arXiv subjects

G. Date

Publications and source records attributed to G. Date.

17 recordsLinked to original sources

arXiv2003

The Landau electron problem on a cylinder

We consider the quantum mechanics of an electron confined to move on an infinite cylinder in the presence of a uniform radial magnetic field. This problem is in certain ways very similar to the corresponding problem on the infinite plane. Unlike the plane however, the group of symmetries of the magnetic field, namely, rotations about the axis and the axial translations, is {\em not} realized by the quantum electron but only a subgroup comprising rotations and discrete translations along the axial direction, is. The basic step size of discrete translations is such that the flux through the `unit cylinder cell' is quantized in units of the flux quantum. The result is derived in two different ways: using the condition of projective realization of symmetry groups and using the more familiar approach of determining the symmetries of a given Hamiltonian.

math-ph↗

arXiv2003

Classical and Quantum Mechanics of Anyons

We review aspects of classical and quantum mechanics of many anyons confined in an oscillator potential. The quantum mechanics of many anyons is complicated due to the occurrence of multivalued wavefunctions. Nevertheless there exists, for arbitrary number of anyons, a subset of exact solutions which may be interpreted as the breathing modes or equivalently collective modes of the full system. Choosing the three-anyon system as an example, we also discuss the anatomy of the so called ``missing'' states which are in fact known numerically and are set apart from the known exact states by their nonlinear dependence on the statistical parameter in the spectrum. Though classically the equations of motion remains unchanged in the presence of the statistical interaction, the system is non-integrable because the configuration space is now multiply connected. In fact we show that even though the number of constants of motion is the same as the number of degrees of freedom the system is in general not integrable via action-angle variables. This is probably the first known example of a many body pseudo-integrable system. We discuss the classification of the orbits and the symmetry reduction due to the interaction. We also sketch the application of periodic orbit theory (POT) to many anyon systems and show the presence of eigenvalues that are potentially non-linear as a function of the statistical parameter. Finally we perform the semiclassical analysis of the ground state by minimizing the Hamiltonian with fixed angular momentum and further minimization over the quantized values of the angular momentum.

cond-mat↗

arXiv2002

A Discrete Time Presentation of Quantum Dynamics

Inspired by the discrete evolution implied by the recent work on loop quantum cosmology, we obtain a discrete time description of usual quantum mechanics viewing it as a constrained system. This description, obtained without any approximation or explicit discretization, mimics features of the discrete time evolution of loop quantum cosmology. We discuss the continuum limit, physical inner product and matrix elements of physical observables to bring out various issues regarding viability of a discrete evolution. We also point out how a continuous time could emerge without appealing to any continuum limit.

gr-qc↗

arXiv2002

Quantum Geometric Description of Cosmological Models

This is a written version of the review talk given at the meeting on "Interface of Gravitational and Quantum Realms" at IUCAA, Pune during December 2001. The talk reviewed the recent work of Martin Bojowald on Loop Quantum Cosmology.

gr-qc↗

arXiv2001

Isolated Horizon, Killing Horizon and Event Horizon

We consider space-times which in addition to admitting an isolated horizon also admit Killing horizons with or without an event horizon. We show that an isolated horizon is a Killing horizon provided either (1) it admits a stationary neighbourhood or (2) it admits a neighbourhood with two independent, commuting Killing vectors. A Killing horizon is always an isolated horizon. For the case when an event horizon is definable, all conceivable relative locations of isolated horizon and event horizons are possible. Corresponding conditions are given.

gr-qc↗

arXiv2001

Semi-classical States in the Context of Constrained Systems

Algebraic quantization scheme has been proposed as an extension of the Dirac quantization scheme for constrained systems. Semi-classical states for constrained systems is also an independent and important issue, particularly in the context of quantum geometry. In this work we explore this issue within the framework of algebraic quantization scheme by means of simple explicit examples. We obtain semi-classical states as suitable coherent states a la Perelomov. Remarks on possible generalizations are also included.

quant-ph↗

arXiv2000

On a Peculiar Family of Static, Axisymmetric, Vacuum Solutions of the Einstein Equations

The Zipoy-Voorhees family of static, axisymmetric vacuum solutions forms an interesting family in that it contains the Schwarzschild black hole excepting which all other members have naked singularity. We analyze some properties of the region near singularity by studying a natural family of 2-surfaces. We establish that these have the topology of the 2-sphere by an application of the Gauss-Bonnet theorem. By computing the area, we establish that the singular region is `point-like'. Isometric embedding of these surfaces in the three dimensional Euclidean space is used to distinguish the two types of deviations from spherical symmetry.

gr-qc↗

arXiv2000

Notes on Isolated Horizons

A general analysis for characterizing and classifying `isolated horizons' is presented in terms of null tetrads and spin coefficients. The freely specifiable spin coefficients corresponding to isolated horizons are identified and specific symmetry classes are enumerated. For isolated horizons admitting at least one spatial isometry, a standard set of spherical coordinates are introduced and associated metric is obtained. An angular momentum is also defined.

gr-qc↗

arXiv1999

Anyons: Pseudo-integrability, Symmetry reduction and Semiclassical Spectrum

At the classical level anyons with harmonic confinement are known to exhibit two important properties namely partial separability and pseudo-integrability. These stem from the fact that this system is locally identical to isotropic oscillator system but differs in the global topology of the phase space. We clarify the meaning of pseudo-integrability and show that it amounts to a definite reduction of the symmetry group. We elaborate on the role of the fundamental group of the phase space and pseudo-intrgrability in the context of periodic orbit theory and obtain evidence of non-exactly known eigenvalues from the semiclassical trace formula. We also discuss an ambiguity regarding the `half period' trajectories suggested by classical modeling and exhibited by the exactly known propagator for two anyons.

