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We study chaotic behavior and diffusion in the 2D periodic Lorentz gas in the finite-horizon regime. The dynamical observable which we consider is the length of single particle's trajectories, which moves in a triangular array of rigid disks. To analyze the chaoticity of this system, we construct a matrix of the length of trajectories and perform a level spacing analysis of the spectrum of this matrix. We find that a universal behavior occurs both in level spacing distribution and spectral rigidity. In order to examine diffusion in this classical system, we investigate the variance of the length of trajectories versus number of collisions to disks. In the case where there is a finite-horizon, such a variance scales linearly with number of bounces. This shows that a normal diffusion exists and a central limit theorem is maintained in this regime., Comment: 14 pages, 13 figures

We address an old problem in lattice gauge theory - the computation of the spectrum and wave functions of excited states. Our method is based on the Hamiltonian formulation of lattice gauge theory. As strategy, we propose to construct a stochastic basis of Bargmann link states, drawn from a physical probability density distribution. Then we compute transition amplitudes between stochastic basis states. From a matrix of transition elements we extract energy spectra and wave functions. We apply this method to U(1)$_{2+1}$ lattice gauge theory. We test the method by computing the energy spectrum, wave functions and thermodynamical functions of the electric Hamiltonian of this theory and compare them with analytical results. We observe a reasonable scaling of energies and wave functions in the variable of time. We also present first results on a small lattice for the full Hamiltonian including the magnetic term.

Local field potentials (LFPs) are routinely measured experimentally in brain tissue, and exhibit strong low-pass frequency filtering properties, with high frequencies (such as action potentials) being visible only at very short distances ($\approx$10~$\mu m$) from the recording electrode. Understanding this filtering is crucial to relate LFP signals with neuronal activity, but not much is known about the exact mechanisms underlying this low-pass filtering. In this paper, we investigate a possible biophysical mechanism for the low-pass filtering properties of LFPs. We investigate the propagation of electric fields and its frequency dependence close to the current source, i.e. at length scales in the order of average interneuronal distance. We take into account the presence of a high density of cellular membranes around current sources, such as glial cells. By considering them as passive cells, we show that under the influence of the electric source field, they respond by polarisation, i.e., creation of an induced field. Because of the finite velocity of ionic charge movement, this polarization will not be instantaneous. Consequently, the induced electric field will be frequency-dependent, and much reduced for high frequencies. Our model establishes that with respect to frequency attenuation properties, this situation is analogous to an equivalent RC-circuit, or better a system of coupled RC-circuits. We present a number of numerical simulations of induced electric field for biologically realistic values of parameters, and show this frequency filtering effect as well as the attenuation of extracellular potentials with distance. We suggest that induced electric fields in passive cells surrounding neurons is the physical origin of frequency filtering properties of LFPs., Comment: 10 figs, revised tex file and revised figs

We consider a mixed chaotic Hamiltonian system and compare classical with quantum chaos. As alternative to the methods of enegy level spacing statistics and trace formulas, we construct a quantum action and a quantum analogue phase space to analyse quantum chaos., Comment: talk by Helmut Kr\"oger at Conf. on Found. of Quant. Mech., Vaxjo (Sweden) 2003

We investigate chaotic behavior in a 2-D Hamiltonian system - oscillators with anharmonic coupling. We compare the classical system with quantum system. Via the quantum action, we construct Poincar\'e sections and compute Lyapunov exponents for the quantum system. We find that the quantum system is globally less chaotic than the classical system. We also observe with increasing energy the distribution of Lyapunov exponts approaching a Gaussian with a strong correlation between its mean value and energy., Comment: 25 pages, 10 figs

We investigate supervised learning in neural networks. We consider a multi-layered feed-forward network with back propagation. We find that the network of small-world connectivity reduces the learning error and learning time when compared to the networks of regular or random connectivity. Our study has potential applications in the domain of data-mining, image processing, speech recognition, and pattern recognition., Comment: Text completely revised (14 pages), all new figures (7 figs)

We investigate dynamics of scalar field with non-minimal kinetic term. Nontrivial behavior of the field in the vicinity of singular points of kinetic term is observed. In particular, the singular points could serve as attractor for classical solutions., Comment: 6 pages, 3 figures, title is changed, some refs added, to be published in Gen. Rel. and Grav

