Transformations In Phase Space Research Articles

The Evolution of the Phase Space Structure Along Pitchfork and Period-Doubling Bifurcations in a 3D-Galactic Bar Potential

We investigate how the phase space structure of a Three-Dimensional (3D) autonomous Hamiltonian system evolves across a series of successive Two-Dimensional (2D) and 3D pitchfork and period-doubling bifurcations, as the transition of the parent families of Periodic Orbits (POs) from stability to simple instability leads to the creation of new stable POs. Our research illustrates the consecutive alterations in the phase space structure near POs as the stability of the main family of POs changes. This process gives rise to new families of POs within the system, either maintaining the same or exhibiting higher multiplicity compared to their parent families. Tracking such a phase space transformation is challenging in a 3D system. By utilizing the color and rotation technique to visualize the Four-Dimensional (4D) Poincaré surfaces of section of the system, i.e. projecting them onto a 3D subspace and employing color to represent the fourth dimension, we can identify distinct structural patterns. Perturbations of parent and bifurcating stable POs result in the creation of tori characterized by a smooth color variation on their surface. Furthermore, perturbations of simple unstable parent POs beyond the bifurcation point, which lead to the birth of new stable families of POs, result in the formation of figure-8 structures of smooth color variations. These figure-8 formations surround well-shaped tori around the bifurcated stable POs, losing their well-defined forms for energies further away from the bifurcation point. We also observe that even slight perturbations of highly unstable POs create a cloud of mixed color points, which rapidly move away from the location of the PO. Our study introduces, for the first time, a systematic visualization of 4D surfaces of section within the vicinity of higher multiplicity POs. It elucidates how, in these cases, the coexistence of regular and chaotic orbits contributes to shaping the phase space landscape.

Interpreting Symplectic Linear Transformations in a Two-Qubit Phase Space

Interpreting Symplectic Linear Transformations in a Two-Qubit Phase Space

Evolutionary transfer optimization-based approach for automated ictal pattern recognition using brain signals.

The visual scrutinization process for detecting epileptic seizures (ictal patterns) is time-consuming and prone to manual errors, which can have serious consequences, including drug abuse and life-threatening situations. To address these challenges, expert systems for automated detection of ictal patterns have been developed, yet feature engineering remains problematic due to variability within and between subjects. Single-objective optimization approaches yield less reliable results. This study proposes a novel expert system using the non-dominated sorting genetic algorithm (NSGA)-II to detect ictal patterns in brain signals. Employing an evolutionary multi-objective optimization (EMO) approach, the classifier minimizes both the number of features and the error rate simultaneously. Input features include statistical features derived from phase space transformations, singular values, and energy values of time-frequency domain wavelet packet transform coefficients. Through evolutionary transfer optimization (ETO), the optimal feature set is determined from training datasets and passed through a generalized regression neural network (GRNN) model for pattern detection of testing datasets. The results demonstrate high accuracy with minimal computation time (<0.5 s), and EMO reduces the feature set matrix by more than half, suggesting reliability for clinical applications. In conclusion, the proposed model offers promising advancements in automating ictal pattern recognition in EEG data, with potential implications for improving epilepsy diagnosis and treatment. Further research is warranted to validate its performance across diverse datasets and investigate potential limitations.

A Wheeler–DeWitt Non-Commutative Quantum Approach to the Branch-Cut Gravity

In this contribution, motivated by the quest to understand cosmic acceleration, based on the theory of Hořava–Lifshitz and on the branch-cut gravitation, we investigate the effects of non-commutativity of a mini-superspace of variables obeying the Poisson algebra on the structure of the branch-cut scale factor and on the acceleration of the Universe. We follow the guiding lines of a previous approach, which we complement to allow a symmetrical treatment of the Poisson algebraic variables and eliminate ambiguities in the ordering of quantum operators. On this line of investigation, we propose a phase-space transformation that generates a super-Hamiltonian, expressed in terms of new variables, which describes the behavior of a Wheeler–DeWitt wave function of the Universe within a non-commutative algebraic quantum gravity formulation. The formal structure of the super-Hamiltonian allows us to identify one of the new variables with a modified branch-cut quantum scale factor, which incorporates, as a result of the imposed variable transformations, in an underlying way, elements of the non-commutative algebra. Due to its structural character, this algebraic structure allows the identification of the other variable as the dual quantum counterpart of the modified branch-cut scale factor, with both quantities scanning reciprocal spaces. Using the iterative Range–Kutta–Fehlberg numerical analysis for solving differential equations, without resorting to computational approximations, we obtained numerical solutions, with the boundary conditions of the wave function of the Universe based on the Bekenstein criterion, which provides an upper limit for entropy. Our results indicate the acceleration of the early Universe in the context of the non-commutative branch-cut gravity formulation. These results have implications when confronted with information theory; so to accommodate gravitational effects close to the Planck scale, a formulation à la Heisenberg’s Generalized Uncertainty Principle in Quantum Mechanics involving the energy and entropy of the primordial Universe is proposed.

