Volume 37, Issue 6, June 1996
Index of content:

Skyrme models with self‐dual limits: d=2,3
View Description Hide DescriptionThe most general Skyrme–Sigma models in two and three Euclidean dimensions described by O(3) and O(4) fields, respectively, are studied first by numerical methods, and analytic proofs of existence are subsequently given. Particular emphasis is given to the special cases of these models, where the topological inequalities can be saturated by self‐duality equations. The O(d+1) models in d dimensions exhibit qualitatively similar features.

Correlation properties of quantum measurements
View Description Hide DescriptionThe kind of information provided by a measurement is determined in terms of the correlation established between observables of the apparatus and the measured system. Using the framework of quantum measurement theory, necessary and sufficient conditions for a measurementinteraction to produce strong correlations are given and are found to be related to properties of the final object and apparatus states. These general results are illustrated with reference to the standard model of the quantum theorymeasurement.

Canonical quantization of spontaneously broken topologically massive gauge theory
View Description Hide DescriptionIn this paper we investigate the canonical quantization of a non‐Abelian topologically massive Chern–Simons theory in which the gauge fields are minimally coupled to a multiplet of scalar fields in such a way that the gauge symmetry is spontaneously broken. Such a model produces the Chern–Simons–Higgs mechanism in which the gauge excitations acquire mass both from the Chern–Simons term and from the Higgs–Kibble effect. The symmetry breaking is chosen to be only partially broken, in such a way that in the broken vacuum there remains a residual non‐Abelian symmetry. We develop the canonical operator structure of this theory in the broken vacuum, with particular emphasis on the particle‐content of the fields involved in the Chern–Simons–Higgs mechanism. We construct the Fock space and express the dynamical generators in terms of creation and annihilation operator modes. The canonical apparatus is used to obtain the propagators for this theory, and we use the Poincaré generators to demonstrate the effect of Lorentz boosts on the particle states.

The q‐Coulomb problem
View Description Hide DescriptionThere exist operator solutions of the q‐Coulomb problem in both configuration and momentum space. Since the arguments of the corresponding amplitudes are noncommuting, however, there are problems of physical interpretation. Here we answer the question of physical interpretation by associating with the operator amplitude in momentum space a numerically valued amplitude lying in a Hilbert space defined by the SU_{ q }(2) algebra. This new amplitude, now depending on commuting arguments, may be interpreted by the usual rules of quantum mechanics and may also be Fourier transformed to yield the amplitude in configuration space.

Lattice properties of quantum effects
View Description Hide DescriptionSufficient conditions for the existence of the infimum AΛB of two quantum effectsA and B are given. The existence of AΛB is characterized for commuting A and B with pure point spectrum. Properties of a generalized infimum and supremum are studied. Some previous finite dimensional, commutative results are extended to the infinite dimensional and noncommutative case.

On the diagonalization of quantum Birkhoff–Gustavson normal form
View Description Hide DescriptionAn application to quantum mechanics of one of classical perturbation theory methods, the Birkhoff–Gustavson normal form (BGNF), is described. In the quantum case it results in the Van Vleck perturbation theory performed upon Wick normal ordered operators. Algebraic aspects of this procedure and formal construction of invariants (integrals of motion) for a perturbed system are considered. It turned out that a larger set of such operators existed in the quantum mechanics, rather than in the classical one. It is demonstrated that, according to general results of the quantum mechanical perturbation theory, the quantum BGNF may always be diagonalized, and two formal processes for such diagonalization are constructed. In the opposite case, the classical BGNF is, in general, nondiagonalizable. This reflects the fact that the classical perturbation theory cannot handle a system with two or more resonances. Possible reasons for such different behavior of two very close, in spirit, perturbation procedures are discussed. Results of the described procedure, entirely performed upon the Wick normal ordered operators, are equivalent to those of Rayleigh–Schrödinger perturbation expansion.

