Index of content:
Volume 35, Issue 1, January 1994

Quantum flows associated to master equations in quantum optics
View Description Hide DescriptionA stochastic dilation of master equation for a maser is constructed using the technique of quantum stochastic differential equations with unbounded operator coefficients. The associated quantum Markov flow is characterized and turns out to extend a classical birth and death process. Furthermore, the stationary distribution of the aforementioned process, which is connected to the stationary state of the micromaser, is obtained in a closed form.

Vertex operators in solvable lattice models
View Description Hide DescriptionThe basic properties of q‐vertex operators are formulated in the context of the Andrews–Baxter–Forrester (ABF) series, as an example of face interaction models, the q‐difference equations satisfied by their correlation functions are derived, and their connection with representation theory established. The q‐difference equations of the Kashiwara–Miwa (KM) series are discussed as an example of edge interaction models. Next, the Ising model, the simplest special case of both ABF and KM series, is studied in more detail using the Jordan–Wigner fermions. In particular, all matrix elements of vertex operators are calculated.

Exact and avoided crossings of adiabatic hyperspherical potential curves
View Description Hide DescriptionExact crossing of potential curves related to the adiabatic hyperspherical (AH) approach to a three‐body Coulomb system is studied. Analytic structure of the AH potential curves, harmonics, and coupling matrix elements near crossing points in the complex plane of hyperradius is investigated. Results are applied to derive some basic features of avoided crossings of the AH potential curves at real hyperradii.

Generalized string‐flip model for quantum cluster scattering
View Description Hide DescriptionThe generalized string‐flip model for quantum scattering in a system of two N‐body clusters is under consideration. The extra channel interpreted as the quark compound bag is considered which generates energy‐dependent integral boundary conditions in the effective boundary problem. It is shown that the standard string‐flip model can be generalized for N identical particles in a cluster. The effective Hamiltonian and effective integral equations for the partial s‐wave S matrix are proven to be form invariant with respect to N. The effective configuration space is shown to be two‐dimensional for every N and its geometry for different N is discussed. Results of the numerical calculations of s‐wave scattering phases, inelasticity parameters, and scattering lengths for N=2 (meson–meson scattering) and N=3 (baryon–baryon scattering) are presented. The resonance influence of the extra channel is investigated. Generalizations including spin–spin interaction and quark color group SU(n) are also presented.

Cocycles for boson and fermion Bogoliubov transformations
View Description Hide DescriptionUnitarily implementable Bogoliubov transformations for charged, relativistic bosons and fermions are discussed, and explicit formulas for the two‐cocycles appearing in the group product of their implementers are derived. In the fermion case this provides a simple field theoretic derivation of the well‐known cocycle of the group of unitary operators on a Hilbert space modeled on the Hilbert–Schmidt class and closely related to the loop groups. In the boson case the cocycle is obtained for a similar group of pseudo‐unitary (symplectic) operators. Formulas are also given for the phases of one‐parameter groups of implementers and, more generally, families of implementers which are unitary propagators with parameter‐dependent generators.

Random time evolution and direct integrals: Constants of the motion and the mass operator
View Description Hide DescriptionThe members H _{ω}=p ^{2}+V _{ω}(x) of ergodic families of random operators, defined in H=L ^{2}(R ^{ d }), are considered as acting in fibers in a direct integral decomposition of K=L ^{2} (Ω,P(dω);H), (Ω,P) being the underlying probability space. Such a formulation may turn out to be useful in providing a rigorous background for theoretical physical methods employed in treating random systems. As an example a rigorous definition of the mass operator Σ(z) is presented and the formulation of transport equations in a functional analytic setting is briefly indicated. In case there is translational invariance in the probability law the operator H is shown to be unitarily equivalent to an operator Ĥ, which can be decomposed, Ĥ→Ĥ(k), in a way that diagonalizes p. Thus problems about averaged time evolution can be formulated entirely in terms of Ĥ(k) in L ^{2}(Ω,P).

Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli
View Description Hide DescriptionLinear processes in media with dissipation arising in conductivity, optics, viscoelasticity, etc. are considered. Time‐periodic fields in such media are described by linear differential equations for complex‐valued potentials. The properties of the media are characterized by complex valued tensors, for example, by complex conductivity or complex elasticitytensors. Variational formulations are suggested for such problems: The functionals whose Euler equations coincide with the original ones are constructed. Four equivalent variational principles are obtained: two minimax and two minimal ones. The functionals of the obtained minimal variational principles are proportional to the energy dissipation averaged over the period of oscillation. The last principles can be used in the homogenization theory to obtain the bounds on the effective properties of composite materials with complex valued propertiestensors.

A non‐Lagrangian approach to gauge symmetry in three dimensions
View Description Hide DescriptionA non‐Lagrangian approach to three‐dimensional field theory is discussed. The general classification and construction of Lorentz‐covariant, local fields carrying massless and massive particles of arbitrary spin are presented. The ‘‘minimal’’ massive (massless) fields for (half) integer spin are explicitly constructed. Following Weinberg’s analysis in the 4D case it is shown that the gauge symmetry for massive fields results from the group theoretical considerations and the condition that there should exist a long‐range interaction.

Reduction of space–time by a constraint for C ^{(0)} metrics
View Description Hide DescriptionIt is shown how a constraint imposed by the physical relevance of the solutions reduces the dimension of space–time. The C ^{(0)} metric given by Nutku is used to describe gravitational shock waves and the trivial case is looked at. It is found that imposing triviality on the solutions reduces the dimension of space–time by one.

