Table of contents

Classical mechanics is formulated in complex Hilbert space with the introduction of a commutative product of operators, an antisymmetric bracket and a quasidensity operator that is not positive definite. These are analogues of the star product, the Moyal bracket, and the Wigner function in the phase space formulation of quantum mechanics. Quantum mechanics is then viewed as a limiting form of classical mechanics, as Planck's constant approaches zero, rather than the other way around. The forms of semiquantum approximations to classical mechanics, analogous to semiclassical approximations to quantum mechanics, are indicated.

The correlation functions of the spin-1/2 XXZ chain in the ground state are expressed in the form of multiple integrals. For −1 < Δ < 1, they were obtained by Jimbo and Miwa in 1996. In particular, the next-nearest-neighbour correlation functions are given as certain three-dimensional integrals. We show that these can be reduced to one-dimensional integrals and thereby we evaluate the values of the next-nearest-neighbour correlation functions. We have also found that the remaining one-dimensional integrals can be evaluated analytically, when ν = cos⁻¹(Δ)/π is a rational number.

Two new families of exact solutions of the wave equation u_xx + u_yy + u_zz − c⁻²u_tt = 0 generalizing Bessel–Gauss pulses and Bateman–Hillion relatively undistorted progressive waves, respectively are presented. In each of these families new simple solutions describing localized wave propagation are found. The approach is based on a kind of separation of variables.

We show that the one-dimensional spin-1 XY model has an additional SU(2) symmetry for the open boundary condition and for an artificial one. We can explain some degeneracies of excitation states which were reported in previous numerical studies.

We investigate the dynamics and geometric phases of a time-dependent singular oscillator. We construct certain Gaussian wave packet solutions of the corresponding Schrödinger equation, relate the latter with the classical equation of motion and explore the relationship between the associated quantum and phase angles. It is shown by a simple geometrical approach that the geometrical phase is connected with the classical nonadiabatic Hannay angle of the generalized harmonic oscillator. Our geometric approach is based on a rule for a 'natural transport' of the complex two-dimensional vector in the phase space and the results obtained are quite suggestive of similarities to the quantum mechanical two-state evolution.

We present a stochastic model of dynamically interacting grains in one dimension, in the presence of a low vibrational intensity, to investigate the effect of shape on the statics and dynamics of the compaction process. Regularity and irregularity in grain shapes are shown to be centrally important in determining the statics of close-packing states, as well as the nature of zero- and low-temperature dynamics in this columnar model.

The standard result for the force per unit length on a vortex is derived for an arbitrary configuration of vortices and transport currents in the London limit. The sign reversal between this force, and the Lorentz force on a current in a magnetic field, is shown to be because of the fact that the vortex and the currents driving it are embedded in a single condensate.

Consider a closed lipid membrane (vesicle), modelled as a two-dimensional surface, described by a geometrical Hamiltonian that depends on its extrinsic curvature. The vanishing of its first variation determines the equilibrium configurations for the system. In this paper, we examine the second variation of the Hamiltonian about any given equilibrium, using an explicitly surface covariant geometrical approach. We identify the operator which determines the stability of equilibrium configurations.

The Yang–Lee, Fisher and Potts zeros of the one-dimensional Q-state Potts model are studied using the theory of dynamical systems. An exact recurrence relation for the partition function is derived. It is shown that zeros of the partition function may be associated with neutral fixed points of the recurrence relation. Further, a general equation for zeros of the partition function is found and a classification of the Yang–Lee, Fisher and Potts zeros is given. It is shown that the Fisher zeros in a nonzero magnetic field are located on several lines in the complex temperature plane and that the number of these lines depends on the value of the magnetic field. Analytical expressions for the densities of the Yang–Lee, Fisher and Potts zeros are derived. It is shown that densities of all types of zeros of the partition function are singular at the edge singularity points with the same critical exponent σ = −½.

The dynamics of a class of zero-range processes exhibiting a condensation transition in the stationary state is studied. The system evolves in time starting from a random disordered initial condition. The analytical study of the large-time behaviour of the system in its mean-field geometry provides a guide for the numerical study of the one-dimensional version of the model. Most qualitative features of the mean-field case are still present in the one-dimensional system, both in the condensed phase and at criticality. In particular the scaling analysis, valid for the mean-field system at large time and for large values of the site occupancy, still holds in one dimension. The dynamical exponent z, characteristic of the growth of the condensate, is changed from its mean-field value 2 to 3. In the presence of a bias, the mean-field value z = 2 is recovered. The dynamical exponent z_c, characteristic of the growth of critical fluctuations, is changed from its mean-field value 2 to a larger value, z_c ≃ 5. In the presence of a bias, z_c ≃ 3.

