Volume 45, Issue 12, December 2004
 METHODS OF MATHEMATICAL PHYSICS


Extremal covariant positive operator valued measures
View Description Hide DescriptionWe consider the convex set of positive operator valued measures (POVM) which are covariant under a finite dimensional unitary projective representation of a group. We derive a general characterization for the extremal points, and provide bounds for the ranks of the corresponding POVM densities, also relating extremality to uniqueness and stability of optimized measurements. Examples of applications are given.
 Top

 GENERAL RELATIVITY AND GRAVITATION


A new geometrical look at gravity coupled with Yang–Mills fields
View Description Hide DescriptionA new geometrical framework for tetradaffine formulation of gravity, pure or coupled with Yang–Mills fields, is proposed. After analyzing the geometrical properties of the new mathematical setting, field equations are deduced from a variational principle in the Poincaré–Cartan formalism. A generalized Noether Theorem is stated and classical relationship between symmetries and conserved quantities are recovered in the newer scheme. Some explicit examples are given.

Hyperbolic Kac Moody algebras and Einstein billiards
View Description Hide DescriptionWe identify the hyperbolic Kac Moody algebras for which there exists a Lagrangian of gravity, dilatons, and forms which produces a billiard that can be identified with their fundamental Weyl chamber. Because of the invariance of the billiard upon toroidal dimensional reduction, the list of admissible algebras is determined by the existence of a Lagrangian in three space–time dimensions, where a systematic analysis can be carried out since only zeroforms are involved. We provide all highest dimensional parent Lagrangians with their full spectrum of forms and dilaton couplings. We confirm, in particular, that for the rank 10 hyperbolic algebra,, also known as the dual of , the maximally oxidizedLagrangian is ninedimensional and involves besides gravity, 2 dilatons, a 2form, a 1form, and a 0form.
 Top

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Lifshits tails for random smooth magnetic vortices
View Description Hide DescriptionWe study the density of states of the Pauli Hamiltonian with a Poisson random distribution of smooth finitewidth vortices and we obtain classical bounds for the Lifshits tails for them. These Hamiltonians are smooth approximations to the selfadjoint extensions of the Aharonov–Bohm Hamiltonian. In this case because pairs of impurities are coupled by the magnetic field we cannot use the Laplace characteristic functional.
 Top

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Coherent solutions for relativistic vectorial fields
View Description Hide DescriptionApproximate interacting localized solutions of a vectorial massive nonlinear equation are obtained by using the asymptotic perturbation (AP) method, based on Fourier expansion and spatiotemporal rescaling. The amplitude slow modulation of Fourier modes is described by a system of nonlinear evolution equations solvable via an appropriate change of variables. Various types of localized solutions (dromions, lumps, ring solitons, and breathers) as well as multiple soliton and instanton solutions can be explicitly constructed and their interaction is completely elastic, because they pass through each other and preserve their shape, the only change being a phase shift.

Avoiding superluminal propagation of higher spin waves via projectors onto invariant subspaces
View Description Hide DescriptionWe propose to describe higher spins as invariant subspaces of the Casimir operators of the Poincaré Group, , and the squared Pauli–Lubanski operator, , in a properly chosen representation, (in momentum space), of the Homogeneous Lorentz Group. The resulting equation of motion for any field with is then just a specific combination of the respective covariant projectors. We couple minimally electromagnetism to this equation and show that the corresponding wave fronts of the classical solutions propagate causally. Furthermore, for representations, the formalism predicts the correct gyromagnetic factor, . The advocated method allows us to describe any higher spin without auxiliary conditions and by one covariant matrix equation alone. This master equation is only quadratic in the momenta and its dimensionality is that of . We prove that the suggested master equation avoids the Velo–Zwanziger problem of superluminal propagation of higher spin waves and points toward a consistent description of higher spin quantum fields.

