus. Torus splitting leads to the creation of a pair of spin $j=1$ particles. In the paper strip model, these particles are represented by $T=\pm 2$ twists in the $\left(2\pi \right)$-realm, respectively. Randomness emerges as the location of the $RR$ and the $LL$ twist is arbitrary. Upon measurement, the context defines what is considered as $RR$, and what is $LL$. Only correlation remains between both parts due to the common origin.

#### Appendix A.2. Generalized Hopf Mapping

#### Appendix A.3. Pair Creation of Two Spin j = 1 Fields from the Unknot

#### Appendix A.4. Generalized HS-States

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**Figure 1.**If a closed paper strip without any knots (

**1**) is rotated once around, two loops emerge with opposite twists R and L (

**2**). If both loops are separated (

**3**,

**4**), they can be distinguished and labeled as $(A,B)$ (

**5**). The assignment of A and B to the left and right $L,R$ twist is arbitrary: Before doing so, there is no left or right twist—there is only one closed paper strip without any twists. Starting with many copies of the original paper strip without knots, we will obtain with $50\%$ probability R or L for A in a series of experiments, and anticorrelation for the random results of B. An observer who is only able to see R and L twisted knots would conclude the Black Box produces random results.

**Figure 2.**Twofold cover of the special linear group and the part of the Lorentz group $SO{(3,1)}^{\mathrm{id}}$ connected to the idendity. Crucially, a rotation of $4\pi $ in $SL(2,\mathbb{C})$ is mapped to two traversals of $2\pi $ in $SO(3,1)$. As the gluing can be done in two equivalent ways, the resulting geometry in the $2\pi $-realm is a torus. Arrow at the edges are identified via gluing.

**Figure 3.**The operations $IJK={e}^{\left(2\pi i\right)/2}$ and $JKI=1$ shown in the Heegard-splitting of the algebra $su\left(2\right)$: The quaternions $I,J,K$ generate rotations around the $x,y,z$-axis, respectively. $|*\rangle $ is any kind of test object to be rotated. In quantum theory, these objects will be interpreted as quantum states.

**Figure 4.**The path of an arbitrary quaternion ${Q}^{4}=1$ is homotopically equivalent to the identity. In the real space (the $2\pi $-realm), the $4\pi $-rotation induced by the quaternion ${Q}^{4}$ can be deformed to the identity (see e.g., the program Antitwister (http://ariwatch.com/VS/Algorithms/Antitwister.htm (accessed on 31 March 2021))). This is the famous Dirac-belt trick. In the $2\pi $-realm, this is equivalent to the situation shown in Figure 1(1),(2), since four inner twists induced by ${Q}^{4}$ in the $4\pi $-realm are equivalent to one virtual Dehn twist with rotation angle $\pi $ in the $2\pi $-realm.

**Figure 5.**Torus knots and corresponding paper strip models for spin $1/2$ (the Dirac belt) and spin 1 (the Hopf link). In general, a spin j representation is related to a $(2j,2)$ torus knot. In the $\left(2\pi \right)$-realm, identification of the parts $(0,2\pi )$ and $(2\pi ,4\pi )$ leads to a Klein Bottle for spin $j=1/2$, and a torus with one full ($2\pi $) Dehn twist for spin $j=1$. In the paper strip model, for simplicity, the re-gluing is only done in one of the two possible manners; compare also Figure 2.

**Figure 6.**A virtual (half) Dehn twist (

**I.**) leads to a topologically equivalent configuration of the unknot (the trivial torus). Torus splitting leads to two opposite Klein-bottles (

**II.**), corresponding to two distinguishable spin $1/2$-particles. Superposition of these distinct particles $(A,B)$ leads back to the entangled state. The paper strip model shown in Figure 1 displays the same transformation, if we think of the edges of the paper strip as identified to form a torus, compare also Figure 2.

**Figure 7.**Transition of the entangled state $\frac{1}{\sqrt{2}}\left(\right|{0}_{A}\phantom{\rule{4pt}{0ex}}{1}_{B}\rangle +|{1}_{B}\phantom{\rule{4pt}{0ex}}{0}_{A}\rangle )$ to the distinct (product) states $|{0}_{A}\rangle |{1}_{B}\rangle $ or $|{1}_{B}\rangle |{0}_{A}\rangle $ in the $\left(4\pi \right)$-realm. Compare to Figure 6 for the equivalent virtual Dehn twist. In the paper strip model, the unknot is simply rotated by $\pi $. The edges of the paper strips must be identified in the $2\pi $-realm; compare also to Figure 2, leading to a representation of spin $1/2$-particles via $\pm \pi $-Dehn twists. Crucially, not all steps of this transition allow for the 2:1-mapping to particles in space-time, since free particles correspond to closed tori, modeled in the $\left(2\pi \right)$-realm as a paper strip with left (L) or right (R) twist. For the intermediate configurations shown in the red box, such a mapping is impossible.

