Invariant Phase Space Research Articles

Phase space descriptions for simplicial 4D geometries

Starting from the canonical phase space for discretized (4D) BF theory, we implement a canonical version of the simplicity constraints and construct phase spaces for simplicial geometries. Our construction allows us to study the connection between different versions of Regge calculus and approaches using connection variables, such as loop quantum gravity. We find that on a fixed triangulation the (gauge invariant) phase space associated with loop quantum gravity is genuinely larger than the one for length and even area Regge calculus. Rather, it corresponds to the phase space of area–angle Regge calculus, as defined in [1] (prior to the imposition of gluing constraints, which ensure the metricity of the triangulation). Finally, we show that for a subclass of triangulations one can construct first-class Hamiltonian and diffeomorphism constraints leading to flat 4D spacetimes.

Minimal length in quantum gravity, equivalence principle and holographic entropy bound

A possible discrepancy has been found between the results of a neutron interferometry experiment and quantum mechanics. This experiment suggests that the weak equivalence principle is violated at small length scales, which quantum mechanics cannot explain. In this paper, we investigated whether the generalized uncertainty principle (GUP), proposed by some approaches to quantum gravity such as string theory and doubly special relativity theories, can explain the violation of the weak equivalence principle at small length scales. We also investigated the consequences of the GUP on the Liouville theorem in statistical mechanics. We have found a new form of invariant phase space in the presence of GUP. This result should modify the density states and affect the calculation of the entropy bound of local quantum field theory, the cosmological constant, black body radiation, etc. Furthermore, such modification may have observable consequences at length scales much larger than the Planck scale. This modification leads to a -type correction to the bound of the maximal entropy of a bosonic field which would definitely shed some light on the holographic theory.

Transition state geometry near higher-rank saddles in phase space

We present a detailed analysis of invariant phase space structures near higher-rank saddles of Hamiltonian systems. Using the theory of pseudo-hyperbolic invariant surfaces, we show the existence of codimension-one normally hyperbolic invariant manifolds that govern transport near the higher-rank saddle points. Such saddles occur in a number of problems in celestial mechanics, chemical reactions, and atomic physics. As an example, we consider the problem of double ionization of helium in an external electric field, a basis of many modern ionization experiments. In this example, we illustrate our main results on the geometry and transport properties near a rank-two saddle.

Time reversible molecular dynamics algorithms with holonomic bond constraints in the NPH and NPT ensembles using molecular scaling

A modification of the constrained equations of motion of Kalibaeva et al. [Mol. Phys. 101, 765 (2003)] in the NPH and NPT ensembles is presented. The modified equations of motion are discretized using central-difference techniques, and the derived integrators are time reversible and conserve the invariant phase space measure. The constraint algorithm builds on the work of Toxvaerd et al. [J. Chem. Phys. 131, 064102 (2009)] in the NVE and NVT ensembles: it thus conserves the holonomic bond constraints at the finite machine precision level in the NPH and NPT ensembles. The algorithms were tested on a system of n=320 ortho-terphenyl molecules, arriving at the target temperature and pressure in a low and high pressure state. Isobaric heat capacities in the NPH and NPT ensembles were calculated for comparison using the fluctuation formulas as well as the thermodynamic definition. The heat capacities agree within the estimated uncertainties.

From the BRST invariant Hamiltonian to the field–antifield formalism

We study the relation between the Lagrangian field–antifield formalism and the BRST invariant phase–space formulation of gauge theories. Starting from the Batalin–Fradkin–Vilkovisky unitarized action, we demonstrate in a deductive way the equivalence of the phase–space, and the Lagrangian field–antifield partition functions for the case of irreducible first rank theories.

NON-ABELIAN GAUGED CHIRAL BOSON WITH A GENERALIZED FADDEEVIAN REGULARIZATION

We consider non-Abelian gauged version of chiral boson with a generalized Faddeevian regularization. It is a second-class constrained theory. We quantize the theory and analyze the phase space. It is shown that in spite of the lack of manifest Lorentz invariance in the action, it has a consistent and Poincaré invariant phase space structure.

Local virial relation and velocity anisotropy in self-gravitating system

We investigate the merging process in N-body self-gravitating system from the viewpoints of the local virial relation which is the relation between the local kinetic energy and the local potential. We compare both the density profile and the phase space density profile in cosmological simulations with the critical solutions of collisionless static state satisfying the local virial (LV) relation. We got the results that the critical solution can explain the characteristic density profile with the appropriate value of anisotropy parameter β ∼ 0.5. It can also explain the power law of the phase space density profile in the outer part of a bound state. However, it fails in explaining the central low temperature part which is connected to the scale invariant phase space density. It can be well fitted to the critical solution with the higher value of β ∼ 0.75. These results indicate that the LV relation is not compatible with the scale invariant phase space density in cosmological simulation.

