A TOPOLOGICAL QUANTUM THEORY INTERPRETATION OF INTEGRABLE MODELS

1991 ◽  
Vol 06 (19) ◽  
pp. 1797-1806
Author(s):  
ÖMER F. DAYI
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
Hamiltonian System ◽  
Initial Phase ◽  
Toda Lattice ◽  
Integrable Model ◽  
Constrained Hamiltonian System ◽  
Topological Quantum ◽  
Constants Of Motion ◽  
Theory Interpretation

An integrable model can be interpreted as a constrained Hamiltonian system by treating constants of motion of the former as constraints of the latter. The new constrained Hamiltonian system, when we deal with a finite initial phase space, after quantization does not have local excitations if operator ordering does not cause anomalies. So that it is a topological quantum theory. As an example, operator quantization of the Toda lattice where the ordering is important, is studied.

scholarly journals EQUIVARIANT LOCALIZATION, SPIN SYSTEMS AND TOPOLOGICAL QUANTUM THEORY ON RIEMANN SURFACES

1994 ◽  
Vol 09 (29) ◽  
pp. 2705-2718 ◽  
Author(s):  
GORDON W. SEMENOFF ◽  
RICHARD J. SZABO
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
Riemann Surfaces ◽  
Quantum Dynamics ◽  
Projective Representation ◽  
Finite Dimensional ◽  
Topological Quantum ◽  
Group Orbit ◽  
Canonical Quantum ◽  
State Path

We study equivariant localization formulas for phase space path-integrals when the phase space is a multiply connected compact Riemann surface. We consider the Hamiltonian systems to which the localization formulas are applicable and show that the localized partition function for such systems is a topological invariant which represents the non-trivial homology classes of the phase space. We explicitly construct the coherent states in the canonical quantum theory and show that the Hilbert space is finite-dimensional with the wave functions carrying a projective representation of the discrete homology group of the phase space. The corresponding coherent state path-integral then describes the quantum dynamics of a novel spin system given by the quantization of a nonsymmetric coadjoint Lie group orbit. We also briefly discuss the geometric structure of these quantum systems.

Topology knots and hierarchy problem

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Theory ◽  
String Theory ◽  
Quantum Gravity ◽  
Dimensional Space ◽  
Space Time ◽  
Elementary Particles ◽  
Hierarchy Problem ◽  
Topological Quantum ◽  
Dimensional Space Time ◽  
New Formula

Many approaches to quantum gravity consider the revision of the space-time geometry and the structure of elementary particles. One of the main candidates is string theory. It is possible that this theory will be able to describe the problem of hierarchy, provided that there is an appropriate Calabi-Yau geometry. In this paper we will proceed from the traditional view on the structure of elementary particles in the usual four-dimensional space-time. The only condition is that quarks and leptons should have a common emerging structure. When a new formula for the mass of the hierarchy is obtained, this structure arises from topological quantum theory and a suitable choice of dimensional units.

scholarly journals Quantum gravity, dynamical phase-space and string theory

2014 ◽  
Vol 23 (12) ◽  
pp. 1442006 ◽  
Author(s):  
Laurent Freidel ◽  
Robert G. Leigh ◽  
Djordje Minic
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
String Theory ◽  
Quantum Gravity ◽  
Momentum Space ◽  
Symplectic Geometry ◽  
Complex Geometry ◽  
Hamiltonian Dynamics ◽  
Natural Extension ◽  
Dynamical Phase

In a natural extension of the relativity principle, we speculate that a quantum theory of gravity involves two fundamental scales associated with both dynamical spacetime as well as dynamical momentum space. This view of quantum gravity is explicitly realized in a new formulation of string theory which involves dynamical phase-space and in which spacetime is a derived concept. This formulation naturally unifies symplectic geometry of Hamiltonian dynamics, complex geometry of quantum theory and real geometry of general relativity. The spacetime and momentum space dynamics, and thus dynamical phase-space, is governed by a new version of the renormalization group (RG).