We discuss the recently proposed quantum action - its interpretation, its motivation, its mathematical properties and its use in physics: quantum mechanical tunneling, quantum instantons and quantum chaos., Comment: inv. talk, 10th Int. Conf. Comput. and Appl. Math., Leuven, Belgium (2002)

We investigate chaotic behavior in a 2-D Hamiltonian system - oscillators with anharmonic coupling. We compare the classical system with quantum system. Via the quantum action, we construct Poincar\'{e} sections and compute Lyapunov exponents for the quantum system. We find that the quantum system is globally less chaotic than the classical system. We also observe with increasing energy the distribution of Lyapunov exponts approaching a Gaussian with a strong correlation between its mean value and energy., Comment: text (LaTeX) + 7 figs.(ps)

In order to solve quantum field theory in a non-perturbative way, Lagrangian lattice simulations have been very successful. Here we discuss a recently proposed alternative Hamiltonian lattice formulation - the Monte Carlo Hamiltonian. In order to show its working in the case of the scalar $\Phi^{4}_{1+1}$ model, we have computed thermodynamic functions like free energy, average energy, entropy and specific heat. We find good agreement between the results from the Monte Carlo Hamiltonian and standard Lagrangian lattice computations. However, the Monte Carlo Hamiltonian results show less fluctuations under variation of temperature. We address properties of the MC Hamiltonian, like a finite temperature window, and scaling properties. Also we discuss possible future applications - like quantum chaos in many-body systems, the non-perturbative computation of wave functions of elementary particles, as well as scattering amplitudes in high energy physics., Comment: H. Kr\"oger invited, Frontiers of Fundamental Physics 5

We discuss the concept of the quantum action with the purpose to characterize and quantitatively compute quantum chaos. As an example we consider in quantum mechanics a 2-D Hamiltonian system - harmonic oscillators with anharmonic coupling - which is classically a chaotic system. We compare Poincar\'e sections obtained from the quantum action with those from the classical action., Comment: Talk given at 4th Int. Conf. Dyn. Syst. and Diff. Eqs., Wilmington NC (2002), Text (LaTeX), Style file dcds2

We present a numerical study of the quantum action previously introduced as a parametrisation of Q.M. transition amplitudes. We address the questions: Is the quantum action possibly an exact parametrisation in the whole range of transition times ($0 < T < \infty$)? Is the presence of potential terms beyond those occuring in the classical potential required? What is the error of the parametrisation estimated from the numerical fit? How about convergence and stability of the fitting method (dependence on grid points, resolution, initial conditions, internal precision etc.)? Further we compare two methods of numerical determination of the quantum action: (i) global fit of the Q.M. transition amplitudes and (ii) flow equation. As model we consider the inverse square potential, for which the Q.M. transition amplitudes are analytically known. We find that the relative error of the parametrisation starts from zero at T=0 increases to about $10^{-3}$ at $T=1/E_{gr}$ and then decreases to zero when $T \to \infty$. Second, we observe stability of the quantum action under variation of the control parameters. Finally, the flow equation method works well in the regime of large $T$ giving stable results under variation of initial data and consistent with the global fit method., Comment: Text (LaTeX), Figures(ps)

We present a new way to compute thermodynamical observables on the lattice. We compute excited states and thermodynamical functions in the scalar model via the Monte Carlo Hamiltonian technique. We find agreement with standard Lagrangian lattice calculations, but observe lesser fluctuations in the results from the MC Hamiltonian., Comment: talk at Lattice'02, Boston [non-zero temperature and density] Text (LaTeX), Figures(eps), style file

We investigate an influence of virtual particles on the classical motion of a system in Minkowski and Euclidean spaces. Our results indicate that fluctuations of fields different from the main field decelerate significantly its motion at high energies. The formalism described here is applicable to the majority of inflationary models. We show also that the quantum fluctuations suppress false vacuum decay in first-order phase transitions at high energies., Comment: 5 pages, 1 figure

The Monte Carlo (MC) Hamiltonian is a new stochastic method to solve many-body problems. The MC Hamiltonian represents an effective Hamiltonian in a finite energy window. We construct it from the classical action via Monte Carlo with importance sampling. The MC Hamiltonian yields the energy spectrum and corresponding wave functions in a low energy window. This allows to compute thermodynamical observables in a low temperature window. We show the working of the MC Hamiltonian by an example from lattice field theory (Klein-Gordon model)., Comment: inv. talk given by H. Kr\"oger at 3. IMACS Seminar on Monte Carlo Methods Salzburg (2001), Text (LaTeX), style file