Phase-space description of momentum spectra in relativistic heavy-ion collisions

We consider a phase-space model for particle production in nuclear collisions. Once the multiplicities of the individual particle species are known, single-inclusive momentum spectra can be computed after making simplifying assumptions for the matrix element for multiparticle production. Comparison of the calculated spectra with data for pions and kaons from central Pb + Pb collisions at E Lab = 158 A GeV reveals a residual longitudinal phase-space dominance in the final state of the reaction. We account for this by modifying the isotropic, relativistic invariant phase space in a way which retains boost invariance in beam direction but suppresses large transverse momenta. Adjusting a single parameter, we obtain a reasonably good description of transverse momentum and rapidity spectra for both pions and kaons.

Approximating stable and unstable manifolds in experiments.

We introduce a procedure to reveal invariant stable and unstable manifolds, given only experimental data. We assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle. We show that the method is able to capture the fine structure of the manifold, is independent of dimension, and is efficient relative to previous techniques.

PROPERTIES OF THE INVARIANT DISK PACKING IN A MODEL BANDPASS SIGMA–DELTA MODULATOR

In this paper we discuss the packing properties of invariant disks defined by periodic behavior of a model for a bandpass Σ–Δ modulator. The periodically coded regions form a packing of the forward invariant phase space by invariant disks. For this one-parameter family of PWIs, by introducing codings underlying the map operations we give explicit expressions for the centers of the disks by analytic functions of the parameters, and then show that tangencies between disks in the packings are very rare; more precisely they occur on parameter values that are at most countably infinite. We indicate how similar results can be obtained for other plane maps that are piecewise isometries.

Non-Hamiltonian molecular dynamics: Generalizing Hamiltonian phase space principles to non-Hamiltonian systems

The use of non-Hamiltonian dynamical systems to perform molecular dynamics simulation studies is becoming standard. However, the lack of a sound statistical mechanical foundation for non-Hamiltonian systems has caused numerous misconceptions about the phase space distribution functions generated by these systems to appear in the literature. Recently, a rigorous classical statistical mechanical theory of non-Hamiltonian systems has been derived, [M. E. Tuckerman, et al., Europhys. Lett. 45, 149 (1999)]. In this paper, the new theoretical formulation is employed to develop the non-Hamiltonian generalization of the usual Hamiltonian based statistical mechanical phase space principles. In particular, it is shown how the invariant phase space measure and the complete sets of conservation laws of the dynamical system can be combined with the generalized Liouville equation for non-Hamiltonian systems to produce a well defined expression for the phase space distribution function. The generalization provides a systematic, controlled procedure for designing non-Hamiltonian molecular dynamics algorithms which can be used to generate nonmicrocanonical ensembles, stationary nonequilibrium flows, and/or the dynamics of constrained systems. In light of this new general analysis, molecular dynamics algorithms for the canonical and isothermal–isobaric ensembles are examined, potential difficulties are illuminated, and the limitations of previous theoretical treatments are elucidated.

Chaos-assisted tunneling with cold atoms

In the context of quantum chaos, both theory and numerical analysis predict large fluctuations of the tunneling transition probabilities when irregular dynamics is present at the classical level. Here we consider the nondissipative quantum evolution of cold atoms trapped in a time-dependent modulated periodic potential generated by two laser beams. We give some precise guidelines for the observation of chaos-assisted tunneling between invariant phase space structures paired by time-reversal symmetry.

Blowout bifurcations and the onset of magnetic activity in turbulent dynamos.

The transition to magnetic-field self-generation in a turbulent, electrically conducting fluid is shown to exhibit intermittent bursting characterized by distinct scaling laws. This behavior is predicted on the basis of prior analysis of a type of bifurcation (called a blowout bifurcation) occurring in chaotic systems with an invariant phase space submanifold. The predicted scalings are shown to be consistent with numerical solutions of the governing magnetohydrodynamic equations, and implications for recently implemented experimental programs are discussed.

Dynamics of the velocity gradient tensor invariants in isotropic turbulence

The evolution of the invariants (R and Q) of the velocity gradient tensor in homogeneous isotropic turbulence is investigated using data from direct numerical simulation (DNS). The concepts of conditional average time rate of change of the invariants and conditional mean trajectories (CMT) in invariant phase space are introduced to study the dynamics of this flow. The resulting dynamical system in the (R,Q) phase space is a clockwise spiral with a stable focus at the origin, illustrating that in the mean, the cyclic sequence of topological evolution following a fluid particle is unstable-node/saddle/saddle (UN/S/S)→stable-node/saddle/saddle (SN/S/S)→stable-focus/stretching (SF/S)→unstable-focus/contracting (UF/C). The mean rates of change of R and Q, i.e., Ṙ, Q̇, are found to be negligible near the right branch of the null discriminant (D=0) curve, indicating that this curve is an attractor in the (R,Q) space. The effects of both the diffusion term and the anisotropic part of the pressure Hessian term on the dynamics of the invariants have also been analyzed using the conditional averages. Both contributions are found to be important in the dynamics of the velocity gradient invariants. Based on these results the extent of the validity of the model equations governing the evolution of R and Q proposed by Cantwell [Phys. Fluids A 4, 782 (1992)] and Dopazo et al. [“Velocity gradients in turbulent flows. Stochastic models,” Ninth Symposium on “Turbulent Shear Flows,” Kyoto, Japan, 1993, pp. 26-2-1–26-2-5] are discussed.

