Invariant Phase Space Research Articles

Reduced dimensional Monte Carlo method: Preliminary integrations.

A technique for reducing the number of integrals in a Monte Carlo calculation is introduced. For integrations relying on classical or mean-field trajectories with local weighting functions, it is possible to integrate analytically at least half of the integration variables prior to setting up the particular Monte Carlo calculation of interest, in some cases more. Proper accounting of invariant phase space structures shows that the system's dynamics is reducible into composite stable and unstable degrees of freedom. Stable degrees of freedom behave locally in the reduced dimensional phase space exactly as an analogous integrable system would. Classification of the unstable degrees of freedom is dependent upon the degree of chaos present in the dynamics. The techniques for deriving the requisite canonical coordinate transformations are developed and shown to block diagonalize the stability matrix into irreducible parts. In doing so, it is demonstrated how to reduce the amount of sampling directions necessary in a Monte Carlo simulation. The technique is illustrated by calculating return probabilities and expectation values for different dynamical regimes of a two-degrees-of-freedom coupled quartic oscillator within a classical Wigner method framework.

Search for Electron Bursts in the Inner Van Allen Belts with the CSES and NOAA POES Satellites

Earthquake monitoring plays a key role in human life protection, especially in highly populated areas of the Earth. First indications have been found in SAMPEX and NOAA satellite data that particle fluxes of trapped electrons in the Van Allen belts can be correlated to seismic activity on the ground. Within the framework of the CSES mission, a systematic analysis of the electron flux, within the invariant phase space in the L-shell and equatorial pitch angle, has started with the goal to identify short-term variations of the flux (particle bursts) in conjunction with seismic activity. This analysis is based on the statistical evaluation of the flux measurements and built to be potentially implemented in an online monitoring system. The first milestone in that direction has been achieved, providing a stable background estimation. With the injection of artificial signals, the efficiency of the method was evaluated and found to be better than 95% for isolated (one per day) and short (∼3 min) signals with a significance ≥ 5σ above the background. The developed method is set up in a manner to be easily applicable to data from different instruments and satellites. This article presents the studied datasets of the low-energy HEPP instrument onboard the CSES-01 satellite and the electron telescopes of the MEPED detector onboard the NOAA POES-19 satellite, introduces the method for the background estimation, and discusses the first correlation studies of particle bursts with geomagnetic indices obtained within this framework.

Hamiltonian chemical kinetics for studying roaming in formaldehyde dissociation: Linear and nonlinear models

A recently developed theory for formulating chemical kinetics with a Hamiltonian theory similar to classical mechanics is applied to study linear and nonlinear kinetic models for the photodissociation of formaldehyde ( H 2 CO ) via roaming pathways. Chemical kinetic equations are proposed by mapping time invariant phase space structures, such as stationary points of the potential energy surface and periodic orbits, to relevant intermediate chemical species. Solving kinetic equations in the Hamiltonian framework, we calculate reaction paths to asymptotically stable equilibria for closed reactions and varying the initial concentration of H2CO at constant temperature and pressure. The reaction paths may be classified by defining a thermodynamic distance between initial and equilibrium states, as well as the entropy production. A brusselator-type nonlinear model is proposed as the phenomenological equivalent to the periodic orbit description of roaming mechanism given before. Treating the photodissociation of formaldehyde as an open system and properly selecting the rate coefficients of the reactions and the initial concentration of formaldehyde, a periodic orbit emanates as the result of a Hamiltonian Hopf bifurcation. This yields an oscillatory reaction between the two long-range metastable intermediates, { H…CHO } and { H…HCO }, which lead to radical and molecular products.

Implications of the generalized uncertainty principle on the Walecka model equation of state and neutron star structure

We explore the implications of the generalized uncertainty principle (GUP) on the nuclear equation of state (EoS) and on the structure of neutron stars. Two approaches of GUP are used: the quadratic GUP approach, satisfying minimal length and the linear GUP approach, satisfying both minimal length and maximal momentum. The resulting invariant phase space volumes from these GUP approaches are applied to the [Formula: see text]-[Formula: see text] or Walecka model, serving as a starting point for neutron matter in the relativistic mean field theory. We find that linear GUP increases the range of energy densities corresponding to instabilities in the [Formula: see text]-[Formula: see text] EoS, while quadratic GUP decreases it. A stable EoS was constructed from the GUP-modified EoS via Maxwell construction, and this was fed into the Tolman–Oppenheimer–Volkoff equations and the mass–radius relation of neutron stars was obtained. We observe linear GUP to decrease both the maximum mass and limiting radius of the neutron star, while shifting the whole mass–radius relation to the low-radius regime. Meanwhile, quadratic GUP increases the maximum mass and limiting radius, and the mass-radius relation is shifted to the high-radius regime. The effects that are observed for both GUP modifications in the EoS and mass–radius relations get more prominent as we increase the values of the still unknown GUP parameters.