Simultaneous Momentum and Position Measurement and the Instrumental Weyl-Heisenberg Group.

The canonical commutation relation, [Q,P]=iℏ, stands at the foundation of quantum theory and the original Hilbert space. The interpretation of P and Q as observables has always relied on the analogies that exist between the unitary transformations of Hilbert space and the canonical (also known as contact) transformations of classical phase space. Now that the theory of quantum measurement is essentially complete (this took a while), it is possible to revisit the canonical commutation relation in a way that sets the foundation of quantum theory not on unitary transformations but on positive transformations. This paper shows how the concept of simultaneous measurement leads to a fundamental differential geometric problem whose solution shows us the following. The simultaneous P and Q measurement (SPQM) defines a universal measuring instrument, which takes the shape of a seven-dimensional manifold, a universal covering group we call the instrumental Weyl-Heisenberg (IWH) group. The group IWH connects the identity to classical phase space in unexpected ways that are significant enough that the positive-operator-valued measure (POVM) offers a complete alternative to energy quantization. Five of the dimensions define processes that can be easily recognized and understood. The other two dimensions, the normalization and phase in the center of the IWH group, are less familiar. The normalization, in particular, requires special handling in order to describe and understand the SPQM instrument.

Mapas padrão twist e não-twist

Symplectic maps can provide a straightforward and accurate way to visualize and quantify the dynamics of conservative systems with two degrees of freedom. These maps can be easily iterated from the simplest computers to obtain trajectories with great accuracy. Their usage arises in many fields, including celeste mechanics, plasma physics, chemistry, and so on. In this paper we introduce two examples of symplectic maps, the standard map and the standard non-twist map, exploring the phase space transformation as their control parameters are varied. Furthermore, for the non-twist map, we also present, that its chaotic motion is separated in phase space by the non-twist transport barrier.

Geometry of nonrelativistic string

The nonrelativistic bosonic string theory in a curved manifold is formulated here using gauging of symmetry approach (Galilean Gauge theory). The corresponding model in flat space has some global symmetries. By localizing these symmetries as per Galilean Gauge theory, the action for the nonrelativistic string interacting with gravity is obtained. A canonical analysis of the model has been performed which demonstrate that the transformations of the basic field variables under gauge transformations in phase space are equivalent to the diffeomorphism parameters by an exact mapping. Thus complete consistency of our results from both Lagrangian and Hamiltonian procedures are established.

A Multi-Frequency Tomographic Inverse Scattering Using Beam Basis Functions.

We present an overview of a beam-based approach to ultra-wide band (UWB) tomographic inverse scattering, where beam-waves are used for local data-processing and local imaging, as an alternative to the conventional plane-wave and Green’s function approaches. Specifically, the method utilizes a phase–space set of iso-diffracting beam-waves that emerge from a discrete set of points and directions in the source domain. It is shown that with a proper choice of parameters, this set constitutes a frame (an overcomplete generalization of a basis), termed “beam frame”, over the entire propagation domain. An important feature of these beam frames is that they need to be calculated once and then used for all frequencies, hence the method can be implemented either in the multi-frequency domain (FD), or directly in the time domain (TD). The algorithm consists of two phases: in the processing phase, the scattering data is transformed to the beam domain using windowed phase–space transformations, while in the imaging phase, the beams are backpropagated to the target domain to form the image. The beam-domain data is not only localized and compressed, but it is also physically related to the local Radon transform (RT) of the scatterer via a local Snell’s reflection of the beam-waves. This expresses the imaging as an inverse local RT that can be applied to any local domain of interest (DoI). In previous publications, the emphasis has been set on TD data processing using a special class of localized space–time beam-waves (wave-packets). The goal of the present paper is to present the imaging scheme in the UWB FD, utilizing simpler Fourier-based data-processing tools in the space and time domains.

Numerical solution of quadratic SDE with measurable drift

In this paper we are interested in solving numerically quadratic SDEs with non-necessary continuous drift of the from Xt = x + ? t 0 b(s,Xs)ds + ? t 0 f (Xs)?2(Xs)ds + ? t 0 ?(Xs)dWs, where, x is the initial data b and ? are given coefficients that are assumed to be Lipschitz and bounded and f is a measurable bounded and integrable function on the whole space R. Numerical simulations for this class of SDE of quadratic growth and measurable drift, induced by the singular term f (x)?2(x), is implemented and illustrated by some examples. The main idea is to use a phase space transformation to transform our initial SDEs to a standard SDE without the discontinuous and quadratic term. The Euler-Maruyama scheme will be used to discretize the new equation, thus numerical approximation of the original equation is given by taking the inverse of the space transformation. The rate of convergence are shown to be of order 1/2.

