Table of contents

Recently, a generalization of Emden's equation has been proposed by one of us in a form which embraces Thomas-Fermi-like theories. Here some specific solutions are presented and discussed.

The critical thermodynamics of an MN-component field model with cubic anisotropy relevant to the phase transitions in certain crystals with complicated ordering is studied within the four-loop ε expansion using the minimal subtraction scheme. Investigation of the global structure of RG flows for the physically significant cases M = 2, N = 2 and M = 2, N = 3 shows that the model has an anisotropic stable fixed point with new critical exponents. The critical dimensionality of the order parameter is proved to be equal to N_c^C = 1.445(20), that is exactly half its counterpart in the real hypercubic model.

The perimeter and area generating functions of exactly solvable polygon models satisfy q-functional equations, where q is the area variable. The behaviour in the vicinity of the point where the perimeter generating function diverges can often be described by a scaling function. We develop the method of q-linear approximants in order to extract the approximate scaling behaviour of polygon models when an exact solution is not known. We test the validity of our method by approximating exactly solvable q-linear polygon models. This leads to scaling functions for a number of q-linear polygon models, notably generalized rectangles, Ferrers diagrams, and stacks.

A number of authors have proposed stochastic versions of the Schrödinger equation, either as effective evolution equations for open quantum systems or as alternative theories with an intrinsic collapse mechanism. Here we discuss two directions for the generalization of these equations. First, we study a general class of norm preserving stochastic evolution equations, and show that even after making several specializations there is an infinity of possible stochastic Schrödinger equations for which state vector collapse is provable. Second, we explore the problem of formulating a relativistic stochastic Schrödinger equation, using a manifestly covariant equation for a quantum field system based on the interaction picture of Tomonaga and Schwinger. The stochastic noise term in this equation can couple to any local scalar density that commutes with the interaction energy density, and leads to collapse onto spatially localized eigenstates. However, as found in a similar model by Pearle, the equation predicts an infinite rate of energy nonconservation proportional to δ³(), arising from the local double commutator in the drift term.

To classify the classical field theories with W-symmetry one has to classify the symplectic leaves of the corresponding W-algebra, which are the intersection of the defining constraint and the coadjoint orbit of the affine Lie algebra if the W-algebra in question is obtained by reducing a Wess-Zumino-Novikov-Witten (WZNW) model. The fields that survive the reduction will obey nonlinear Poisson bracket (or commutator) relations in general. For example, the Toda models are well known theories which possess such a nonlinear W-symmetry and many features of these models can only be understood if one investigates the reduction procedure. In this paper we analyse the SL(n,) case from which the so-called W_n-algebras can be obtained. One advantage of the reduction viewpoint is that it gives a constructive way to classify the symplectic leaves of the W-algebra - for the n = 2 case corresponding to the coadjoint orbits of the Virasoro algebra and for the n = 3 case which gives rise to the Zamolodchikov algebra. Our method, in principle, is capable of constructing explicit representatives on each leaf. Another attractive feature of this approach is the fact that the global nature of the W-transformations can be explicitly described. The reduction method also enables one to determine the classical highest-weight (HW) states which are the stable minima of the energy on a W-leaf. These are important as only to those leaves can a HW representation space of the W-algebra be associated which contains a classical HW state.

Within the generalized definition of coherent states as group orbits we study the orbit spaces and the orbit manifolds in the projective spaces constructed from linear representations. Invariant functions are suggested for arbitrary groups. The group SU(2) is studied in particular and the orbit spaces of its j = 1/2 and 1 representations completely determined. The orbits of SU(2) in CP^N can be either two- or three-dimensional, the former being either isomorphic to S² or to RP² and the latter being isomorphic to quotient spaces of RP³. We end with a look from the same perspective to the quantum mechanical space of states in particle mechanics.

We develop a simple approach based on time-independent perturbation theory that leads to a frequency operator for quantum mechanical models. This method is suitable for the calculation of time-dependent physical properties free from secular terms. In particular we obtain the first-order perturbation correction to the frequency operator for a class of anharmonic oscillators.

A class of indecomposable representations of U_q(sℓ_n) is considered for q, an even root of unity (q^h = -1) exhibiting a similar structure as (height h) indecomposable lowest weight Kac-Moody modules associated with chiral conformal field theory. In particular, U_q(sℓ_n) counterparts of the Bernard-Felder BRS operators are constructed for n = 2,3. For n = 2 a pair of dual d₂(h) = h-dimensional U_q(sℓ₂) modules gives rise to a 2h-dimensional indecomposable representation including those studied earlier in the context of tensor-product expansions of irreducible representations. For n = 3 the interplay between the Poincaré-Birkhoff-Witt and (Lusztig) canonical bases is exploited in the study of d₃(h) = (1/6)h(h + 1)(2h + 1)-dimensional indecomposable modules and of the corresponding intertwiners.

