Quantum knots and knotted zeros

Louis H. Kauffman, Samuel J. Lomonaco

Research output: Chapter in Book/Report/Conference proceedingConference contributionResearchpeer-review

Abstract

In 2001, Michael Berry4 published the paper "Knotted Zeros in the Quantum States of Hydrogen" in Foundations of Physics. In this paper we show how to place Berry's discovery in the context of general knot theory and in the context of our formulations for quantum knots. Berry gave a time independent wave function for hydrogen, as a map from three space R3 to the complex plane and such that the inverse image of 0 in the complex plane contains a knotted curve in R3. We show that for knots in R3 this is a generic situation in that every smooth knot K in R3 has a smooth classifying map f: R3-→ C (the complex plane) such that f-1(0) = K. This leaves open the question of characterizing just when such f are wave-functions for quantum systems. One can compare this result with the work of Mark Dennis and his collaborators and with the work of Lee Rudolph. Our approach provides great generality to the structure of knotted zeros of a wavefunction and opens up many new avenues for research in the relationships of quantum theory and knot theory. We show how this classifying construction can be related our previous work on two dimensional and three dimensional mosaic and lattice quantum knots.

Original languageEnglish
Title of host publicationQuantum Information Science, Sensing, and Computation XI
EditorsEric Donkor, Michael Hayduk, Michael R. Frey, Samuel J. Lomonaco, John M. Myers
PublisherSPIE
Number of pages9
ISBN (Electronic)9781510626331
DOIs
Publication statusPublished - 1 Jan 2019
EventQuantum Information Science, Sensing, and Computation XI 2019 - Baltimore, United States
Duration: 18 Apr 2019 → …

Publication series

NameProceedings of SPIE - The International Society for Optical Engineering
Volume10984
ISSN (Print)0277-786X
ISSN (Electronic)1996-756X

Conference

ConferenceQuantum Information Science, Sensing, and Computation XI 2019
CountryUnited States
CityBaltimore
Period18.04.2019 → …

Keywords

  • Ambient group
  • Braids
  • Classifying map
  • Fibration
  • Fundamental group
  • Graphs
  • Groups
  • Hamiltonian
  • Knot complement
  • Knots
  • Link of singularity
  • Links
  • Quantum computing
  • Quantum knots
  • Schrodinger equation
  • Unitary transformation
  • quantum knots
  • link of singularity
  • classifying map
  • graphs
  • unitary transformation
  • LINK
  • links
  • fibration
  • knot complement
  • fundamental group
  • braids
  • groups
  • ambient group
  • quantum computing
  • knots

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