Search for first integrals in relativistic time‐dependent Hamiltonian systems
View Description Hide DescriptionIn this paper, first integrals of one‐dimensional time‐dependent relativistic Hamiltonians are looked for. In a first step, potentials for which the corresponding motion equations are completely integrable are found. In a second step, a more realistic potential described by the time‐dependent relativistic harmonic oscillator is studied. It appears that it is not analytically integrable. However its asymptotic evolution systematically converges towards a classical evolution for which we can exhibit a first integral.

On convergence to equilibrium for Kac’s caricature of a Maxwell gas
View Description Hide DescriptionThe trend towards equilibrium of the solution to the one‐dimensional Kac’s caricature of a Maxwell gas is investigated. When the initial value has finite energy and finite Linnik’s entropy, it is proven that the solution converges in L _{1}∩L _{∞} to the equilibrium Maxwellian distribution.

Supercritical bifurcation theory and stability of point attractors
View Description Hide DescriptionA novel approach is employed for studying a paramagnetic/ferromagnetic phase transition. Here, the Fokker–Planck transport equation is used to describe the time dependence of the spin distribution function (order parameter) for the XYmodel in mean‐field theory. The evolution of the phase‐space trajectories from initial nonequilibrium states is obtained. This is then applied in explaining the otherwise well‐known behavior of the XYmodel using a fully dynamical‐systems approach. The attractors of this infinite‐dimensional dynamical system are determined and their stability for any value of a system parameter δ, that plays the role of the absolute temperature, is obtained. A supercritical bifurcation occurs for δ=1/2, and this bifurcation corresponds to the paramagnetic/ferromagnetic phase transition. For an arbitrary initial spin density, a unique equilibrium magnetization is shown as being due to a continuum of fixed points existing in the temperature range 0<δ<1/2. These fixed points attract all the phase‐space trajectories, except those lying on the stable manifold of a trivial fixed point. The trivial fixed point at the origin is stable if δ≳1/2, otherwise it is a saddle point. These two types of fixed points determine the limit behavior of the dynamical system and therefore also the equilibrium state of the XYmodel in the approximation used here.

On a ferromagnetic spin chain. II. Thermodynamic limit
View Description Hide DescriptionThe existence of the thermodynamic limit of the free energy for the ferromagnetic spin chain connected with the Riemann zeta function is proven.

Modular invariant partition functions and method of shift vector
View Description Hide DescriptionUsing the shift vector method a large class of self‐dual lattices of dimension (l,l) is obtained, which has a one to one correspondence with modular invariants of free bosonic theory compactified on a coroot lattice of a rank lLie group. Then a large number of modular invariants of affine Lie algebras are derived explicitly. Two applications of this method are a direct derivation of the D series of SU(N) and a new proof for the A‐D‐E classification of the SU(2)_{ k } partition functions.

New symmetry reductions and exact solutions of the Davey–Stewartson system. I. Reductions to ordinary differential equations
View Description Hide DescriptionIn this article symmetry reductions and exact solutions are presented for the (2+1)‐dimensional Davey–Stewartson (DS) system which has the completely integrable DSI and DSII systems as special cases. These symmetry reductions are obtained using the direct method originally developed by Clarkson and Kruskal to study symmetry reductions of the Boussinesq equation which involves no group theoretic techniques. The DS system is reduced directly to ordinary differential equations, with no intermediate step. Using these reductions exact solutions of the DS system including some expressible in terms of the second and fourth Painlevé equations and elementary functions are obtained.

Symmetries of the Ablowitz–Kaup–Newell–Segur hierarchy
View Description Hide DescriptionNonlocal symmetries of the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy are introduced and it is shown that the symmetry algebra of the AKNS hierarchy is isomorphic to the loop algebra sl(2,C)⊗C[λ, λ^{−1}]. As a special case, the symmetry algebra of the nonlinear Schrödinger equation is determined and is shown to be isomorphic to the loop algebra su(2)⊗R[λ, λ^{−1}] or g⊗R[λ, λ^{−1}] corresponding to the sign of the nonlinear term, where g is a noncompact real form of sl(2,C).

A twistorial description of the processes of mass scattering of Dirac fields
View Description Hide DescriptionA set of twistorial formulas for the null graphs that describe the mass scattering of Dirac fields in real Minkowski space is explicitly derived. By making particular use of two holomorphic expressions for the generalized mass‐scattering differential forms the scattering integrals are transcribed directly into the framework of twistor theory.

Invariant connections with torsion on group manifolds and their application in Kaluza–Klein theories
View Description Hide DescriptionInvariant connections with torsion on simple group manifoldsS are studied and an explicit formula describing them is presented. This result is used for the dimensional reduction in a theory of multidimensional gravity with curvature squared terms on M ^{4}×S. The potential of scalar fields emerging from extra components of the metric and torsion is calculated, and the role of the torsion for the stability of spontaneous compactification is analyzed.

Geodesic multiplication and the theory of gravity
View Description Hide DescriptionNonassociative algebraic systems called local geodesic loops and their tangent Akivis algebras are considered. The construction of geo‐odular structure of a manifold with an affine connection is briefly reviewed. A possible role of these algebraic structures in classical and quantum gravity is discussed.

Hamiltonian formulation of the teleparallel description of general relativity
View Description Hide DescriptionThe Hamiltonian formulation of the teleparallel description of Einstein’s general relativity is established. Under a particular gauge fixing the Hamiltonian of the theory is written in terms of first class constraints. The algebra of the Hamiltonian and vector constraints resembles that of the standard Arnowitt–Deser–Misner formulation. This geometrical framework might be relevant as it is known that in manifolds with vanishing curvature tensor but with nonzero torsion tensor it is possible to carry out a simple construction of Becchi–Rouet–Stora–Tyutin‐like operators.