We obtain the bilinear form of the supersymmetric sine-Gordon equation. Through the Hirota approach we contruct multisoliton solutions for this equation. We find that, in contrast to the purely bosonic case, the solitons are `dressed' through their mutual interaction and we compute this dressing explicitly. Using the Ablowitz, Ramani and Segur algorithm we verify that the supersymmetric sine-Gordon equation possesses the Painlevé property.

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or sections along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper.

The present, first, part of this investigation is devoted to the introduction of basic concepts on which the fibre bundle approach to quantum mechanics rests. We show that the evolution of pure quantum mechanical states can be described as a suitable linear transport along paths, called evolution transport, of the state liftings in the Hilbert bundle of states of a considered quantum system.

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced by an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or section along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper.

In the second part of this investigation are derived several forms of the bundle (analogue of the) Schrödinger equation governing the time evolution of state liftings of paths or sections along paths. We prove that up to a constant the matrix-bundle Hamiltonian, entering the bundle analogue of the matrix form of the conventional Schrödinger equation, coincides with the matrix of coefficients of the evolution transport. This allows us to interpret the Hamiltonian as a gauge field. We apply the bundle approach to the description of observables. It is shown that to any observable there corresponds a unique Hermitian lifting of paths or morphism along paths in corresponding bundles.

We propose a new systematic fibre bundle formulation of nonrelativistic quantum mechanics. The new form of the theory is equivalent to the usual one and is in harmony with the modern trends in theoretical physics and potentially admits new generalizations in different directions. In it the Hilbert space of a quantum system (from conventional quantum mechanics) is replaced with an appropriate Hilbert bundle of states and a pure state of the system is described by a lifting of paths or sections along paths in this bundle. The evolution of a pure state is determined through the bundle (analogue of the) Schrödinger equation. Now the dynamical variables and density operators are described via liftings of paths or morphisms along paths in suitable bundles. The mentioned quantities are connected by a number of relations derived in this paper.

In the third part of our series we investigate the bundle analogues of the conventional pictures of motion. In particular, we find the state liftings and observable liftings corresponding to state vectors and observables respectively in the different pictures of motion. The equations of motion for these quantities are derived. Using the results obtained, problems concerning the integrals of motion are considered from the bundle viewpoint. Necessary and sufficient invariant bundle conditions for a dynamical variable to be an integral of motion are found.

The properties of the entangled pure states in phase space are analysed using the classical entropy introduced by Wehrl. The general entropy inequalities for non-entangled pure states are derived, which are violated by any entangled state. To measure the strength of intermode correlations in phase space the parameters related to these inequalities are proposed. As an example we study the correlations between amplitudes and phases for two-mode Fock states. We find that the amplitudes as well as phases of different modes are correlated. It is also shown that the degree of the intermode correlation strongly depends on the photon number difference in two-mode Fock states.

A Painlevé (P) test of the coupled Gross-Pitaevskii equations has been carried out with the result that the coupled equations pass the P test only if a special relation containing system parameters (masses and scattering lengths) is satisfied. Computer algebra is applied to evaluate the j = 4 compatibility condition for admissible external potentials. The appearance of an arbitrary real potential embedded in the external potentials is shown to be the consequence of the coupling. The connection with recent experiments related to the stability of two-component Bose-Einstein condensates of Rb atoms is discussed.

All local generalized symmetries (including x,t-dependent ones) of the Bakirov system are found. In particular, it is shown that its only non-Lie-point local generalized symmetry is the sixth order one found by Bakirov. This result generalizes a similar result of Beukers, Sanders and Wang on x,t-independent symmetries and completes the refutation of the popular conjecture stating that the existence of one non-Lie-point local generalized symmetry for a (1+1)-dimensional system of PDEs implies the existence of infinitely many such symmetries.

Completeness of the Dirac oscillator eigenfunctions is proved in one and three spatial dimensions. Proofs are based on standard properties of the Hermite and the generalized Laguerre polynomials.

The angle-averaged distribution function of action variables is studied by means of projection operators on the basis of the Liouville equation for the single-particle phase-space distribution of weakly non-integrable Hamiltonian systems. It is shown that the angle-averaged distribution function is governed by a kinetic equation similar to the Fokker-Planck equation but with a memory integral. An explicit form of the memory kernel is derived by a second-order perturbation expansion using the adiabatic and averaging approximations. For localized nonlinear perturbations, the kinetic equation takes the form of a functional map which can further be reduced to a moment map by using the Gaussian approximation. To examine the validity of this treatment, the evolution of actions is studied with examples of non-integrable systems. It is found that the result from the moment map agrees very well with the results from multi-particle tracking.