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AIP conference proceedings

The path integral formalism applied on the quantum rigid rotator

Ioan Zaharie, Viorel Laurentiu Cartas

Published Jan 1, 2012• 0 citations
Publisher

Abstract

In this paper we apply the path integral formalism for the quantum rigid rotator in 2+1 space time dimension related to the knot theory. The knots theory put the knots into an one to one relation with the braids considered in the braid group theory where braids are actually path into the path integrals formalism. The present work aims the connection between such theories and the anyons behavior.

DOI
10.1063/1.4748074

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In this paper we apply the path integral formalism for the quantum rigid rotator in 2+1 space time dimension related to the knot theory. The knots theory put the knots into an one to one relation with the braids considered in the braid group theory where braids are actually path into the path integrals formalism. The present work aims the connection between such theories and the anyons behavior.
Abstract We introduce the idea of the path integral and use it to study the properties of identical particles under exhange. We explain why the topology of paths through space-time, the fundamental group of the configuration space (the braid group in 2+1 dimensions and the permutation group in 3+1 dimensions) determines the possible particle types. We argue that in 3+1 dimensions one only has bosons and fermions but in 2+1 dimensions other particles, known as anyons, can exist. We describe both the abelian and nonabelian type of anyon and argue why nonabelian anyons could be useful for quantum computation. We introduce the notion of the “quantum dimension” of an anyon which describes the size of the many anyon Hilbert space.
Topological quantum computing is an emerging field that leverages the mathematical structures of knots and braids to encode and manipulate quantum information. However, the topological nature of quantum tunneling remains largely unexplored. In this work, we introduce a novel paradigm for quantum tunneling based on the knot representation of quantum wavefunctions, where Reidemeister moves—interpreted as braid group operators—govern the wavefunction’s evolution. Specifically, we propose four distinct models for topological quantum tunneling: (1) tunneling via Reidemeister transitions, (2) tunneling on a ring-domain, (3) tunneling under space-time curvature, and (4) tunneling influenced by gravitational potential. By establishing a rigorous connection between knot theory, braid group structures, and quantum tunneling, our approach provides a new framework for understanding quantum transport in complex environments. The broader impact of this work lies in bridging topological quantum mechanics with quantum biology and life sciences, opening new avenues for interdisciplinary research in quantum information processing and biophysical modeling.
In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the "quantum embodiment" of a closed knotted physical piece of rope. A quantum knot, as a state of this system, represents the state of such a knotted closed piece of rope, i.e., the particular spatial configuration of the knot tied in the rope. Associated with a quantum knot system is a group of unitary transformations, called the ambient group, which represents all possible ways of moving the rope around (without cutting the rope, and without letting the rope pass through itself.) Of course, unlike a classical closed piece of rope, a quantum knot can exhibit non-classical behavior, such as quantum superposition and quantum entanglement. This raises some interesting and puzzling questions about the relation between topological and quantum entanglement. The knot type of a quantum knot is simply the orbit of the quantum knot under the action of the ambient group. We investigate quantum observables which are invariants of quantum knot type. We also study the Hamiltonians associated with the generators of the ambient group, and briefly look at the quantum tunneling of overcrossings into undercrossings. A basic building block in this paper is a mosaic system which is a formal (rewriting) system of symbol strings. We conjecture that this formal system fully captures in an axiomatic way all of the properties of tame knot theory.
These notes review a description of quantum mechanics in terms of the topology of spaces, basing on the axioms of Topological Quantum Field Theory and path integral formalism. In this description quantum states and operators are encoded by the topology of spaces that are used as modules to build the quantum mechanical model, while expectation values and probabilities are given by topological invariants of spaces, knots and links. The notes focus on the specific way the topology encodes quantum mechanical features, or, equivalently, on how these features can be controlled through the topology. A topological classification of entanglement is discussed, as well as properties of entanglement entropy and basic quantum protocols. The primary aim is to build a less conventional diagrammatic intuition about quantum mechanics, expanding the paradigm of ``Quantum Picturalism".
The model of a topological quantum computer is a promising one due to its natural resistance to noise and other errors. Operations in such a computer are implemented by braiding the trajectories of anyons. While it is easy to understand how to build one-qubit operations, two-qubit operations are more difficult. In arXiv:2412.20931 we suggested an approach to build such operations for a topological quantum computer based on SU(2) Chern-Simons theory with arbitrary level using cabling of knots. In this paper we discuss how this approach should be generalized to the SU(N) case, what the differences are, and which new problems arise.