We use the quantum action to study the dynamics of quantum system at finite temperature. We construct the quantum action non-perturbatively and find temperature dependent action parameters. Here we apply the quantum action to study quantum chaos. We present a numerical study of a classically chaotic 2-D Hamiltonian system - harmonic oscillators with anharmonic coupling. We compare Poincar\'e sections for the quantum action at finite temperature with those of classical action., Comment: 14 pages, DCDIS international conference in London Canada,2001

We construct an effective low-energy Hamiltonian from the classical action via Monte Carlo with importance sampling. We use Monte Carlo (i) to compute matrix elements of the transition amplitude and (ii) to construct stochastically a basis. The MC Hamiltonian allows to obtain energies and wave functions of low-lying states. It allows also to compute thermodynamical observables in some temperature window (starting from temperature zero). We present examples from lattice field theory (Klein-Gordon model)., Comment: inv. talk given by H. Kr\"oger at Quantum Monte Carlo, Trento, Italy (3.-6.7.2001), Text (LaTeX), style file

We have recently suggested a quantum action, which has the form of a classical action and takes into account quantum effects via renormalized action parameters. Here we apply it to quantum chaos. We study a system in 2-D with weak anharmonic coupling ($V_{coupl} \propto x^{2}y^{2}$) being classically chaotic. We construct the quantum action at finite temperature. We compute Poincar\'e sections of the quantum action at that temperature and compare it with those of the classical action. We observe chaotic behavior in both cases., Comment: H. Kr\"{o}ger invited, Frontiers Fund. Phys. IV, Hyderabad (2000). Text (LaTeX), 4 Figs. (ps)

We address the problem of quantum chaos: Is there a rigorous, physically meaningful definition of chaos in quantum physics? Can the tools of classical chaos theory, like Lyapunov exponents, Poincar\'e sections etc. be carried over to quantum systems? Can quantitative predictions be made? We show that the recently proposed quantum action is well suited to answer those questions. As an example we study chaotic behavior of the 2-D anharmonic oscillator and compare classical with quantum chaos. Moreover, we study quantum chaos as function of temperature (the classical system can be considered as the limit where temperature goes to infinity)., Comment: talk given by H. Kr\"{o}ger at ICSSUR, Boston. Text (LaTeX), 5 Figs. (ps)

We use the quantum action to study quantum chaos at finite temperature. We present a numerical study of a classically chaotic 2-D Hamiltonian system - harmonic oscillators with anharmonic coupling. We construct the quantum action non-perturbatively and find temperature dependent quantum corrections in the action parameters. We compare Poincar\'{e} sections of the quantum action at finite temperature with those of the classical action., Comment: Text (LaTeX), Figs. (ps)

We have previously proposed a conjecture stating that quantum mechanical transition amplitudes can be parametrized in terms of a quantum action. Here we give a proof of the conjecture and establish the existance of a local quantum action in the case of imaginary time in the Feynman-Kac limit (when temperature goes to zero). Moreover we discuss some symmetry properties of the quantum action., Comment: revised version, Text (LaTeX)

We study a recently proposed quantum action depending on temperature. We construct a renormalisation group equation describing the flow of action parameters with temperature. At zero temperature the quantum action is obtained analytically and is found free of higher time derivatives. It makes the quantum action an ideal tool to investigate quantum chaos and quantum instantons., Comment: replaced version with new figs. Text (LaTeX), 3 Figs. (ps)

We suggest how to construct non-perturbatively a renormalized action in quantum mechanics. We discuss similarties and differences with the standard effective action. We propose that the new quantum action is suitable to define and compute quantum instantons and quantum chaos., Comment: talk given by H. Kr\"{o}ger at WCNA, Catania. Text (LaTeX), 3 Figures, style file elsart

We discuss $QCD$ in the Hamiltonian frame work. We treat finite density $QCD$ in the strong coupling regime. We present a parton-model inspired regularisation scheme to treat the spectrum ($\theta$-angles) and distribution functions in $QED_{1+1}$. We suggest a Monte Carlo method to construct low-dimensionasl effective Hamiltonians. Finally, we discuss improvement in Hamiltonian $QCD$., Comment: Proceedings of Hadrons and Strings, invited talk given by H. Kr\"{o}ger; Text (LaTeX file), 3 Figures (ps file)

