Phase Transition in Two-Dimensional Classical Coulomb Plasma by Mapping to Quantum sine-Gordon System in Euclidean space

The phase transition of the two-dimensional (20) Coulomb plasma is studied with the help of the equivalence to a quantum sine-Gordon system in 2D Euclidean space. The minimum and the stability of the grand canonical potential are derived by making use of the variational effective potential method. The phase diagram thus obtained shows a larger region of plasma phase in the parameter plane than the parallel result in the (l+l)-dimensional Minkowski space. The origin of this difference is discussed in connection with the manner of the large momentum cut off being taken.     

Rigid body dynamics in non-Euclidean spaces

In this paper, we focus on the study of two-dimensional plate dynamics on the Lobachevskii plane L ². First of all, we consider the free motion of such a plate, which is a pseudospherical analog of dynamics of the Euler top, and also present an analog of the Euler–Poisson equations enabling us to study the motion of the body in potential force fields having rotational symmetry. We present a series of integrable cases, having analogs in Euclidean space, for different fields. Moreover, in the paper, a partial qualitative analysis of the dynamics of free motion of a plate under arbitrary initial conditions is made and a number of computer illustrations are presented which show a substantial difference of the motion from the case of Euclidean space. The study undertaken in the present paper leads to interesting physical consequences, which enable us to detect the influence of curvature on the body dynamics.     

Two-Dimensional Krall–Sheffer Polynomials andQuantum Systems on Spaces with Constant Curvature

Krall and Sheffer found in 1967 that there exists at most nine different types of two-dimensional orthogonal polynomials which are eigensolutions of a second-order linear differential operator with polynomial coefficients. We show that, for all these types, there correspond quantum mechanical systems on a Euclidean (pseudo-Eeuclidean) plane, two-dimensional sphere, or hyperboloid.     

Grassmann variables on quantum spaces

Attention is focused on antisymmetrized versions of quantum spaces that are of particular importance in physics, i.e. two-dimensional quantum plane, q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski space. For each case standard techniques for dealing with q-deformed Grassmann variables are developed. Formulae for multiplying supernumbers are given. The actions of symmetry generators and fermionic derivatives upon antisymmetrized quantum spaces are calculated. The complete Hopf structure for all types of quantum space generators is written down. From the formulae for the coproduct a realization of the L-matrices in terms of symmetry generators can be read off. The L-matrices together with the action of symmetry generators determine how quantum spaces of different type have to be fused together. Arrival of the final proofs: 6 December 2005     

The generalized Fubini instanton

We show that (1+2) nonlinear Klein-Gordon equation with negative coupling admits an exact solution which appears to be the linear superposition of the plane wave and the nonsingular rational soliton. We show that the same approach allows to construct the solution of similar properties for the Euclidean ?⁴ model with broken symmetry. Interestingly, this regular solution will be of instanton type only in the D?5 Euclidean space. Thus one can use the generalized Fubini instantons (in quantum cosmology for example) only for the case of the single infinite extra dimension.     

A Non-Zero Hadronic Elliptic Flow with a Vanished Partonic Elliptic Flow in a Coalescence Scenario

The elliptic flow of a hadron is calculated using a quark coalescence model based on the quark phase space distribution produced by a free streaming locally thermalized quark in a two-dimensional transverse plane at initial time. Without assuming the quark＇s elliptic flow, it is shown that the hadron obtains a non-zero elliptic flow in this model. The elliptic flow of the hadron is shown to be sensitive to both space momentum correlation and the hadron＇s internal structure. Quark number scaling is obtained only for some special cases.     

Quantum mechanics on manifolds embedded in Euclidean space

Quantum particles confined to surfaces in higher-dimensional spaces are acted upon by forces that exist only as a result of the surface geometry and the quantum mechanical nature of the system. The dynamics are particularly rich when confinement is implemented by forces that act normal to the surface. We review this confining potential formalism applied to the confinement of a particle to an arbitrary manifold embedded in a higher-dimensional Euclidean space. We devote special attention to the geometrically induced gauge potential that appears in the effective Hamiltonian for motion on the surface. We emphasize that the gauge potential is only present when the space of states describing the degrees of freedom normal to the surface is degenerate. We also distinguish between the effects of the intrinsic and extrinsic geometry on the effective Hamiltonian and provide simple expressions for the induced-scalar potential. We discuss examples including the case of a three-dimensional manifold embedded in a five-dimensional Euclidean space.     