Finite size effects in thermal field theory
View Description Hide DescriptionWe consider a neutral selfinteracting massive scalar field defined in a dimensional Euclidean space. Assuming thermal equilibrium, we discuss the oneloop perturbative renormalization of this theory in the presence of rigid boundary surfaces (two parallel hyperplanes), which break translational symmetry. In order to identify the singular parts of the oneloop twopoint and fourpoint Schwinger functions, we use a combination of dimensional and zetafunction analytic regularization procedures. The infinities which occur in both the regularized oneloop twopoint and fourpoint Schwinger functions fall into two distinct classes: local divergences that could be renormalized with the introduction of the usual bulk counterterms, and surface divergences that demand counterterms concentrated on the boundaries. We present the detailed form of the surface divergences and discuss different strategies that one can assume to solve the problem of the surface divergences. We also briefly mention how to overcome the difficulties generated by infrared divergences in the case of Neumann–Neumann boundary conditions.
 Top

 METHODS OF MATHEMATICAL PHYSICS


On a conjecture of Givental
View Description Hide DescriptionThese brief notes record our puzzles and findings surrounding Givental’s recent conjecture which expresses higher genus Gromov–Witten invariants in terms of the genus0 data. We limit our considerations to the case of a complex projective line, whose Gromov–Witten invariants are wellknown and easy to compute. We make some simple checks supporting his conjecture.
 Top

 GENERAL RELATIVITY AND GRAVITATION


Space–time slices and surfaces of revolution
View Description Hide DescriptionUnder certain conditions, a dimensional slice of a spherically symmetric black hole space–time can be equivariantly embedded in dimensional Minkowski space. The embedding depends on a real parameter that corresponds physically to the surface gravity of the black hole horizon. Under conditions that turn out to be closely related, a real surface that possesses rotational symmetry can be equivariantly embedded in threedimensional Euclidean space. The embedding does not obviously depend on a parameter. However, the Gaussian curvature is given by a simple formula: If the metric is written , then . This note shows that metrics and occur in dual pairs, and that the embeddings described above are orthogonal facets of a single phenomenon. In particular, the metrics and their respective embeddings differ by a Wick rotation that preserves the ambient symmetry. Consequently, the embedding of depends on a real parameter. The ambient space is not smooth, and is inversely proportional to the cone angle at the axis of rotation. Further, the Gaussian curvature of is given by a simple formula that seems not to be widely known.
 Top

 METHODS OF MATHEMATICAL PHYSICS


On the resolvent and spectral functions of a second order differential operator with a regular singularity
View Description Hide DescriptionWe consider the resolvent of a second order differential operator with a regular singularity, admitting a family of selfadjoint extensions. We find that the asymptotic expansion for the resolvent in the general case presents unusual powers of which depend on the singularity. The consequences for the pole structure of the function, and for the small asymptotic expansion of the heat kernel, are also discussed.

Deformations of loop algebras and integrable systems: hierarchies of integrable equations
View Description Hide DescriptionUsing special quasigraded Lie algebras, that could be viewed as deformations of loop algebras, we obtain new hierarchies of integrable nonlinear equations admitting zerocurvature representations. In particular, we obtain integrable hierarchies that generalize the Heisenberg magnet, Landau–Lifshitz, and anisotropic chiral field hierarchies. We also obtain a new type of anisotropic chiral field equation along with its higher rank generalization.

Moduli of quantum Riemannian geometries on points
View Description Hide DescriptionWe classify parallelizable noncommutative manifold structures on finite sets of small size in the general formalism of framed quantum manifolds and vielbeins introduced previously [S. Majid, Commun. Math. Phys. 225, 131 (2002)]. The full moduli space is found for points, and a restricted moduli space for 4 points. Generalized Levi–Cività connections and their curvatures are found for a variety of models including models of a discrete torus. The topological part of the moduli space is found for points based on the known atlas of regular graphs. We also remark on aspects of quantum gravity in this approach.
 Top

 DYNAMICAL SYSTEMS


Nonintegrability of nonhomogeneous nonlinear lattices
View Description Hide DescriptionWe study the integrability of nonlinear lattices with nonhomogeneous polynomial potentials. We prove a nonintegrability theorem for these dynamical systems.
 Top

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


The invariant charges of the Nambu–Goto string and canonical quantization
View Description Hide DescriptionIt is shown that the algebra of diffeomorphisminvariant charges of the Nambu–Goto string cannot be quantized in the framework of canonical quantization. The argument is shown to be independent of the dimension of the underlying Minkowski space.
 Top

