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

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.

A Knot Theoretic Extension of the Bloch Sphere Representation for Qubits in Hilbert Space and Its Application to Contextuality and Many-Worlds Theories

“…While |Ψ + is part of the spin triplet with j = 1, |Ψ − is the antisymmetric spin singlet state with j = 0, which is basis invariant. As discussed in detail in Reference [3], for any maximally entangled state, there exists a homotopic loop where the amplitude of |Ψ + remains constant. For |Ψ + , this is the rotation in z-direction…”

“…Obviously, the complete group can be characterized by three parameters-the direction of the rotation axis, indicated by the unit vector e with e = 1, and the rotation angle −π ≤ ϑ < π. The π-ball (with radius π) defined by eϑ represents all possible group elements of SO(3) on the level of the Lie algebra so (3). Note that the rotations ±π e are identical rotations in R 3 , therefore, antipodes of the π-ball are identified, see Figures 1 and 2.…”

“…One merit of the paper strip model is the possibility to visualize topologically equivalent configurations which represent the same quantum state, or to be more precise, the same amplitude on a certain homotopic loop in Hilbert space. We discuss the homotopic loop with constant amplitude, because this provides a nice representation for entanglement-as discussed in detail in Reference [3], this constant phase is topologically equivalent to a combination of one right twist R and one left twist L (see Figure 9).…”

“…In (28), the superposition of product states is needed to wash out the significance of the labels A, B, that is, to undo the possibility of two 'parts' of the wave function. Indeed, the entangled state can be represented just as a constant phase, which is topologically equivalent to the combination of two opposite twists [3]. It is impossible to uniquely label A and B to these twists, as all homotopically equivalent variants coexist, as shown in Figure 10.…”

“…In Section 4, we review the topological model of entanglement recently introduced in Reference [3], and apply this model in Section 5 to give a geometric interpretation of the Kochen-Specker theorem, which states that even for commuting observables, the eigenvalues of a given quantum state depend on the context. It turns out that the difference between 2π and 4π-rotations, which cannot be resolved in the usual Bloch-sphere representation, lies at the heart of contextuality in quantum physics.…”