**Figure 9.**The time evolution with $H=\omega (K\times K)$ induces an oscillation between the product state $|++\rangle $ and an entangled Bell-state. On the $\alpha $-loop as defined here, the phase of the Bell state is trivial.

**Figure 10.**Knot structure of the W-state along the $(0/1)$-axis. Deletion of one crossing corresponds to the ${\mathrm{tr}}_{\mathrm{A}}$-operation.

**Figure 11.**Starting with the GHZ-state, we show the relation between superposition states and mixtures using knot theory. In the Bloch sphere representation, a spin j states is characterized by $2j$ nodes, indicated as crosses. Note that nodes in the $2\pi $-realm are mapped to knots in the $4\pi $-realm via Hopf mapping.

**Figure 12.**Change of topology during the transition of the unknot to the distinct (product) states $|{000}_{A}\rangle |{111}_{B}\rangle $. Similar as for Figure 7, the intermediate steps of this transition shown in the red box do not allow for a 2:1-mapping to particles in space-time. The edges of the paper strips must be identified; compare also to Figure 2.

**Figure 13.**(

**I.**) The knots corresponding to the colors red, green blue are obtained by a virtual Dehn twist with $\pi /3$, respectively. There are two different ways to construct a color singulet: (

**II.**) One virtual $\pi $-Dehn twists leads to the six knots $ABAaba\simeq ABAbab$. This configuration can be interpreted as a combination of three antisymmetric Bell-states $|{\mathsf{\Psi}}_{ij}^{-}\rangle $ times a “spectator” color $|{q}_{k}\rangle $. In the grey box, we highlight for example $|{q}_{3}{\mathsf{\Psi}}_{12}^{-}\rangle $. All other terms can be read of from the knot in a similar manner. (

**III.**) Three virtual Dehn twists leads to 18 knots, which can be viewed as superposition of any of the three color-anticolor quark combinations $|{q}_{i}{\overline{q}}_{i}\rangle $.

**Figure 14.**As shown in Figure 3, we may map operations of quaternions $I,J,K$ to motions of a (Dirac)-belt. Here, we generalize this ansatz up to octonions. We show $1,2,3$ belts suited for 1:1, 2:1, 3:1 mapping to observables. If only one complex unit is available, only inner twist operations are possible in the $\left(4\pi \right)$-realm (

**1a**,

**1b**). The paper strips laced up on a ring can be braided in case of $U\left(2\right),U\left(3\right)$ by appropriate motions of the ring. Here, we show some of the (infinitely many) possible configurations which can be obtained starting with the unknot (

**2a**). In particular, the braid (

**2b**) can be transformed to the configuration (

**2c**) with $\pm 2$ inner twists. While for $SU\left(2\right)$, interchaging braiding and twisting is quite trivial, Sundance et al. [25] have shown that also for ${B}_{3}^{c}$, braiding can always be transformed to a configuration with inner twists only. In the example shown, (

**3a**–

**3c**) are topologically equivalent. (

**3b**) corresponds to a Preon configuration.

**Figure 15.**While unitary (time) development in Hilbert space (the $4\pi $-realm) is smooth and corresponds to various topological changes of the knot structures, in space-time (the $2\pi $-realm), only $(2j,2)$-torus knots can be observed, see Figure 7 and Figure 12, Figure A2 in Appendix A for explicit examples. Therefore, (ontic) transitions during interactions between quantum states are (epistemically) unobservable in space-time. Here, we display interactions as tube $T({K}_{1},{K}_{2})$ connecting the two tori with knot structures ${K}_{1},{K}_{2}$. Topological changes are mediated via this tube. Equivalently, we may think of virtual Dehn twist and torus splitting as interaction between the particles. Randomness emerges as we only have a stroboscopic view to this change of topology, since topologically indistinguishable states as (entangled) intermediate states wash our any information about individual properties of the particles in the final states. Only correlations survive the transition.

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**MDPI and ACS Style**

Heusler, S.; Schlummer, P.; Ubben, M.S.
The Topological Origin of Quantum Randomness. *Symmetry* **2021**, *13*, 581.
https://doi.org/10.3390/sym13040581

**AMA Style**

Heusler S, Schlummer P, Ubben MS.
The Topological Origin of Quantum Randomness. *Symmetry*. 2021; 13(4):581.
https://doi.org/10.3390/sym13040581

**Chicago/Turabian Style**

Heusler, Stefan, Paul Schlummer, and Malte S. Ubben.
2021. "The Topological Origin of Quantum Randomness" *Symmetry* 13, no. 4: 581.
https://doi.org/10.3390/sym13040581