scholarly journals MATRIX QUANTIZATION OF TURBULENCE

2012 ◽  
Vol 22 (09) ◽  
pp. 1250213 ◽  
Author(s):  
EMMANUEL FLORATOS
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Evolution Equations ◽  
Lorenz System ◽  
Quantum Mechanical ◽  
Noncommutative Phase Space ◽  
Commutation Relations ◽  
The Matrix ◽  
Lorenz Attractors ◽  
Volume Preserving

Based on our recent work on Quantum Nambu Mechanics [Axenides & Floratos 2009], we provide an explicit quantization of the Lorenz chaotic attractor through the introduction of noncommutative phase space coordinates as Hermitian N × N matrices in R3. For the volume preserving part, they satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction, it violates the quantum commutation relations. We demonstrate that the Heisenberg–Nambu evolution equations for the Matrix Lorenz system develop fast decoherence to N independent Lorenz attractors. On the other hand, there is a weak dissipation regime, where the quantum mechanical properties of the volume preserving nondissipative sector survive for long times.

scholarly journals Über der Kopplung von Schwingungsfreiheitsgraden an die Koordinate intramolekularer Umlagerungen / Coupling of Vibrational Degrees of Freedom lo the Coordinate of Intramolecular Rearrangements

1973 ◽  
Vol 28 (11) ◽  
pp. 1759-1781 ◽  
Author(s):  
J. Brickmann
Keyword(s):  
Degrees Of Freedom ◽  
Energy Surface ◽  
Quantum Mechanical ◽  
Matrix Elements ◽  
Damping Term ◽  
Intramolecular Rearrangements ◽  
Quantum Numbers ◽  
The Matrix ◽  
Localized Quantum States ◽  
Internal Degrees Of Freedom

Intramolecular rearrangements A ⇌ B are investigated, which can be described in terms of the motion of an effective quantum mechanical ‘‘particle’’ on an energy surface with at least two minima. We regard the energy surface as a function of a limited number of relevant internal degrees of freedom. The rate of isomerization is calculated from the matrix elements of a transition operator W with respect to the localized quantum states of the two isomers, and the coupling to the inter- and intramolecular degrees of freedom, not explizitely considered in the energy surface. It is shown that the matrixelements of W be reduced to integrals over functions of the adiabatic reaction coordinate of the isomerization, and selection rules for the vibrational quantum numbers for the motion perpendicular to this coordinate. The degrees of freedom not relevant for the reaction are summarily taken into account by introducing a heath bath in thermodynamic equilibrium and a simple damping term. Applications are discussed.

DEFORMED BOSON ALGEBRA AND PROJECTION OPERATOR OF VACUUM IN NONCOMMUTATIVE PHASE SPACE

2009 ◽  
Vol 24 (08) ◽  
pp. 591-600 ◽  
Author(s):  
BINGSHENG LIN ◽  
YONG GUAN ◽  
SICONG JING
Keyword(s):  
Phase Space ◽  
Coherent State ◽  
Projection Operator ◽  
Fock Space ◽  
Space Structure ◽  
Completeness Relation ◽  
Noncommutative Phase Space ◽  
New Formalism ◽  
Commutation Relations ◽  
Vacuum Structure

In this paper we introduce a new formalism to analyze Fock space structure of noncommutative phase space (NCPS). Based on this new formalism, we derive deformed boson commutation relations and study corresponding deformed Fock space, especially its vacuum structure, which leads to get a form of the vacuum projection operator. As an example of applications of such an operator, we define two-mode coherent state in the NCPS and show its completeness relation.

scholarly journals ACTION WITH ACCELERATION I: EUCLIDEAN HAMILTONIAN AND PATH INTEGRAL

2013 ◽  
Vol 28 (27) ◽  
pp. 1350137 ◽  
Author(s):  
BELAL E. BAAQUIE
Keyword(s):  
Boundary Conditions ◽  
Path Integral ◽  
Degrees Of Freedom ◽  
Similarity Transformation ◽  
Quantum Mechanical ◽  
Matrix Elements ◽  
Quantum Mechanical System ◽  
The Matrix ◽  
Correct Boundary Conditions ◽  
Acceleration Term

An action having an acceleration term in addition to the usual velocity term is analyzed. The quantum mechanical system is directly defined for Euclidean time using the path integral. The Euclidean Hamiltonian is shown to yield the acceleration Lagrangian and the path integral with the correct boundary conditions. Due to the acceleration term, the state space depends on both position and velocity — and hence the Euclidean Hamiltonian depends on two degrees of freedom. The Hamiltonian for the acceleration system is non-Hermitian and can be mapped to a Hermitian Hamiltonian using a similarity transformation; the matrix elements of the similarity transformation are explicitly evaluated.

Shape Dynamics and the Linking Theory

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Line Element ◽  
Hamiltonian Formulation ◽  
Operational Definition ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Problem Of Time ◽  
Definition Of ◽  
Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

scholarly journals Semi-Classical Localisation Properties of Quantum Oscillators on a Noncommutative Configuration Space

2015 ◽  
Vol 22 (04) ◽  
pp. 1550021 ◽  
Author(s):  
Fabio Benatti ◽  
Laure Gouba
Keyword(s):  
Phase Space ◽  
Configuration Space ◽  
Classical Limit ◽  
Coherent States ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Wigner Functions ◽  
Quantum Mechanical Oscillators ◽  
Mechanical Oscillators ◽  
Space Noncommutativity

When dealing with the classical limit of two quantum mechanical oscillators on a noncommutative configuration space, the limits corresponding to the removal of configuration-space noncommutativity and position-momentum noncommutativity do not commute. We address this behaviour from the point of view of the phase-space localisation properties of the Wigner functions of coherent states under the two limits.

scholarly journals Algebraic Approach to Exact Solution of the (2 + 1)-Dimensional Dirac Oscillator in the Noncommutative Phase Space

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
H. Panahi ◽  
A. Savadi
Keyword(s):  
Exact Solution ◽  
Phase Space ◽  
Algebraic Approach ◽  
Wave Functions ◽  
Dirac Oscillator ◽  
Noncommutative Phase Space ◽  
Energy Eigenvalues ◽  
Good Agreement

We study the (2 + 1)-dimensional Dirac oscillator in the noncommutative phase space and the energy eigenvalues and the corresponding wave functions of the system are obtained through the sl(2) algebraization. It is shown that the results are in good agreement with those obtained previously via a different method.

scholarly journals Quantum speed limit for a relativistic electron in the noncommutative phase space

2017 ◽  
Vol 32 (23n24) ◽  
pp. 1750143 ◽  
Author(s):  
Kang Wang ◽  
Yu-Fei Zhang ◽  
Qing Wang ◽  
Zheng-Wen Long ◽  
Jian Jing
Keyword(s):  
Relativistic Electron ◽  
Uniform Magnetic Field ◽  
Lorentz Invariance ◽  
Relativistic Quantum ◽  
Quantum Mechanical ◽  
Speed Limit ◽  
Speed Of Light ◽  
Minimum Evolution ◽  
Noncommutative Phase Space ◽  
Average Speed

The influence of the noncommutativity on the average speed of a relativistic electron interacting with a uniform magnetic field within the minimum evolution time is investigated. We find that it is possible for the wave packet of the electron to travel faster than the speed of light in vacuum because of the noncommutativity. It is a clear signature of violating Lorentz invariance in the noncommutative relativistic quantum mechanical region.

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