Geometries of quantum states
View Description Hide DescriptionThe quantum analog of the Fisher information metric of a probability simplex is searched and several Riemannian metrics on the set of positive definite density matrices are studied. Some of them appeared in the literature in connection with Cramér–Rao‐type inequalities or the generalization of the Berry phase to mixed states. They are shown to be stochastically monotone here. All stochastically monotone Riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them. A class of metrics can be extended to pure states and a constant multiple of the Fubini–Study metric appears in the extension.

Transformation brackets between U(ν+1)⊃U(ν)⊃SO(ν) and U(ν+1)⊃SO(ν+1)⊃SO(ν)
View Description Hide DescriptionWe derive a general expression for the transformation brackets between the chains U(ν+1)⊇U(ν)⊇SO(ν) and U(ν+1)⊇SO(ν+1)⊇SO(ν) for ν≥2.

Applications of quantum and classical Fisher information to two‐level complex and quaternionic and three‐level complex systems
View Description Hide DescriptionIn the Bayesiantheory of statistical inference, as first suggested by Harold Jeffreys in highly influential work, one can employ the square root of the determinant of an n×n Fisher information matrix as a reparametrization‐invariant prior (generally unnormalized) measure over an n‐dimensional family (Riemannian manifold) of probability distributions. Jeffreys’ ansatz is adopted here to the quantum context, that is, with regard to density matrices rather than probability distributions, by computing the quantum Fisher information matrices (associated with Helstrom and Holevo) for the three‐, five‐, and eight‐dimensional convex sets of two‐level complex, two‐level quaternionic, and three‐level complex systems, respectively. In both the two‐level cases, the priors have been normalized to probability distributions over the 2×2 density matrices, while, in the much more computationally demanding three‐level situation, no such normalization has been accomplished. An argument is made for the general form, in terms of eigenvalues, that the (unnormalized) prior should assume over the (n ^{2}−1)‐dimensional convex set of n×n density matrices.

Free quantum fields on the Poincaré group
View Description Hide DescriptionA class of free quantum fields defined on the Poincaré group is described by means of their two‐point vacuum expectation values. They are not equivalent to fields defined on the Minkowski space–time and they are ‘‘elementary’’ in the sense that they describe particles that transform according to irreducible unitary representations of the symmetry group, given by the product of the Poincaré group and of the group SL(2,C) considered as an internal symmetry group. Some of these fields describe particles with positive mass and arbitrary spin and particles with zero mass and arbitrary helicity or with an infinite helicity spectrum. In each case the allowed SL(2,C) internal quantum numbers are specified. The properties of local commutativity and the limit in which one recovers the usual field theories in Minkowski space–time are discussed. By means of a superposition of elementary fields, one obtains an example of a field that presents a broken symmetry with respect to the group Sp(4,R) that survives in the short‐distance limit. Finally, the interaction with an accelerated external source is studied and it is shown that, in some theories, the average number of particles emitted per unit of proper time diverges when the acceleration exceeds a finite critical value.

The Stückelberg–Kibble model as an example of quantized symplectic reduction
View Description Hide DescriptionRecently, it has been observed that a certain class of classical theories with constraints can be quantized by a mathematical procedure known as Rieffel induction. After a short exposition of this idea, we apply the new quantization theory to the Stückelberg–Kibble model. We explicitly construct the physical state space H_{phys}, which carries a massive representation of the Poincaré group. The longitudinal one‐particle component arises from a particular Bogoliubov transformation of the five (unphysical) degrees of freedom one has started with. Our discussion exhibits the particular features of the proposed constrained quantization theory in great clarity.

The global flow of the Manev problem
View Description Hide DescriptionThe Manev problem (a two‐body problem given by a potential of the form A/r+B/r ^{2}, where r is the distance between particles and A,B are positive constants) comprises several important physical models, having its roots in research done by Isaac Newton. We provide its analytic solution, then completely describe its global flow using McGehee coordinates and topological methods, and offer the physical interpretation of all solutions. We prove that if the energy constant is negative, the orbits are, generically, precessional ellipses, except for a zero‐measure set of initial data, for which they are ellipses. For zero energy, the orbits are precessional parabolas, and for positive energy they are precessional hyperbolas. In all these cases, the set of initial data leading to collisions has positive measure.