Meson Photo-Production in GEANT4 for $E_\gamma = 0.225$Eγ=0.225-3.0 GeV using the $\gamma + p \rightarrow p + \pi^{0}$γ+p→p+π0 reaction

The study of nuclear resonances in the intermediate energy regime requires accurate modeling of the experimental physics background. The cross sections in the GEANT4 toolkit do not adequately treat meson photo-production. The purpose of this study is to compare experimental data and theoretical models implemented in version 9.6.p02 of the GEANT4 Monte Carlo simulation toolkit for the exclusive and differential cross sections of $ \pi^0$ photo-production. Two models were studied: the Chiral Invariant Phase Space (CHIPS) and Bertini Cascade (BERT). The cross section data from the models were compared to the experimental data from the CEBAF Large Acceptance Spectrometer (CLAS) of Jefferson Lab and the phenomenological Scattering Analysis Interactive Dial-in (SAID) model. Relativistic Breit-Wigner fits were used to compare the $\Delta(1232)$, $N(1520)$, $N(1650)$ and $N(1875)$ resonances from SAID, CLAS and CHIPS, giving agreements to within 3% for the mass of each resonance. The limited tabulated data set of BERT does not allow for an accurate description of the energy distribution, even following its improvement in version 10p02. The angular distributions of the cross sections show significant differences between CHIPS and both CLAS and SAID data, the latter being the most accurate model to describe the $\gamma p\rightarrow p\pi^0$ exclusive cross section. The physics in CHIPS appears to be correct but it needs further development for the individual resonances contributions that could be useful since it is also the only model able to treat electromagnetically induced meson production for $A > 1$.

Harmonic Oscillator Chain in Noncommutative Phase Space with Rotational Symmetry

We consider a quantum space with a rotationally invariant noncommutative algebra of coordinates and momenta. The algebra contains the constructed tensors of noncommutativity involving additional coordinates and momenta. In the rotationally invariant noncommutative phase space, the harmonic oscillator chain is studied. We obtain that the noncommutativity affects the frequencies of the system. In the case of a chain of particles with harmonic oscillator interaction, we conclude that, due to the noncommutativity of momenta, the spectrum of the center-of-mass of the system is discrete and corresponds to the spectrum of a harmonic oscillator.

Independently parameterised momenta variables and Monte Carlo IR subtraction

We introduce a system of parameters for the Monte Carlo generation of Lorentz invariant phase space that is particularly well-suited to the treatment of the infrared divergences that occur in the most singular, Born-like configurations of 1 → n QCD processes. A key feature is that particle momenta are generated independently of one another, leading to a simple parameterisation of all such IR limits. We exemplify the use of these variables in conjunction with the projection to Born subtraction technique at next-to-next-to-leading order. The geometric origins of this parameterisation lie in a coordinate chart on a Grassmannian manifold.

System of interacting harmonic oscillators in rotationally invariant noncommutative phase space

Rotationally invariant space with noncommutativity of coordinates and noncommutativity of momenta of canonical type is considered. A system of N interacting harmonic oscillators in uniform field and a system of N particles with harmonic oscillator interaction are studied. We analyze effect of noncommutativity on the energy levels of these systems. It is found that influence of coordinates noncommutativity on the energy levels of the systems increases with increasing of the number of particles. The spectrum of N free particles in uniform field in rotationally invariant noncommutative phase space is also analyzed. It is shown that the spectrum corresponds to the spectrum of a system of N harmonic oscillators with frequency determined by the parameter of momentum noncommutativity.