Two-Dimensional Chiral Chromodynamics with Faddeevian Anomaly

We consider the non-Abelian generalization of chiral electrodynamics with Faddeevian anomaly and show that despite the lack of manifest Lorentz invariance in the action, it has a consistent and Poincaré invariant phase space structure. We also discuss a modification of this system by imposing a constraint that removes the non-interacting chirality sector.

Role of parametric noise in nonintegrable quantum dynamics.

Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland 21218(Received 3 August 1993; revised manuscript received 19 January 1994)Theeffect ofparametric perturbations on the quantum dynamics ofa system with nonintegrableclassical limit and a mixed phase space is shown to depend crucially on the local invariant phasespace structures sampled by the evolution. Using a probe coherent state four distinct scenarios areconsidered: (i) large regular regions surrounding primary fixed points; (ii) smaller regular regionscorresponding to higher order orbits; (iii) scarred states on hyperbolic fixed points; (iv) chaoticregions. The result that scarring stabilizes quantum dynamics against fluctuations is discussed inthegeneralcontext ofinstabilityin thequantumevolution as well asinrelationtorecent experimentson microwave ionization ofRydberg atoms.PACS number(s): 05.45.+b, 03.65.SqIn all cases

Meson-nucleus dynamics

A diagrammatic perturbation theory for the interaction of a pion with a finite nucleus is developed. The theory allows us to identify the various pieces of the physics which are believed to be important and to put into a unifying language the understanding which is emerging from apparently diverse studies. The theory proposed here is arranged so that it maintains the use of the nearly free pion-nucleon T-matrix and contemporary nuclear structure work which utilizes a nucleon-nucleon Bethe-Goldstone g-matrix. Within this framework, we calculate in momentum space and without approximation the first-order optical potential. This potential includes: (1) an exact performance of the Fermi-averaging integral (delta propagation), (2) covariant kinematics including the recoil of the nuclear target, (3) the use of invariant amplitudes and invariant normalizations and phase space factors, (4) the inclusion of the delta-nucleus interaction through a mean-spectral approximation, and (5) a realistic off-shell extrapolation of the pion-nucleon amplitude. The elastic scattering cross sections predicted by this potential are remarkably close to the data. The higher-order effects are classified and their inter-relationships discussed. Three of the higher-order corrections, Pauli effects, pion true-absorption, and short range correlation effects, are calculated and the success of the first-order potential is found to be validated by a large cancellation between the Pauli and true-absorption corrections. A detailed discussion of and comparison to alternate approaches to investigating pion-nucleus dynamics is provided.

(2+1)-dimensional quantum gravity with spinor field

The (2+1)-dimensional quantum gravity coupled with a spinor field is considered in the connection representation. The canonical structure of the theory is discussed and gauge invariant phase space variables are identified. The authors find a physical observable and its eigenstate. It is shown that the trace of the holonomy is a non-trivial solution of the Wheeler-De Witt equation.

Quantization ambiguities for Chern-Simons topological theories

We point out that the singularities of the gauge invariant phase space of the three-dimensional topological Chern-Simons theory with compact gauge group G give rise to ambiguities of the canonical quantization which are classified by the one-dimensional representations of the Weyl group of G. We show that the quantum Hilbert space associated with the non-trivial representation. of the Weyl group has the unitary structure of the space of current blocks of two-dimensional Wess-Zumino-Witten theory on the group manifold G. This provides a topological understanding of the “shift” of the central charge in the two-dimensional Sugawara construction.

Fractal dimensions from a three-dimensional intermittency analysis in e +e − annihilation

The intermittency structure of multihadronic e +e − annihilation is analyzed by evaluating the factorial moments F 2− F 5 in three-dimensional Lorentz invariant phase space as a function of the resolution scale. We interpret our data in the language of fractal objects. It turns out that the fractal dimension depends on the resolution scale in a way that can be attributed to geometrical resolution effects and dynamical effects, such as the π 0 Dalitz decay. The LUND 7.2 hadronization model provides an excellent description of the data. There is no indication of unexplained multiplicity fluctuations in small phase space regions.