We consider atoms interacting with each other through the topological structure of a complex network and investigate lattice vibrations of the system, the quanta of which we call netons for convenience. The density of neton levels, obtained numerically, reveals that unlike a local regular lattice, the system develops a gap of finite width, manifesting extreme rigidity of the network structure at low energies. Two different network models, the small-world network and the scale-free network, are compared: the characteristic structure of the former is described by an additional peak in the level density whereas a power-law tail is observed in the latter, indicating excitability of netons at arbitrarily high energies. The gap width is also found to vanish in the small-world network when the connection range r = 1.

In this paper we consider an energy operator (a free Hamiltonian), in the second-quantized approach, for the multiparameter quon algebras: a_ia†_j − q_ija†_ja_i = δ_ij, i, j ∊ I with (q_ij)_i,j∊I any Hermitian matrix of deformation parameters. We obtain an elegant formula for normally ordered (sometimes called Wick-ordered) series expansions of number operators (which determine a free Hamiltonian). As a main result (see theorem 1) we prove that the number operators are given, with respect to a basis formed by 'generalized Lie elements', by certain normally ordered quadratic expressions with coefficients given precisely by the entries of the inverses of Gram matrices of multiparticle weight spaces. (This settles a conjecture by Meljanac S and Perica A (1994 J. Phys. A: Math. Gen.27 4737–44).) These Gram matrices are Hermitian generalizations of the Varchenko matrices, associated with a quantum (symmetric) bilinear form of diagonal arrangements of hyperplanes. The solution of the inversion problem of such matrices in Meljanac S and Svrtan D (1996 Math. Commun.1 1–24 (theorem 2.2.17)), leads to an effective formula for the number operators studied in this paper. The one-parameter case, in the monomial basis, was studied by Zagier, Stanciu and Møller.

We investigate the ground states of classical Heisenberg spin systems which have point group symmetry. Examples are the regular polygons (spin rings) and the seven quasi-regular polyhedra including the five Platonic solids. For these examples, ground states with special properties, e.g. coplanarity or symmetry, can be completely enumerated using group-theoretical methods. For systems having coplanar (anti-) ground states with vanishing total spin we also calculate the smallest and largest energies of all states having a given total spin S. We find that these extremal energies depend quadratically on S and prove that, under certain assumptions, this happens only for systems with coplanar S = 0 ground states. For general systems the corresponding parabolas represent lower and upper bounds for the energy values. This provides strong support and clarifies the conditions for the so-called rotational band structure hypothesis which has been numerically established for many quantum spin systems.

We study the correspondence between phase-space localization of quantum (quasi-)energy eigenstates and classical correlation decay, given by Ruelle–Pollicott resonances of the Frobenius–Perron operator. It will be shown that scarred (quasi-)energy eigenstates are correlated: pairs of eigenstates strongly overlap in phase space (scar in same phase-space regions) if the difference of their eigenenergies is close to the phase of a leading classical resonance. Phase-space localization of quantum states will be measured by L² norms of their Husimi functions.

For a singularity free gradient field in an open set of an oriented Euclidean space of dimension three we define a natural principal bundle out of an immanent complex line bundle. The fibres of this bundle encode information. The elements of both bundles are called internal variables. Several other natural bundles are associated with the principal bundle and, in turn, determine the vector field. Two examples are given and it is shown that for a constant vector field circular polarized waves with values in the principal bundle are associated with the vector field. These waves transmit information encoded in internal variables and, moreover, determine a Schrödinger representation. On U(2) a relation between spin representations and Schrödinger representations is established. The link between the spin ½ model and the Schrödinger representations yields a connection between a microscopic and a macroscopic viewpoint. Quantization and its link to information is derived out of the Schrödinger representation.

Working in the framework of Sym's soliton surfaces approach we point out that some simple assumptions about the structure of linear (spectral) problems of the theory of solitons lead uniquely to the geometry of some special immersions. In this paper we consider general su(2) spectral problems. Under some very weak assumptions they turn out to be associated with hyperbolic surfaces (surfaces of negative Gaussian curvature) immersed in three-dimensional Euclidean space, and especially with the so-called Bianchi surfaces.

Escape to infinity is proved for a great variety of potentials, including the potential created by an infinite number of sources. Relativistic escape is studied. Escape in the presence of a constant electromagnetic field and a potential is also considered.

The factorization problem for the group of canonical transformations close to the identity and the corresponding twistor equations for an ample family of canonical variables are considered. A method to deal with these reductions is developed for the construction of classes of nontrivial solutions of the dKP equation.

A third-order linear homogeneous differential equation is first derived for the local density of s-states only, N_s(r, E), for a general central potential V(r). The major example presented is then for the bare Coulomb potential −Ze²/r, for which analytical forms are obtained for both N_s(r, E) and for the total density of states N(r, E). In the appendix, the repulsive linear potential case is also solved analytically.