Rotationally invariant noncommutative phase space of canonical type with recovered weak equivalence principle

We study the influence of noncommutativity of coordinates and noncommutativity of momenta on the motion of a particle (macroscopic body) in uniform and nonuniform gravitational fields in noncommutative phase space of canonical type with preserved rotational symmetry. It is shown that because of noncommutativity the motion of a particle in a gravitational field is determined by its mass. The trajectory of motion of a particle in a uniform gravitational field corresponds to the trajectory of a harmonic oscillator with frequency determined by the value of the parameter of momentum noncommutativity and mass of the particle. The equations of motion of a macroscopic body in a gravitational field depend on its mass and composition. From this follows a violation of the weak equivalence principle caused by noncommutativity. We conclude that the weak equivalence principle is recovered in rotationally invariant noncommutative phase space if we consider the tensors of noncommutativity to be dependent on mass. So, finally we construct noncommutative algebra which is rotationally invariant, equivalent to noncommutative algebra of canonical type, and does not lead to violation of the weak equivalence principle.

Effect of noncommutativity on the spectrum of free particle and harmonic oscillator in rotationally invariant noncommutative phase space

We consider rotationally invariant noncommutative algebra with tensors of noncommutativity constructed with the help of additional coordinates and momenta. The algebra is equivalent to the well-known noncommutative algebra of canonical type. In the noncommutative phase space, rotational symmetry influence of noncommutativity on the spectrum of free particle and the spectrum of harmonic oscillator is studied up to the second-order in the parameters of noncommutativity. We find that because of momentum noncommutativity, the spectrum of free particle is discrete and corresponds to the spectrum of harmonic oscillator in the ordinary space (space with commutative coordinates and commutative momenta). We obtain the spectrum of the harmonic oscillator in the rotationally invariant noncommutative phase space and conclude that noncommutativity of coordinates affects its mass. The frequency of the oscillator is affected by the coordinate noncommutativity and the momentum noncommutativity. On the basis of the results, the eigenvalues of squared length operator are found and restrictions on the value of length in noncommutative phase space with rotational symmetry are analyzed.

Composite system in rotationally invariant noncommutative phase space

Composite system is studied in noncommutative phase space with preserved rotational symmetry. We find conditions on the parameters of noncommutativity on which commutation relations for coordinates and momenta of the center-of-mass of composite system reproduce noncommutative algebra for coordinates and momenta of individual particles. Also, on these conditions, the coordinates and the momenta of the center-of-mass satisfy noncommutative algebra with effective parameters of noncommutativity which depend on the total mass of the system and do not depend on its composition. Besides, it is shown that on these conditions the coordinates in noncommutative space do not depend on mass and can be considered as kinematic variables, the momenta are proportional to mass as it has to be. A two-particle system with Coulomb interaction is studied and the corrections to the energy levels of the system are found in rotationally invariant noncommutative phase space. On the basis of this result the effect of noncommutativity on the spectrum of exotic atoms is analyzed.

Noncommutative phase space with rotational symmetry and hydrogen atom

We construct algebra with noncommutativity of coordinates and noncommutativity of momenta which is rotationally invariant and equivalent to noncommutative algebra of canonical type. Influence of noncommutativity on the energy levels of hydrogen atom is studied in rotationally invariant noncommutative phase space. We find corrections to the levels up to the second order in the parameters of noncommutativity and estimate the upper bounds of these parameters.

Black hole shadows and invariant phase space structures

Utilizing concepts from dynamical systems theory, we demonstrate how the existence of light rings, or fixed points, in a spacetime will give rise to families of periodic orbits and invariant manifolds in phase space. It is shown that these structures define the shape of the black hole shadow as well as a number of salient features of the spacetime lensing. We illustrate this through the analysis of lensing by a hairy black hole.

Detecting kinematic boundary surfaces in phase space: particle mass measurements in SUSY-like events

We critically examine the classic endpoint method for particle mass determination, focusing on difficult corners of parameter space, where some of the measurements are not independent, while others are adversely affected by the experimental resolution. In such scenarios, mass differences can be measured relatively well, but the overall mass scale remains poorly constrained. Using the example of the standard SUSY decay chain tilde{q}to {tilde{chi}}_2^0to tilde{ell}to {tilde{chi}}_1^0 , we demonstrate that sensitivity to the remaining mass scale parameter can be recovered by measuring the two-dimensional kinematical boundary in the relevant three-dimensional phase space of invariant masses squared. We develop an algorithm for detecting this boundary, which uses the geometric properties of the Voronoi tessellation of the data, and in particular, the relative standard deviation (RSD) of the volumes of the neighbors for each Voronoi cell in the tessellation. We propose a new observable, overline{Sigma} , which is the average RSD per unit area, calculated over the hypothesized boundary. We show that the location of the overline{Sigma} maximum correlates very well with the true values of the new particle masses. Our approach represents the natural extension of the one-dimensional kinematic endpoint method to the relevant three dimensions of invariant mass phase space.