 DYNAMICAL SYSTEMS


A technique to identify solvable dynamical systems, and another solvable extension of the goldfish manybody problem
View Description Hide DescriptionWe take advantage of the simple approach, recently discussed, which associates to (solvable) matrix equations (solvable) dynamical systems interpretable as (interesting) manybody problems, possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles. Starting from a solvable matrix evolution equation, we obtain the corresponding manybody model and note that in one case the auxiliary variables can be altogether eliminated, obtaining thereby an (also Hamiltonian) extension of the “goldfish” model. The solvability of this novel model, and of its isochronous variant, is exhibited. A related, as well solvable, model, is also introduced, as well as its isochronous variant. Finally, the small oscillations of the isochronous models around their equilibrium configurations are investigated, and from their isochronicity certain diophantine relations are evinced.
 Top

 METHODS OF MATHEMATICAL PHYSICS


An algebraic Birkhoff decomposition for the continuous renormalization group
View Description Hide DescriptionThis paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing us to formulate perturbative renormalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited.
 Top

 GENERAL RELATIVITY AND GRAVITATION


Asymptotic quasinormal frequencies for black holes in nonasymptotically flat space–times
View Description Hide DescriptionThe exact computation of asymptotic quasinormal frequencies is a technical problem which involves the analytic continuation of a Schrödingertype equation to the complex plane and then performing a method of monodromy matching at several poles in the plane. While this method was successfully used in asymptotically flat space–time, as applied to both the Schwarzschild and Reissner–Nordstro/m solutions, its extension to nonasymptotically flat space–times has not been achieved yet. In this work it is shown how to extend the method to this case, with the explicit analysis of Schwarzschild–de Sitter and large Schwarzschild–anti–de Sitter black holes, both in four dimensions. We obtain, for the first time, analytic expressions for the asymptotic quasinormal frequencies of these black hole space–times, and our results match previous numerical calculations with great accuracy. We also list some results concerning the general classification of asymptotic quasinormal frequencies in dimensional space–times.
 Top

 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Composite systems and the role of the complex numbers in quantum mechanics
View Description Hide DescriptionAn axiomatic approach to the mathematical formalism of quantum mechanics, based upon a certain concept of conditional probability, has been proposed in two recent papers by the author. It leads to Jordan operator algebras and thus comes rather close to the standard Hilbert spacemodel of quantum mechanics, but still includes the socalled exceptional Jordan algebras, for which a Hilbert space representation does not exist. This approach is now extended by defining a mathematical model of composite systems. Such a model is required for the study of the joint distribution of two quantum observables. A very general type of observables (not only the realvalued observables corresponding to the selfadjoint operators) is considered. The joint distribution is defined, using the concept of conditional probability, and exhibits a certain dependence on the succession of the observations which is different from the classical case and unknown so far in quantum mechanics. Finally, it turns out that, at least in the finitedimensional case, a really satisfying model of the composite system exists only if each single system is modeled by a complex Jordan matrix algebra (or a direct sum), and the model then becomes the tensor product. This provides some reasoning why the exceptional Jordan algebras can be ruled out, why quantum mechanics needs the complex numbers and the complex Hilbert space, and why the tensor product is the right choice for the model of a composite system.
 Top

 GENERAL RELATIVITY AND GRAVITATION


On spherically symmetric solutions with horizon in model with multicomponent anisotropic fluid
View Description Hide DescriptionA family of spherically symmetric solutions in the model with component multicomponent anisotropic fluid is considered. The metric of the solution depends on parameters , , relating radial pressures and the densities and contains parameters corresponding to Ricciflat “internal space” metrics and obeying certain (“orthogonality”) relations. For (for all ) and certain equations of state the metric coincides with the metric of intersecting black branesolution in the model with antisymmetric forms. A family of solutions with (regular) horizon corresponding to natural numbers is singled out. Certain examples of “generalized simulation” of intersecting branes in supergravity are considered. The postNewtonian parameters and corresponding to the fourdimensional section of the metric are calculated.
 Top

 METHODS OF MATHEMATICAL PHYSICS


On the Treves theorem for the Ablowitz–Kaup–Newell–Segur equation
View Description Hide DescriptionAccording to a theorem of Treves, the conserved functionals of the Ablowitz, Kaup, Newell, and Segur (AKNS) equation vanish on all pairs of formal Laurent series of a specified form, both of them with a pole of the first order. We propose a new and very simple proof for this statement, based on the theory of Bäcklund transformations; using the same method, we prove that the AKNS conserved functionals vanish on other pairs of Laurent series. The spirit is the same as in our previous paper on the Treves theorem for the Korteweg–de Vries hierarchy, with some nontrivial technical differences.