From Euclidean to Minkowski space with the Cauchy–Riemann equations

We present an elementary method to obtain Green’s functions in non-perturbative quantum field theory in Minkowski space from Green’s functions calculated in Euclidean space. Since in non-perturbative field theory the analytical structure of amplitudes often is unknown, especially in the presence of confined fields, dispersive representations suffer from systematic uncertainties. Therefore, we suggest to use the Cauchy–Riemann equations, which perform the analytical continuation without assuming global information on the function in the entire complex plane, but only in the region through which the equations are solved. We use as example the quark propagator in Landau gauge quantum chromodynamics, which is known from lattice and Dyson–Schwinger studies in Euclidean space. The drawback of the method is the instability of the Cauchy–Riemann equations against high-frequency noise,which makes it difficult to achieve good accuracy. We also point out a few curious details related to the Wick rotation.     

Fabrication of two-dimensional elliptic photonic lattices in photorefractive crystal by optical induction method

We fabricate two-dimensional elliptic photonic lattices in iron-doped lithium niobate photorefractive crystal for the first time with optical induction method. The experimental setup of our method is very simple and flexible without complicated optical adjustment system. We analyze and verify the two-dimensional elliptic photonic lattices by plane wave guiding, far field diffraction pattern imaging, and Brillouin-zone spectroscopy. Induced elliptic photonic lattices can stably exist for a long time in the iron-doped lithium niobate crystal. The induced two-dimensional elliptic photonic lattices might offer an easy method to study generic band gap phenomena in anisotropic periodic structures.     

Stability of Covariant Relativistic Quantum Theory

In this paper we study the relativistic quantum-mechanical interpretation of the solution of the inhomogeneous Euclidean Bethe-Salpeter equation. Our goal is to determine conditions on the input to the Euclidean Bethe-Salpeter equation so the solution can be used to construct a model Hilbert space and a dynamical unitary representation of the Poincaré group. We prove three theorems that relate the stability of this construction to properties of the kernel and driving term of the Bethe-Salpeter equation. The most interesting result is that the positivity of the Hilbert space norm in the non-interacting theory is not stable with respect to Euclidean covariant perturbations defined by Bethe-Salpeter kernels. The long-term goal of this work is to understand which model Euclidean Green functions preserve the underlying relativistic quantum theory of the original field theory. Understanding the constraints imposed on the Green functions by the existence of an underlying relativistic quantum theory is an important consideration for formulating field-theory motivated relativistic quantum models.This work supported in part by the U.S. Department of Energy, under contract DE-FG02-86ER40286     

Quark structure and weak decays of heavy mesons

We construct a cellular space which has as a continuous limit the Euclidean spaceR ^N . We consider quantum mechanics on this cellular space and we examine in particular an harmonic oscillator and a free particle on the cellularR ¹,R ² respectively. In both cases we find that the energy spectrum is bounded from above.Partially supported by CEC Science project No SC1-CT91-0729     

A new kind of representations on noncommutative phase space

We introduce new representations to formulate quantum mechanics on noncommutative phase space, in which both coordinate-coordinate and momentum-momentum are noncommutative. These representations explicitly display entanglement properties between degrees of freedom of different coordinate and momentum components. To show their potential applications, we derive explicit expressions of Wigner function and Wigner operator in the new representations, as well as solve exactly a two-dimensional harmonic oscillator on the noncommutative phase plane with both kinetic coupling and elastic coupling.     

Space-Time Grains: Roots of Special and Doubly Special Relativity

We show that the special relativistic dynamics when combined with quantum mechanics and the concept of superstatistics can be interpreted as arising from two interlocked non-relativistic stochastic processes that operate at different energy scales. This interpretation leads to Feynman amplitudes that are in the Euclidean regime identical to transition probability of a Brownian particle propagating through a granular space. Some kind of spacetime granularity could be therefore held responsible for the emergence at larger scales of various symmetries. For illustration we consider also the dynamics and the propagator of a spinless relativistic particle. Implications for doubly special relativity, quantum field theory, quantum gravity and cosmology are discussed.     