We construct representations of a q-oscillator algebra by operators on Fock space on positive matrices. They emerge from a multiresolution scaling construction used in wavelet analysis. The representations of the Cuntz algebra arising from this multiresolution analysis are contained as a special case in the Fock space construction.

In static experiments that comprise three conducting spheres suspended by torsion wires and held at constant electric potential, a net angular displacement about their centres has been observed. We demonstrate that the observed rotation is consistent with Coulomb's law of electrical forces complemented by Gauss' surface integrals for electrical potential. Analysis demonstrates that electrostatic torque is the result of electrostatic forces acting on an asymmetric distribution of charges residing on the surfaces of the spheres. The asymptotic value for electrostatic torque is proportional to the inverse of the fourth power of separation distance with the rotation direction, up or down taken perpendicular to a plane passing through sphere centres, given explicitly by the equation for torque. The identification of electrostatic torque prompts further analysis of models of matter at all size scales where electrostatic forces are the dominant operative force.

All possible irreducible representations of the Dirac algebra for a particle constrained to move on a D-dimensional manifold f(x) = 0 are explicitly constructed in terms of canonical operators _α, _α (α = 1, 2, ..., D + 1) in ^D+1 by assuming that the manifold, which is embedded in ^D+1, is diffeomorphic to S^D. It is shown that for D = 1 any irreducible representation is uniquely specified by a real parameter α belonging to [0, 1), while for D ⩾ 2 the irreducible representation is unique. The explicit form of inner products in _α-diagonal representation is given with the help of auxiliary wavefunctions on ^D+1, provided that they satisfy certain boundary conditions on the manifold. Applying it we further examine the hermiticity property of the fundamental operators of the Dirac algebra.

First-order Lagrangian densities with first-order Euler–Lagrange operator are characterized. Variationally trivial Lagrangians and locally variational operators of first order are determined.

A new type of exact solvability is reported. The Schrödinger equation is considered in a very large spatial dimension D ≫ 1 and its central polynomial potential is allowed to depend on 'many' ( = 2q) coupling constants. In a search for its bound states possessing an exact and elementary wavefunction ψ (proportional to a harmonic-oscillator-like polynomial of a freely varying, i.e., not just small, degree N), the 'solvability conditions' are known to form a complicated nonlinear set which requires a purely numerical treatment at a generic choice of D, q and N. Assuming that D is large we discovered and demonstrate that this problem may be completely factorized and acquires an amazingly simple exact solution at all N and up to q = 5 at least.

The discontinuous dependence of the properties of a quantum game on its entanglement has been shown to be very much like phase transitions viewed in the entanglement-payoff diagram (J Du et al 2002 Phys. Rev. Lett.88 137902). In this paper we investigate such phase-transition-like behaviour of quantum games, by suggesting a method which would help to illuminate the origin of such a kind of behaviour. For the particular case of the generalized Prisoners' Dilemma, we find that, for different settings of the numerical values in the payoff table, even though the classical game behaves the same, the quantum game exhibits different and interesting phase-transition-like behaviour.

The Larmor precession of a relativistic neutral spin-½ particle in a uniform constant magnetic field confined to the region of a one-dimensional rectangular potential well is investigated. The spin precession serves as a clock to measure the time spent by a quantum particle dwelling at a potential well. With the help of a general spin coherent state it is explicitly shown that the spin precession time is equal to the dwell time in the first-order approximation of the infinitesimal field limit. The comparison of the time in a potential well with that in free space shows apparent superluminality.

We deal with the motion of a charged particle in a constant magnetic field when immersed in a medium that exerts some type of friction. We analyse the problem classically and also quantum mechanically. In the latter case we use coherent state solutions of the problem that, for large energies, compared with the one associated with the cyclotron frequency, reduce to the classical limit. To study the dynamical behaviour we use time-dependent wave packets that can be constructed by superposition of the coherent states.

The structure of the Euler–Lagrange equations for a general Lagrangian theory (e.g. singular, with higher derivatives) is studied. For these equations we present a reduction procedure to the so-called canonical form. In the canonical form the equations are solved with respect to highest-order derivatives of nongauge coordinates, whereas gauge coordinates and their derivatives enter the right-hand sides of the equations as arbitrary functions of time. The reduction procedure reveals constraints in the Lagrangian formulation of singular systems and, in that respect, is similar to the Dirac procedure in the Hamiltonian formulation. Moreover, the reduction procedure allows one to reveal the gauge identities between the Euler–Lagrange equations. Thus, a constructive way of finding all the gauge generators within the Lagrangian formulation is presented. At the same time, it is proved that for local theories all the gauge generators are local in time operators.