Stochastic Helmholtz problem and convergence in distribution

In the present paper, the solvability of the stochastic Helmholtz problem is investigated in the class of stochastic differential equations equivalent in distribution. Earlier, by additional variables method the Helmholtz problem was investigated in the class of stochastic differential equations equivalent almost surely (a.s.). The study of the stochastic Helmholtz problem in the class of equations equivalent in distribution allows us to significantly expand the region of its solvability. This is due to the possibility of using well-known methods of the theory of stochastic processes, such as the method of the phase space transformation, the method of absolutely continuous change of measure, and the method of random change of time. In that paper stochastic equations of the Lagrangian structure equivalent in distribution are constructed by the given second order Ito stochastic equations using the methods of phase space transformation, absolutely continuous measure transformation and randomtime substitution. The obtained results are illustrated by specific examples.

Quadratic BSDEs with jumps and related PIDEs

In this paper we are interested to solve a class of quadratic BSDEs with jumps (QBSDEJs for short) of the following form: Herein, the terminal data ξ will be assumed to be square integrable. Our study covers the following cases where f is a measurable and integrable function, is a functional of the unknown processes and to be defined later and h and enjoy some classical assumptions. The generators show quadratic growth in the Brownian component and non-linear functional form with respect to the jump term. Existence or uniqueness of solutions as well as a comparison and strict comparison principles are established under no monotonicity condition in the third argument of the generator. Probabilistic representations of solutions to some classes of quadratic PIDE are given by means of solutions of these QBSDEJs. The main idea is to use a phase space transformation to transform our initial QBSDEJ to a standard BSDEJ without quadratic term.

The Transformation of Commutative Phase Space to Noncommutative One, and its Lorentz Transformation-Like Forms

Noncommutative phase space of arbitrary dimension is discussed. We introduce momentum-momentum noncommutativity in addition to co-ordinate-coordinate noncommutativity. We find an exact form for the linear transformation which relates a noncommutative phase space to the corresponding ordinary one. By using this form, we show that a noncommutative phase space of arbitrary dimension can be represented by the direct sum of two-dimensional noncommutative ones. In two-dimension, we obtain the transformation which relates a noncommutative phase space to commutative one. The transformation has the Lorentz transformation-like forms and can also describe the Bopp's shift.

Restoring geometrical optics near caustics using sequenced metaplectic transforms

Geometrical optics (GO) is often used to model wave propagation in weakly inhomogeneous media and quantum-particle motion in the semiclassical limit. However, GO predicts spurious singularities of the wavefield near reflection points and, more generally, at caustics. We present a new formulation of GO, called metaplectic geometrical optics (MGO), that is free from these singularities and can be applied to any linear wave equation. MGO uses sequenced metaplectic transforms of the wavefield, corresponding to symplectic transformations of the ray phase space, such that caustics disappear in the new variables and GO is reinstated. The Airy problem and the quantum harmonic oscillator are described analytically using MGO for illustration. In both cases, the MGO solutions are remarkably close to the exact solutions and remain finite at cutoffs, unlike the usual GO solutions.

Cosmological spinor

We build upon previous investigation of the one-dimensional conformal symmetry of the Friedman-Lema\^ itre-Robertson-Walker (FLRW) cosmology of a free scalar field and make it explicit through a reformulation of the theory at the classical level in terms of a manifestly $\textrm{SL}(2,\mathbb{R})$-invariant action principle. The new tool is a canonical transformation of the cosmological phase space to write it in terms of a spinor, i.e. a pair of complex variables that transform under the fundamental representation of $\textrm{SU}(1,1)\sim\textrm{SL}(2,\mathbb{R})$. The resulting FLRW Hamiltonian constraint is simply quadratic in the spinor and FLRW cosmology is written as a Schr\"odinger-like action principle. Conformal transformations can then be written as proper-time dependent $\textrm{SL}(2,\mathbb{R})$ transformations. We conclude with possible generalizations of FLRW to arbitrary quadratic Hamiltonian and discuss the interpretation of the spinor as a gravitationally-dressed matter field or matter-dressed geometry observable.

T-dualities and Doubled Geometry of the Principal Chiral Model

The Principal Chiral Model (PCM) defined on the group manifold of SU(2) is here investigated with the aim of getting a further deepening of its relation with Generalized Geometry and Doubled Geometry. A one-parameter family of equivalent Hamiltonian descriptions is analysed, and cast into the form of Born geometries. Then O(3, 3) duality transformations of the target phase space are performed and we show that the resulting dual models are defined on the group SB(2, ℂ) which is the Poisson-Lie dual of SU(2) in the Iwasawa decomposition of the Drinfel’d double SL(2, ℂ). A parent action with doubled degrees of freedom and configuration space SL(2, ℂ) is then defined that reduces to either one of the dually related models, once suitable constraints are implemented.