Generalized higher-order Snyder model

In this paper we consider the higher-order generalization of the Snyder algebra. We find the modified inner product for this algebra. We also discuss the one-dimensional hydrogen atom eigenvalue problem. We find the invariant phase space volume for the generalized Snyder algebra. Finally we use the density of states in three dimension to discuss the Maxwellian distribution and cosmological constant.

Evolution of the velocity gradient tensor invariant dynamics in a turbulent boundary layer

In order to improve the physical understanding of the development of turbulent structures, the compressible evolution equations for the first three invariants $P$, $Q$ and $R$ of the velocity gradient tensor have been derived. The mean evolution of characteristic turbulent structure types in the $QR$-space were studied and compared at different wall-normal locations of a compressible turbulent boundary layer. The evolution of these structure types is fundamental to the physics that needs to be captured by turbulence models. Significant variations of the mean evolution are found across the boundary layer. The key features of the changes of the mean trajectories in the invariant phase space are highlighted and the consequences of the changes are discussed. Further, the individual elements of the overall evolution are studied separately to identify the causes that lead to the evolution varying with the distance to the wall. Significant impact of the wall-normal location on the coupling between the pressure-Hessian tensor and the velocity gradient tensor was found. The highlighted features are crucial for the development of more universal future turbulence models.

Spinning geometry = Twisted geometry

It is well known that the SU(2)-gauge invariant phase space of loop gravity can be represented in terms of twisted geometries. These are piecewise-linear-flat geometries obtained by gluing together polyhedra, but the resulting geometries are not continuous across the faces. Here we show that this phase space can also be represented by continuous, piecewise-flat three-geometries called spinning geometries. These are composed of metric-flat three-cells glued together consistently. The geometry of each cell and the manner in which they are glued is compatible with the choice of fluxes and holonomies. We first remark that the fluxes provide each edge with an angular momentum. By studying the piecewise-flat geometries which minimize edge lengths, we show that these angular momenta can be literally interpreted as the spin of the edges: the geometries of all edges are necessarily helices. We also show that the compatibility of the gluing maps with the holonomy data results in the same conclusion. This shows that a spinning geometry represents a way to glue together the three-cells of a twisted geometry to form a continuous geometry which represents a point in the loop gravity phase space.

On the form invariant volume transformation in phase space by focusing neutron guides: An analytic treatment

Tapered guides with supermirror coating are frequently used to focus neutron beams on specimens. The divergence distribution in the focused beam is of a great importance for the quality of neutron instrumentation. Using an analytic approach we derive the tapering which is needed to achieve a form invariant phase space transformation of a rectangular phase volume. In addition we consider the effect of beam attenuation by the finite reflectivity of supermirrors.

Principle of relative locality

We propose a deepening of the relativity principle according to which the invariant arena for non-quantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in spacetimes are constructed by observers, but different observers, separated from each other by translations, construct different spacetime projections from the invariant phase space. Nonetheless, all observers agree that interactions are local in the spacetime coordinates constructed by observers local to them. This framework, in which absolute locality is replaced by relative locality, results from deforming momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of momentum space geometry, such as its curvature, torsion and non-metricity, are reflected in different kinds of deformations of the energy-momentum conservation laws. These are in principle all measurable by appropriate experiments. We also discuss a natural set of physical hypotheses which singles out the cases of momentum space with a metric compatible connection and constant curvature.

Static Isolated Horizons: SU(2) Invariant Phase Space, Quantization, and Black Hole Entropy

We study the classical field theoretical formulation of static generic isolated horizons in a manifestly SU(2) invariant formulation. We show that the usual classical description requires revision in the non-static case due to the breaking of diffeomorphism invariance at the horizon leading to the non-conservation of the usual pre-symplectic structure. We argue how this difficulty could be avoided by a simple enlargement of the field content at the horizon that restores diffeomorphism invariance. Restricting our attention to static isolated horizons we study the effective theories describing the boundary degrees of freedom. A quantization of the horizon degrees of freedom is proposed. By defining a statistical mechanical ensemble where only the area aH of the horizon is fixed macroscopically—states with fluctuations away from spherical symmetry are allowed—we show that it is possible to obtain agreement with the Hawkings area law (S = aH /(4l 2p)) without fixing the Immirzi parameter to any particular value: consistency with the area law only imposes a relationship between the Immirzi parameter and the level of the Chern-Simons theory involved in the effective description of the horizon degrees of freedom.