Quantization of Co-Isotropic Subgroups

The purpose of this Letter is to show that the concept of co-isotropic subgroups of a Poisson–Lie group can be given a natural analogue in the context of quantum homogeneous spaces, explaining some of the special features of this theory. We will give examples related to some previously known and some new quantum homogeneous spaces of the two-dimensional Euclidean quantum groups.     

An existence theorem for multimeron solutions to classical Yang-Mills field equations

In this paper we prove the existence of solutions to a class of boundary value problems for a singular nonlinear elliptic partial differential equation in a half plane. By a recent paper of J. Glimm and A. Jaffe, this proves the existence of multimeron solutions to the classical SU(2) Yang-Mills field equations in Euclidean space.Supported in part by the National Science Foundation under Grant PHY 77-18762Supported in part by the Icelandic Science FoundationSupported in part by Grant MCS 76-06524     

Bosonization,topological solitons and fractional charges in two-dimensional quantum field theory

We further develop the quantization of topological solitons in two-dimensional quantum field theory in terms of Euclidean region functional integrals. Our approach is nonperturbative and mathematically rigorous. We apply it to construct physical states with fractional fermion number in models of interacting bosons and fermions without recurring to a semiclassical approximation. A related issue discussed in this paper is two-dimensional chiral bosonization.     

Property of partially disturbed polarization of Airy beam

We propose methods for the unambiguous discrimination of quantum states encoded in the spatial profile of light modes, in particular in the orbital angular momentum of light. Perfect discrimination between quantum states is possible only between orthogonal states of a known basis system. Generalised measurement strategies have been developed for the distinction between non-orthogonal states in a two-dimensional state space, e.g. the polarisation of light. Here we consider information encoded in superpositions of orbital angular momentum states of light, defined within an infinite-dimensional state space. Generalised measurements will be an important addition to quantum communication in high-dimensional spaces.     

Quantum Monodromy in Prolate Ellipsoidal Billiards

This is the third in a series of three papers on quantum billiards with elliptic and ellipsoidal boundaries. In the present paper we show that the integrable billiard inside a prolate ellipsoid has an isolated singular point in its bifurcation diagram and, therefore, exhibits classical and quantum monodromy. We derive the monodromy matrix from the requirement of smoothness for the action variables for zero angular momentum. The smoothing procedure is illustrated in terms of energy surfaces in action space including the corresponding smooth frequency map. The spectrum of the quantum billiard is computed numerically and the expected change in the basis of the lattice of quantum states is found. The monodromy is already present in the corresponding two-dimensional billiard map. However, the full three degrees of freedom billiard is considered as the system of greater relevance to physics. Therefore, the monodromy is discussed as a truly three-dimensional effect.     

Hyperbolic conservation laws on the sphere. A geometry-compatible finite volume scheme

We consider entropy solutions to the initial value problem associated with scalar nonlinear hyperbolic conservation laws posed on the two-dimensional sphere. We propose a finite volume scheme which relies on a web-like mesh made of segments of longitude and latitude lines. The structure of the mesh allows for a discrete version of a natural geometric compatibility condition, which arose earlier in the well-posedness theory established by Ben-Artzi and LeFloch. We study here several classes of flux vectors which define the conservation law under consideration. They are based on prescribing a suitable vector field in the Euclidean three-dimensional space and then suitably projecting it on the sphere’s tangent plane; even when the flux vector in the ambient space is constant, the corresponding flux vector is a non-trivial vector field on the sphere. In particular, we construct here “equatorial periodic solutions”, analogous to one-dimensional periodic solutions to one-dimensional conservation laws, as well as a wide variety of stationary (steady state) solutions. We also construct “confined solutions”, which are time-dependent solutions supported in an arbitrarily specified subdomain of the sphere. Finally, representative numerical examples and test cases are presented.