The time harmonic electromagnetic field in a disturbed half‐space: An existence theorem and a computational method
View Description Hide DescriptionThe scattering of time harmonic electromagnetic waves by a perfectly conducting surface that is the boundary of a ‘‘disturbed half‐space’’ is considered. This problem is translated in a boundary value problem for an elliptic system of partial differential equations. Under appropriate hypotheses an existence theorem and an integral representation formula for the solution of this boundary value problem is given. Based on this integral representation formula a new method to compute the solution of the boundary value problem is proposed. This method involves only quadratures and is fully parallelizable. Finally some numerical examples of the results obtained on test problems with this computational method are shown.

On group invariant solutions of the Boltzmann equation
View Description Hide DescriptionAfter reviewing the complete Lie group for the full Boltzmann equation, it is shown that projective transformations play a special role in the general case, and the class of invariant solutions giving rise to homoenergetic affine flows is presented. Homoenergetic affine flows in the two‐dimensional case [potential U(r)∝r ^{−2}] are considered in detail: It is shown that the general solution of this problem can be essentially simplified by projective transformations.

Exact solution of the Ising model on group lattices of genus g>1
View Description Hide DescriptionWe discuss how to apply the dimer method to Ising models on group lattices having nontrivial topological genus g. We find that the use of group extension and the existence of both external and internal group isomorphisms greatly reduces the number of distinct Pfaffians and leads to explicit topological formulas for their sign and weight in the expansion of the partition function. The complete solution for the Ising model on the Klein lattice group L(2,7) with g=3 is given.

The successive reflection method in three dimensional particle transport
View Description Hide DescriptionThe free streaming operator T is considered in a convex three dimensional region V, with diffusive multiplying boundary conditions. Some mathematical properties of T are examined by writing the particle density as an infinite series which takes into account successive reflections on ∂V, and by introducing an operator which in some sense annihilates the multiplying effect of ∂V.

Nonlinear discrete systems with nonanalytic dispersion relations
View Description Hide DescriptionA discrete system of coupled waves (with nonanalytic dispersion relation) is derived in the context of the spectral transform theory for the Ablowitz–Ladik spectral problem (discrete version of the Zakharov–Shabat system). This 3‐wave evolution problem is a discrete version of the stimulated Raman scattering equations, and it is shown to be solvable for arbitrary boundary value of the two radiation fields and initial value of the medium state. The spectral transform is constructed on the basis of the ∂‐approach.

Multi‐Hamiltonian structures for a class of degenerate completely integrable systems
View Description Hide DescriptionIn this paper, a class of degenerate (i.e., associated to a degenerate Poisson structure) completely integrable systems is studied which generalizes the so‐called odd and even master systems introduced and studied by Mumford and by Vanhaecke. It is shown that all these completely integrable systems, called the generalized master systems, admit a multi‐Hamiltonian formulation, and a systematic construction of this multi‐Hamiltonian structure is described.

A completely integrable Hamiltonian system
View Description Hide DescriptionThe dynamical system characterized by the Hamiltonian H( q , p )=∑_{ j,k=1} ^{ n } p _{ jp } _{ kf }(q dj−q _{ k }) with f(x)=λ+μ cos(νx)+μ′ sin(ν‖x‖) is completely integrable. Here n is an arbitrary positive integer and λ,μ,μ′,ν are 4 arbitrary constants (λ and μ real, μ′ and ν both real or both imaginary).

Supertraces on the algebras of observables of the rational Calogero model with harmonic potential
View Description Hide DescriptionWe define a complete set of supertraces on the algebraSH _{ N }(ν), the algebra of observables of the N‐body rational Calogero model with harmonic interaction. This result extends the previously known results for the simplest cases of N=1 and N=2 to arbitrary N. It is shown that SH _{ N }(ν) admits q(N) independent supertraces, where q(N) is a number of partitions of N into a sum of odd positive integers, so that q(N)≳1 for N≥3. Some consequences of the existence of several independent supertraces of SH _{ N }(ν) are discussed, such as the existence of ideals in associated W _{∞}‐type Lie superalgebras.