Coupled oscillators in non-commutative phase space: Path integral approach

We present the path integral techniques in a non-commutative phase space and illustrate the calculation in the case of an exact problem of the coupled oscillator in two dimensions. The non-commutativity, with respect to Poisson (classical) and Heisenberg (quantum) brackets, in this phase space, is governed by two small constant parameters. They characterize the geometric deformation of this space. The path integral is formulated in a mixed representation due to the non-commutativity of the coordinates on one hand and those of the momentum on the other hand. Using a canonical linear transformation in this non-commutative phase space, we retrieve the commutative phase space properties by which the study becomes more suitable. The case of the non-commutative coupled harmonic oscillator in two dimensions is treated and the thermodynamics properties of the assembly of such oscillators are derived too.

Polymer quantum cosmology: Lifting quantization ambiguities using a SL(2,R) conformal symmetry

In this letter, we stress that the simplest cosmological model consisting in a massless scalar field minimally coupled to homogeneous and isotropic gravity has an in-built $\SL(2,\mathbb{R})$ symmetry. Protecting this symmetry naturally provides an efficient way to constrain the quantization of this cosmological system whatever the quantization scheme and allows in particular to fix the quantization ambiguities arising in the canonical quantization program. Applying this method to the loop quantization of the FLRW cosmology leads to a new loop quantum cosmology model which preserves the $\SL(2,\mathbb{R})$ symmetry of the classical system. This new polymer regularization consistent with the conformal symmetry can be derived as a non-linear canonical transformation of the classical FLRW phase space, which maps the classical singular dynamics into a regular effective bouncing dynamics. This improved regularization preserves the scaling properties of the volume and Hamiltonian constraint. 3d scale transformations, generated by the dilatation operator, are realized as unitary transformations despite the minimal length scale hardcoded in the theory. Finally, we point out that the resulting cosmological dynamics exhibits an interesting duality between short and long distances, reminiscent of the T-duality in string theory, with the near-singularity regime dual to the semi-classical regime at large volume. The technical details of the construction of this model are presented in a longer companion paper \cite{BenAchour:2019ywl}.

Gauge-free electromagnetic gyrokinetic theory

A new gauge-free electromagnetic gyrokinetic theory is developed, in which the gyrocenter equations of motion and the gyrocenter phase-space transformation are expressed in terms of the perturbed electromagnetic fields, instead of the usual perturbed potentials. Gyrocenter polarization and magnetization are derived explicitly from the gyrocenter Hamiltonian, up to first order in the gyrocenter perturbation expansion. Expressions for the sources in Maxwell's equations are derived in a form that is suitable for simulation studies, as well as kinetic-gyrokinetic hybrid modeling.

Design of a freeform uniformity corrector lens for extended sources in elliptical reflectors

Illumination design usually requires the collection of a large solid angle of radiation from the light source. However, it is known that elliptical reflectors in combination with extended uniform light sources result in a non-uniform irradiance profile at the secondary focus. Within this paper we propose a design method based on phase space transformations, which includes the source extension from the very beginning. We show that an analysis of the local mapping of the source to the target radiance distribution allows a profound understanding of the effects and in consequence a design concept for an additional freeform lens to correct the uniformity at the secondary focus.

Phase-space topography characterization of nonlinear ultrasound waveforms

Fundamental understanding of ultrasound interaction with material discontinuities having closed interfaces has many engineering applications such as nondestructive evaluation of defects like kissing bonds and cracks in critical structural and mechanical components. In this paper, to analyze the acoustic field nonlinearities due to defects with closed interfaces, the use of a common technique in nonlinear physics, based on a phase-space topography construction of ultrasound waveform, is proposed. The central idea is to complement the “time” and “frequency” domain analyses with the “phase-space” domain analysis of nonlinear ultrasound waveforms. A nonlinear time series method known as pseudo phase-space topography construction is used to construct equivalent phase-space portrait of measured ultrasound waveforms. Several nonlinear models are considered to numerically simulate nonlinear ultrasound waveforms. The phase-space response of the simulated waveforms is shown to provide different topographic information, while the frequency domain shows similar spectral behavior. Thus, model classification can be substantially enhanced in the phase-space domain. Experimental results on high strength aluminum samples show that the phase-space transformation provides a unique detection and classification capabilities. The Poincaré map of the phase-space domain is also used to better understand the nonlinear behavior of ultrasound waveforms. It is shown that the analysis of ultrasound nonlinearities is more convenient and informative in the phase-space domain than in the frequency domain.