Action-Angle Variables Inherent in Quantum Dynamics

In this paper, we focus on the problem of searching for complex activities involving multiple, interacting objects in video. We examine the dynamics of formation and dispersal of groups as well as their interactions with other groups and individuals. In order to establish a general formalism, we examine activities using relative distances in phase space via pairwise analysis of all objects. This allows us to characterize interactions directly by modeling multi-object activities with the Multiple Objects, Pairwise Analysis (MOPA) feature vector, which is based upon physical models of complex interactions in phase space; specifically, we model paired motion as a damped oscillator in phase space. We model and recognize more complex interactions by characterizing pairs which are correlated in phase space as groups. We show how this model can be used for recognition of complex activities on the standard CAVIAR, VIVID, and UCR Videoweb datasets capturing a variety of problem settings.

Hamiltonians which are inaccessible in static systems can be engineered in periodically driven many-body systems, i.e., Floquet many-body systems. We propose to use interacting particles in a one-dimensional (1D) harmonic potential with periodic kicking to investigate two-dimensional topological and many-body physics. Depending on the driving parameters, the Floquet Hamiltonian of single kicked harmonic oscillator has various lattice structures in phase space. The noncommutative geometry of phase space gives rise to the topology of the system. We investigate the effective interactions of particles in phase space and find that the point-like contact interaction in quasi-1D real space becomes a long-rang Coulomb-like interaction in phase space, while the hardcore interaction in pure-1D real space becomes a confinement quark-like potential in phase space. We also find that the Floquet exchange interaction does not disappear even in the classical limit, and can be viewed as an effective long-range spin–spin interaction induced by collision. Our proposal may provide platforms to explore new physics and exotic phases by Floquet many-body engineering.

We investigate the interference of a kicked harmonic oscillator in phase space. With the measure of interference defined in Lee and Jeong [Phys. Rev. Lett. 106 (2011) 220401], we show that interference increases more rapidly in the chaotic regime than in the regular regime, and that the sub-Planck structure is of importance for the decoherence time in the chaotic regime. We also find that interference plays an important role in energy transport between the kicking fields and the kicked harmonic oscillator.

In the previous chapter we studied the kinematics of many-body systems, and the form of operators representing the physical properties of a system, all of which were embodied by the quantum field. In this chapter we shall study the quantum dynamics of many-body systems, which can also be embodied by the quantum fields. We shall employ the fact that the quantum dynamics of a system, instead of being described in terms of the dynamics of the states or the evolution operator, i.e. as previously done through the Schrodinger equation, can instead be carried by the quantum fields. The quantum dynamics is therefore expressed in terms of the correlation functions or Green's functions of the quantum fields evaluated with respect to some state of the system. In particular we shall consider the general case of quantum dynamics for arbitrary non-equilibrium states. After introducing various types of Green's functions and relating them to measurable quantities, we will discuss the simplifications reigning for the special case of equilibrium states. Quantum dynamics Quantum dynamics can be described in different ways since quantum mechanics is a linear theory and the dynamics described by a unitary transformation of states. This will come in handy in the next chapter when we study a quantum theory in terms of its perturbative expansion using the so-called interaction picture. Here we first discuss the Schrodinger and Heisenberg pictures. The Schrodinger picture Having the Hamiltonian on the multi-particle space at hand we can consider the dynamics described in the multi-particle space.

Objects with rhythms naturally synchronize. Synchronization is the coordination of events in order to run the system uniformly. Yet the phenomenon went entirely undocumented until 1665. Since the pioneering description of synchronization by Huygens, the phenomenon has been studied by various researchers in an interdisciplinary manner. Many researchers have contributed to the development of synchronization theory proving that synchronization occurs in coupled non-linear dissipative oscillators. Such oscillators range from mechanical clocks and population dynamics to human heart and neural networks. This article aims to explain the basic principles of synchronization theory. The history and applications of synchronization are discussed in real-world scenarios. We address different types of synchronization with a detailed discussion on the simplest type of synchronization. The phenomenon of synchronization applies to oscillations of different forms—periodic, noisy, and chaotic in nature. Here, we specifically discuss the oscillators which can hold synchronization. In particular, we provide an overview of self-sustained periodic and chaotic oscillators with a detailed description of different forms of these oscillators in phase space. Further, a summary of further research challenges has also been given for the future development of advanced applications based on natural synchronization phenomenon.