math-ph↗

arXiv1998

Novel classical ground state of a many body system in arbitrary dimensions

The classical ground state of a D- dimensional many body system with two and three body interactions is studied as a function of the strength of the three body interaction. We prove exactly that beyond a critical strength of the three body interaction, the classical ground state of the system is one in which all the particles are on a line. The positions of the particles in this string configuration are uniquely determined by the zeros of the Hermite polynomials.

cond-mat↗

arXiv1998

Classical Many-particle Clusters in Two Dimensions

We report on a study of a classical, finite system of confined particles in two dimensions with a two-body repulsive interaction. We first develop a simple analytical method to obtain equilibrium configurations and energies for few particles. When the confinement is harmonic, we prove that the first transition from a single shell occurs when the number of particles changes from five to six. The shell structure in the case of an arbitrary number of particles is shown to be independent of the strength of the interaction but dependent only on its functional form. It is also independent of the magnetic field strength when included. We further study the effect of the functional form of the confinement potential on the shell structure. Finally we report some interesting results when a three-body interaction is included, albeit in a particular model.

cond-mat↗

arXiv1997

Geometry of N-particle clusters in two-dimensions: Some exact results

We report on a study of a finite system of classical confined particles in two-dimensions in the presence of a uniform magnetic field and interacting via a two-body repulsive potential. We develop a simple analytical method of analysis to obtain ground state energies and configurations. We prove analytically the minimum energy configurations are independent of the nature of two body interaction and the magnetic field. In particular we prove that the first transition from a single shell occurs when the number of particles changes from five to six. These results are exact.

cond-mat↗

arXiv1997

On a Static Solution to Einstein Equations with incoming and outgoing radiation

Einstein equations with $T_{μν} = k_μk_ν+ \ell_μ\ell_ν$ where $k, \ell$ are null are considered with spherical symmetry and staticity. The solution has naked singularity and is not asymptotically flat. However, it may be interpreted as an envelope for any static spherical body making it more massive. Such an interpretation and some of its implications are detailed.

gr-qc↗

arXiv1996

Classical and Quantum Aspects of Gravitation and Cosmology

These are the proceedings of the XVIII Conference of the Indian Association for General Relativity and Gravitation (IAGRG) held at the Institute of Mathematical Sciences, Madras, INDIA during Feb. 15-17, 1996. The Conference was dedicated the late Prof. S. Chandrasekhar. The proceedings consists of 17 articles on: - Chandrasekhar's work (N. Panchapkesan); - Vaidya-Raychaudhuri Lecture (C.V. Vishveshwara) - Gravitational waves (B.R. Iyer, R. Balasubramanian) - Gravitational Collapse (T.P. Singh) - Accretion on black hole (S. Chakrabarti) - Cosmology (D. Munshi, S. Bharadwaj, G.S. Mohanty, P. Bhattacharjee); - Classical GR (S. Kar, D.C. Srivatsava) - Quantum aspects (J. Maharana, Saurya Das, P. Mitra, G. Date, N.D. Hari Dass) The body of THIS article contains ONLY the title, contents, foreword, organizing committees, preface, list of contributed talks and list of participants. The plenery talks are available at: http://www.imsc.ernet.in/physweb/Conf/ both as post-script files of individual articles and also as .uu source files. For further information please send e-mail to shyam@imsc.ernet.in

gr-qc↗

arXiv1995

Spatial scaling in fracture propagation in dilute systems

The geometry of fracture patterns in a dilute elastic network is explored using molecular dynamics simulation. The network in two dimensions is subjected to a uniform strain which drives the fracture to develop by the growth and coalescence of the vacancy clusters in the network. For strong dilution, it has been shown earlier that there exists a characteristic time $t_c$ at which a dynamical transition occurs with a power law divergence (with the exponent $z$) of the average cluster size. Close to $t_c$, the growth of the clusters is scale-invariant in time and satisfies a dynamical scaling law. This paper shows that the cluster growth near $t_c$ also exhibits spatial scaling in addition to the temporal scaling. As fracture develops with time, the connectivity length $\xi$ of the clusters increses and diverges at $t_c$ as $\xi \sim (t_c-t)^{-\nu}$, with $\nu = 0.83 \pm 0.06$. As a result of the scale-invariant growth, the vacancy clusters attain a fractal structure at $t_c$ with an effective dimensionality $d_f \sim 1.85 \pm 0.05$. These values are independent (within the limit of statistical error) of the concentration (provided it is sufficiently high) with which the network is diluted to begin with. Moreover, the values are very different from the corresponding values in qualitatively similar phenomena suggesting a different universality class of the problem. The values of $\nu$ and $d_f$ supports the scaling relation $z=\nu d_f$ with the value of $z$ obtained before.

cond-mat↗

arXiv1993

Classical Dynamics of Anyons and the Quantum Spectrum

In this paper we show that (a) all the known exact solutions of the problem of N-anyons in oscillator potential precisely arise from the collective degrees of freedom, (b) the system is pseudo-integrable ala Richens and Berry. We conclude that the exact solutions are trivial thermodynamically as well as dynamically.

cond-mat↗

arXiv1993

Asymptotic analysis and spectrum of three anyons

The spectrum of anyons confined in harmonic oscillator potential shows both linear and nonlinear dependence on the statistical parameter. While the existence of exact linear solutions have been shown analytically, the nonlinear dependence has been arrived at by numerical and/or perturbative methods. We develop a method which shows the possibility of nonlinearly interpolating spectrum. To be specific we analyse the eigenvalue equation in various asymptotic regions for the three anyon problem.

cond-mat↗