scholarly journals Classical and Quantum Integrability: A Formulation That Admits Quantum Chaos

2020 ◽  
Author(s):  
Paul Bracken
Keyword(s):  
Phase Space ◽  
Quantum System ◽  
Quantum Chaos ◽  
Quantum Phase ◽  
Distinct Advantage ◽  
Coset Space ◽  
Dynamical Group ◽  
Quantum Integrability ◽  
Carrier Space ◽  
Maximal Stability

The concept of integrability of a quantum system is developed and studied. By formulating the concepts of quantum degree of freedom and quantum phase space, a realization of the dynamics is achieved. For a quantum system with a dynamical group G in one of its unitary irreducible representative carrier spaces, the quantum phase space is a finite topological space. It is isomorphic to a coset space G/R by means of the unitary exponential mapping, where R is the maximal stability subgroup of a fixed state in the carrier space. This approach has the distinct advantage of exhibiting consistency between classical and quantum integrability. The formalism will be illustrated by studying several quantum systems in detail after this development.

scholarly journals On the Stochastic Origin of Quantum Mechanics

2017 ◽  
Vol 01 (03) ◽  
pp. 1750008 ◽  
Author(s):  
Roumen Tsekov
Keyword(s):  
Stochastic Process ◽  
Diffusion Coefficient ◽  
Quantum Mechanics ◽  
Phase Space ◽  
Quantum System ◽  
Liouville Equation ◽  
Point Of View ◽  
Quantum Phase ◽  
Relativistic Mass ◽  
Space Dynamics

The quantum Liouville equation, which describes the phase space dynamics of a quantum system of fermions, is analyzed from stochastic point of view as a particular example of the Kramers–Moyal expansion. Quantum mechanics is extended to relativistic domain by generalizing the Wigner–Moyal equation. Thus, an expression is derived for the relativistic mass in the Wigner quantum phase space presentation. The diffusion with an imaginary diffusion coefficient is discussed. An imaginary stochastic process is proposed as the origin of quantum mechanics.

INTEGRABILITY, ENTROPY AND QUANTUM COMPUTATION

1999 ◽  
Vol 10 (07) ◽  
pp. 1205-1228 ◽  
Author(s):  
E. V. KRISHNAMURTHY
Keyword(s):  
Quantum Computation ◽  
Quantum Chaos ◽  
Quantum Control ◽  
Dynamical Evolution ◽  
Open Problems ◽  
Quantum Integrability ◽  
Feynman Path Integrals ◽  
Exact Integrability ◽  
Time Arrow ◽  
Computational Systems

The important requirements are stated for the success of quantum computation. These requirements involve coherent preserving Hamiltonians as well as exact integrability of the corresponding Feynman path integrals. Also we explain the role of metric entropy in dynamical evolutionary system and outline some of the open problems in the design of quantum computational systems. Finally, we observe that unless we understand quantum nondemolition measurements, quantum integrability, quantum chaos and the direction of time arrow, the quantum control and computational paradigms will remain elusive and the design of systems based on quantum dynamical evolution may not be feasible.

scholarly journals Collective decay induce quantum phase transition in a well-controlled hybrid quantum system

2021 ◽  
Vol 21 ◽  
pp. 103832
Author(s):  
Dong-Yan Lü ◽  
Guang-Hui Wang ◽  
Yuan Zhou ◽  
Li Xu ◽  
Yong-Jin Hu ◽  
...  
Keyword(s):  
Phase Transition ◽  
Quantum System ◽  
Quantum Phase Transition ◽  
Quantum Phase ◽  
Hybrid Quantum System

scholarly journals Quantum phase-space analysis of the pendular cavity

2004 ◽  
Vol 70 (4) ◽  
Author(s):  
M. K. Olsen ◽  
A. B. Melo ◽  
K. Dechoum ◽  
A. Z. Khoury
Keyword(s):  
Phase Space ◽  
Quantum Phase ◽  
Phase Space Analysis ◽  
Space Analysis

One-Dimensional Harmonic Oscillator in Quantum Phase Space

2011 ◽  
Vol 110-116 ◽  
pp. 3750-3754
Author(s):  
Jun Lu ◽  
Xue Mei Wang ◽  
Ping Wu
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Quantum Phase ◽  
Space Representation ◽  
Wave Mechanics ◽  
One Dimensional ◽  
Matrix Mechanics ◽  
The Matrix ◽  
Quantum Wave ◽  
The One

Within the framework of the quantum phase space representation established by Torres-Vega and Frederick, we solve the rigorous solutions of the stationary Schrödinger equations for the one-dimensional harmonic oscillator by means of the quantum wave-mechanics method. The result shows that the wave mechanics and the matrix mechanics are equivalent in phase space, just as in position or momentum space.

TOPOLOGICAL ASPECTS OF THE SPECIFIC HEAT

2009 ◽  
Vol 23 (20n21) ◽  
pp. 4170-4185 ◽  
Author(s):  
C. M. SARRIS ◽  
A. N. PROTO
Keyword(s):  
Phase Space ◽  
Specific Heat ◽  
Quantum System ◽  
Uncertainty Principle ◽  
Positive Definite ◽  
Metric Tensor ◽  
Quantum Nature ◽  
Generalized Phase Space

We describe how the specific heat of a quantum system is related to a positive definite metric defined on the generalized phase space in which the dynamics and thermodynamics of the system take place. This relationship is given through the components of a second-rank covariant metric tensor, enhancing a topological nature of the specific heat. We also present two examples where it can be seen how the uncertainty principle imposes strong constraints on the values achieved by the specific heat showing its inherent quantum nature.

ENERGY SPECTRAL STATISTICS IN THE QUANTUM SYSTEM WITH TWO PARTICLES

2004 ◽  
Vol 18 (17n19) ◽  
pp. 2740-2744 ◽  
Author(s):  
SHIPING YANG ◽  
GUOYONG YUAN ◽  
ZHE LI ◽  
HONG CHANG ◽  
DE LIU
Keyword(s):  
Energy Level ◽  
Quantum System ◽  
Quantum Chaos ◽  
Level Spacing ◽  
Tunnelling Effect ◽  
Spectral Statistics

In this paper, the quantum system with two particles is analyzed and the energy level spacing statistics distribution and Δ3-statistic are given. The results show that hard quantum chaos appear in the system with a certain potential. Tunnelling effect develops quantum chaos.

The Duflo non-commutative Fourier transform for the Lorentz group

2020 ◽  
Vol 17 (supp01) ◽  
pp. 2040011
Author(s):  
Giacomo Rosati
Keyword(s):  
Fourier Transform ◽  
Phase Space ◽  
Quantum System ◽  
Lie Group ◽  
Lorentz Group ◽  
Commutative Algebra ◽  
Cotangent Bundle ◽  
Irreducible Representations ◽  
Algebra Representation

For a quantum system whose phase space is the cotangent bundle of a Lie group, like for systems endowed with particular cases of curved geometry, one usually resorts to a description in terms of the irreducible representations of the Lie group, where the role of (non-commutative) phase space variables remains obscure. However, a non-commutative Fourier transform can be defined, intertwining the group and (non-commutative) algebra representation, depending on the specific quantization map. We discuss the construction of the non-commutative Fourier transform and the non-commutative algebra representation, via the Duflo quantization map, for a system whose phase space is the cotangent bundle of the Lorentz group.

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