The Weyl correspondence for obtaining quantum operators from functions of classical coordinates and momenta is known to be incorrect. To calculate quantum-mechanical expectation values as phase-space averages with the Wigner density function, one cannot use classical functions but must use Weyl transforms. These transforms are defined and their properties derived from quantum mechanics. Their properties are expressed in terms of a Hermitian operator Δ(Q, K) whose Weyl transform is a δ function. The Wigner function is the transform of the density operator. Every Weyl transform is exhibited as a difference of two functions which are nonnegative on the phase spece. Weyl transforms do not obey the algebra of classical functions. In the classical limit ℏ → 0, Weyl transforms become classical functions, the Wigner function becomes nonnegative throughout the phase space, and the Hilbert space is spanned by an orthonormal set of vectors which are simultaneous eigenkets of the commuting coordinate and momentum operators.

We show that there is a close relationship between quantum mechanics and ordinary probability theory. The main difference is that in quantum mechanics the probability is computed in terms of an amplitude function, while in probability theory a probability distribution is used. Applying this idea, we then construct an amplitude model for quantum mechanics on phase space. In this model, states are represented by amplitude functions and observables are represented by functions on phase space. If we now postulate a conjugation condition, the model provides the same predictions as conventional quantum mechanics. In particular, we obtain the usual quantum marginal probabilities, conditional probabilities and expectations. The commutation relations and uncertainty principle also follow. Moreover Schrodinger's equation is shown to be an averaged version of Hamilton's equation in classical mechanics.

What is the meaning of the thermodynamical temperature in quantum mechanics? What is its role in the classical limit? What can we say about the interplay between quantum and thermal fluctuations? Can we impose a constant-temperature constraint within dynamical simulations of quantum systems as we do in simulations of classical systems? These questions will be reviewed here. In particular, it will be shown how a quantum version of the celebrated Nose constant-temperature dynamics can be defined within the phase space formulation of quantum mechanics.

Quasiclassical representations are given for the diffusion (heat) equation and the Schrodinger equation on a Riemannian manifold. These give an unambiguous functional integral expression for the solution of the diffusion (heat) equation and a (formal) unambiguous functional integral expression for the wave function solution of the Schrodinger equation intimately related to a corresponding classical mechanical problem on the manifold. Guided by these expressions we prove that in the curved space background of a Riemannian manifold quantum mechanics tends to classical mechanics as ħ tends to zero.

A diffusion process for charge distributions in a phase space is examined. The corresponding charge moves in a force field and under an action of a random field. There are the diffusion motions for coordinates and for momenta. In our model, an inner state of the charge is defined by a complex vector. The vector rotates with a great constant angular velocity with respect to the proper time of the charge. A state of the diffusion process is a (complex-valued) wave function on the phase space. As in quantum mechanics, we assume that, for the wave functions, the superposition principle holds. The diffusion process averages out vectors of inner states from different points of the phase space. A differential equation for this diffusion process is founded and examined. We demonstrate that the motion (described by this process) decomposes into a fast motion and a slow motion. The fast motion reduces an arbitrary wave function to a function from a subspace whose elements are parameterized by complex-valued functions of coordinates. The slow motion occurs in this subspace and it is described by the Schrodinger equation. The parameters of the suggested model are estimated. The duration of the fast motion is of order 10−11 s. Introduction In [1, 2], Maslov studied the problem of describing the motion of a charge distribution on a phase space under the action of the a random field on the

In this work we present a brief review about quantum mechanics in phase space. The approach discussed is based in the notion of symplectic structure and star-operators. In this sense, unitary representations for the Galilei group are construct, and the Schrodinger equation in phase space is derived. The connection between phase space amplitudes and Wigner function is presented. As a new result we solved the Schrodinger equation in phase space for simple pendulum. PACS Numbers: 11.10.Nx, 11.30.Cp, 05.20.Dd

The notion of brightness is efficiently conveyed in geometric optics as density of rays in phase space. Wigner has introduced his famous distribution in quantum mechanics as a quasiprobability density of a quantum system in phase space. Naturally, the same formalism can be used to represent light including all the wave phenomena as originally done by Walther and for synchrotron radiation by Kim. It provides a natural framework for radiation propagation and optics matching by transferring the familiar ``baggage'' of accelerator physics ($\ensuremath{\beta}$ function, emittance, phase-space transforms, etc.) to synchrotron radiation. More specifically, the use of Wigner distribution formalism allows a rigorous description of partially coherent non-Gaussian sources, which is generally the case for synchrotron radiation from an undulator with a high degree of transverse coherence. This paper reviews many of the properties of the Wigner distribution starting from quantum mechanics and provides examples of how its use enables physically insightful description of partially coherent synchrotron radiation in phase space. The concepts of diffraction limit and coherence are given an exact correspondence to their quantum mechanical counterparts. In particular, it is shown that the undulator radiation on resonance by a single electron is not diffraction limited though fully coherent. An extension of how to account for practically important cases of electron beams with nearly diffraction limited emittances is presented along with a discussion of appropriate figures of merit suitable for comparing future light sources.