In order to extend the recently proposed Monte Carlo Hamiltonian to many-body systems, we suggest to concept of a stochastic basis. We apply it to the chain of $N_s=9$ coupled anharmonic oscillators. We compute the spectrum of excited states in a finite energy window and thermodynamical observables free energy, average energy, entropy and specific heat in a finite temperature window. Comparing the results of the Monte Carlo Hamiltonian with standard Lagrangian lattice calculations, we find good agreement. However, the Monte Carlo Hamiltonian results show less fluctuations under variation of temperature., Comment: revised version, new figures. Text (LaTeX), 4 Figs. (eps), style file

Based on a closed form expression for the path integral of quantum transition amplitudes, we suggest rigorous definitions of both, quantum instantons and quantum chaos. As an example we compute the quantum instanton of the double well potential., Comment: Extended version with new figures. Text (LaTeX), 5 Figures (epsi files)

We suggest a closed form expression for the path integral of quantum transition amplitudes. We introduce a quantum action with renormalized parameters. We present numerical results for the $V \sim x^{4}$ potential. The renormalized action is relevant for quantum chaos and quantum instantons., Comment: Revised text, 1 figure added; Text (LaTeX file), 1 Figure (ps file)

With our recently proposed effective Hamiltonian via Monte Carlo, we are able to compute low energy physics of quantum systems. The advantage is that we can obtain not only the spectrum of ground and excited states, but also wave functions. The previous work has shown the success of this method in (1+1)-dimensional quantum mechanical systems. In this work we apply it to higher dimensional systems., Comment: Lattice 99 (Algorithm), 3 pages, LaTeX using espcrc2.sty

We suggest how to construct an effective low energy Hamiltonian via Monte Carlo starting from a given action. We test it by computing thermodynamical observables like average energy and specific heat for simple quantum systems., Comment: Contribution to Lattice'99 (Theoretical developments) Text (LaTeX file) + 2 figures (ps files)

We demonstrate that front form quantisation with periodicity in a compact light-like direction (discretized light-cone quantisation) violates microcausality., Comment: Text (REVTEX file) + 4 figures (4 epsf files + 4 ps files)

We investigate the critical exponents $\eta_3(\alpha,M), \eta_1(\alpha,M)$ associated with the singularities in the longitudinal and transverse structure factors of the one dimensional antiferromagnetic Heisenberg model with nearest (J_1) and next to nearest (J_2) neighbour coupling of relative strength $\alpha = {J_2}/{J_1}$ and an external field $B$ with magnetization $M(B)$., Comment: LaTeX file (text) + 7 PS files (figures)

We investigate groundstate energies and magnetization curves in the one dimensional XXZ-model with next to nearest neighbour coupling $\alpha>0$ and anisotropy $\Delta$ ($-1 \le \Delta \le 1$) at T=0. In between the familiar ferro- and antiferromagnetic phase we find a transition region -- called metamagnetic phase -- where the magnetization curve is discontinuous at a critical field $B_c(\alpha,\Delta)$., Comment: LaTeX file (text) + 5 PS files (5 figures)

We present a non-perturbative study of the massive Schwinger model. We use a Hamiltonian approach, based on a momentum lattice corresponding to a fast moving reference frame, and equal time quantization., Comment: contribution to Lattice'98 including: 2 style files (espcrc2.sty,psfig.sty) + text file (LaTeX) + 3 figures (ps) + complete paper(ps)

Lepage's improvement scheme is a recent major progress in lattice $QCD$, allowing to obtain continuum physics on very coarse lattices. Here we discuss improvement in the Hamiltonian formulation, and we derive an improved Hamiltonian from a lattice Lagrangian free of $O(a^2)$ errors. We do this by the transfer matrix method, but we also show that the alternative via Legendre transformation gives identical results. We consider classical improvement, tadpole improvement and also the structure of L{\"u}scher-Weisz improvement. The resulting color-electric energy is an infinite series, which is expected to be rapidly convergent. For the purpose of practical calculations, we construct a simpler improved Hamiltonian, which includes only nearest-neighbor interactions., Comment: 30 pages, LaTeX