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

• Main Authors: Stefan Heusler, Malte Ubben
• Format: Article
• Language: English
• Published: MDPI AG 2019-03-01
• Series: Symmetry
• Subjects: knot theory; spin-statistics; haptic model; boson-fermion correspondence
• Online Access: https://www.mdpi.com/2073-8994/11/3/426
• ISSN: 2073-8994

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&#8217;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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi>S</mi> <mi>U</mi> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>-representation of spin-jstates in <inline-formula> <math display="inline"> <semantics> <msup> <mi>C</mi> <mrow> <mn>2</mn> <mi>j</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </semantics> </math> </inline-formula> before Hopf-mapping and the number <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> <mi>j</mi> </mrow> </semantics> </math> </inline-formula> of nodes in the stellar representation in <inline-formula> <math display="inline"> <semantics> <mrow> <mi>C</mi> <msup> <mi>P</mi> <mn>1</mn> </msup> </mrow> </semantics> </math> </inline-formula> after Hopf mapping.