Classical limits of quantum mechanics are studied on the basis of non-standard analysis. Following the idea of ultra eigenpairs in quantum mechanics extended to non-standard numbers, the exact equality in the Schroedinger equation is replaced by the approximate equality by taking Ħ as an infinitesimal (Ħ ≈0) and the classical limit is introduced as the operation represented by taking the standard part with respect to Ħ defined in the non-standard analysis. We find out that the ultra eigenfunctions which result from the extention of the exact equality to the approximate equality in the equation represent classical stationary states in the classical limit. As an explicit example we study harmonic-oscillator motions and derive two types of solutions: one is given by the solutions for stationary states described by ultra eigenfunctions on the extended Hilbert space* þ and the other by the solutions for non-stationary states represented by coherent states which are expressed as the superposition of eigenfunctions given by the non-standard extension of Hermite polynomials. We see that the former represents the statistical nature of quantum wave functions, while the latter their particle nature. Both solutions are shown to reproduce classical motions exactly in the classical limit. All interferences among different classical solutions disappear in the limit. We also see that the ultra eigenfunctions representing the stationary states cover not only the extensions of the eigenfunctions for the original potential, but also those for potentals which differ from the original one by an order smaller than Ħ. That is, they are spanned over many Hilbert spaces which differ one another in quantum-mechanical worlds whereĦ = a non-infinitesimal number is required. In the argument for measurements on the stationary states we see that the crucial difference between quantum mechanics and classical mechanics in expressed as follows; quantum mechanics does not have any infinitesimal physical constants in the theory, while classical mechanics does,i.e. Ħ ≈ 0 in this approach. Relations between detectors in physics and filters in non-standard analysis is also pointed out. The Schroedinger equation with the approximate equality based on the idea of the non-standard analysis is proposed as the fundamental equation for the unified theory describing quantum and classical phenomena simultaneously.

Classical statistical average values are generally generalized to average values of quantum mechanics. It is discovered that quantum mechanics is a direct generalization of classical statistical mechanics, and we generally deduce both a new general continuous eigenvalue equation and a general discrete eigenvalue equation in quantum mechanics, and discover that a eigenvalue of quantum mechanics is just an extreme value of an operator in possibility distribution, the eigenvalue f is just classical observable quantity. A general classical statistical uncertain relation is further given, and the general classical statistical uncertain relation is generally generalized to the quantum uncertainty principle; the two lost conditions in classical uncertain relation and quantum uncertainty principle, respectively, are found. We generally expound the relations among the uncertainty principle, singularity and condensed matter stability, discover that the quantum uncertainty principle prevents the appearance of singularity of the electromagnetic potential between nucleus and electrons, and give the failure conditions of the quantum uncertainty principle. Finally, we discover that the classical limit of quantum mechanics is classical statistical mechanics, the classical statistical mechanics may further be degenerated to classical mechanics and we discover that merely stating that the classical limit of quantum mechanics is classical mechanics is a mistake. As application examples, we deduce both the Schrödinger equation and the state superposition principle, and deduce that there exists a decoherent factor from a general mathematical representation of the state superposition principle; the consistent difficulty between statistical interpretation of quantum mechanics and determinant property of classical mechanics is overcome.

We study the statistical evolution of a charged particle moving in phase space under the action of vacuum fluctuations of the zero-point electromagnetic field. Our starting point is the Liouville equation, from which we derive a classical stochastic Schrodinger like equation for the probability amplitude in configuration space. It should be stressed that we are not deriving the Schrodinger wave equation. An equation formally identical to the Schrodinger equation used in Quantum Mechanics is obtained as a particular case of the classical stochastic Schrodinger like equation. An inconsistency appearing in the standard Schrodinger equation, when vacuum electromagnetic fluctuations and radiation reaction are taken into account, is clearly identified and explained. The classical stochastic Schrodinger like equation, however, is consistently interpreted within the realm of Stochastic Electrodynamics.