A model to determine the initial phase space of a clinical electron beam from measured beam data

2000 ◽  
Vol 46 (2) ◽  
pp. 269-286 ◽  
Author(s):  
J J Janssen ◽  
E W Korevaar ◽  
L J van Battum ◽  
P R M Storchi ◽  
H Huizenga
Keyword(s):  
Electron Beam ◽  
Phase Space ◽  
Initial Phase

A UNIPOTENT GROUP ACTION ON A FLAG MANIFOLD AND "GAP SEQUENCES" OF PERMUTATIONS

2003 ◽  
Vol 02 (02) ◽  
pp. 215-222 ◽  
Author(s):  
BARBARA A. SHIPMAN
Keyword(s):  
Hamiltonian System ◽  
Toda Lattice ◽  
Initial Conditions ◽  
Flag Manifold ◽  
Unipotent Group ◽  
Unipotent Subgroup ◽  
Bruhat Cells ◽  
Generic Orbit ◽  
Gap Sequence ◽  
Gap Sequences

There is a unipotent subgroup of Sl(n, C) whose action on the flag manifold of Sl(n, C) completes the flows of the complex Kostant–Toda lattice (a Hamiltonian system in Lax form) through initial conditions where all the eigenvalues coincide. The action preserves the Bruhat cells, which are in one-to-one correspondence with the elements of the permutation group Σn. A generic orbit in a given cell is homeomorphic to Cm, where m is determined by the "gap sequence" of the permutation, which lists the number inversions of each degree.

scholarly journals Tilting and Squeezing: Phase Space Geometry of Hamiltonian Saddle-Node Bifurcation and its Influence on Chemical Reaction Dynamics

2020 ◽  
Vol 30 (04) ◽  
pp. 2030008 ◽  
Author(s):  
Víctor J. García-Garrido ◽  
Shibabrat Naik ◽  
Stephen Wiggins
Keyword(s):  
Phase Space ◽  
Hamiltonian System ◽  
Potential Energy ◽  
Invariant Manifolds ◽  
Reaction Dynamics ◽  
Potential Energy Function ◽  
Space Structures ◽  
Stable And Unstable Manifolds ◽  
Saddle Node Bifurcation ◽  
Saddle Node

In this article, we present the influence of a Hamiltonian saddle-node bifurcation on the high-dimensional phase space structures that mediate reaction dynamics. To achieve this goal, we identify the phase space invariant manifolds using Lagrangian descriptors, which is a trajectory-based diagnostic suitable for the construction of a complete “phase space tomography” by means of analyzing dynamics on low-dimensional slices. First, we build a Hamiltonian system with one degree-of-freedom (DoF) that models reaction, and study the effect of adding a parameter to the potential energy function that controls the depth of the well. Then, we extend this framework to a saddle-node bifurcation for a two DoF Hamiltonian, constructed by coupling a harmonic oscillator, i.e. a bath mode, to the other reactive DoF in the system. For this problem, we describe the phase space structures associated with the rank-1 saddle equilibrium point in the bottleneck region, which is a Normally Hyperbolic Invariant Manifold (NHIM) and its stable and unstable manifolds. Finally, we address the qualitative changes in the reaction dynamics of the Hamiltonian system due to changes in the well depth of the potential energy surface that gives rise to the saddle-node bifurcation.

WEAK DISSIPATION IN A HAMILTONIAN MAP

1994 ◽  
Vol 04 (04) ◽  
pp. 921-932 ◽  
Author(s):  
RAÚL J. MONDRAGÓN C. ◽  
PETER H. RICHTER
Keyword(s):  
Phase Space ◽  
Hamiltonian System ◽  
Periodic Orbits ◽  
Adiabatic Invariance ◽  
Bifurcation Tree ◽  
Bouncing Ball ◽  
Different Types ◽  
Weak Dissipation ◽  
Hamiltonian Map

The dynamics of a bouncing ball reflected off a harmonic spring is investigated, with weak dissipation of three different types. The phase space is found to be organized into a system of tubes that wind around the branches of the bifurcation tree of periodic orbits of the Hamiltonian system. Instead of attraction towards special periodic orbits we observe a kind of piecewise adiabatic invariance of the tubes, with jumps occurring when the branches penetrate each other.

Exact quantum theory of a time-dependent bound quadratic Hamiltonian system

1993 ◽  
Vol 48 (4) ◽  
pp. 2716-2720 ◽  
Author(s):  
Kyu Hwang Yeon ◽  
Kang Ku Lee ◽  
Chung In Um ◽  
Thomas F. George ◽  
Lakshmi N. Pandey
Keyword(s):  
Quantum Theory ◽  
Hamiltonian System ◽  
Time Dependent ◽  
Exact Quantum ◽  
Quadratic Hamiltonian
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