Volume 41, Issue 10, October 2000
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Spontaneous compactification of twoform gauge fields and the obtainment of Maxwell and Yang–Mills theories
View Description Hide DescriptionWe show that the spontaneous compactification of the Abelian and nonAbelian twoform gauge field theories from to leads to the same theories plus the Maxwell and Yang–Mills ones, respectively. The vector potential comes from the zero mode of the fifth component of the tensor gauge field in Concerning the nonAbelian case, it is necessary to make a more refined definition of the threeform stress tensor in order to be compatible, after the compactification, with the twoform stress tensor of the Yang–Mills theory.

Feynman diagrams of generalized matrix models and the associated manifolds in dimension four
View Description Hide DescriptionThe problem of constructing a quantum theory of gravity has been tackled with very different strategies, most of which rely on the interplay between ideas from physics and from advanced mathematics. On the mathematical side, a central role is played by combinatorial topology, often used to recover the space–time manifold from the other structures involved. An extremely attractive possibility is that of encoding all possible space–times as specific Feynman diagrams of a suitable field theory. In this work we analyze how exactly one can associate combinatorial fourmanifolds with the Feynman diagrams of certain tensor theories.

Lightcone expansion of the Dirac sea in the presence of chiral and scalar potentials
View Description Hide DescriptionWe study the Dirac sea in the presence of external chiral and scalar/pseudoscalar potentials. In preparation, a method is developed for calculating the advanced and retarded Green’s functions in an expansion around the light cone. For this, we first expand all Feynman diagrams and then explicitly sum up the perturbation series. The lightcone expansion expresses the Green’s functions as an infinite sum of line integrals over the external potential and its partial derivatives. The Dirac sea is decomposed into a causal and a noncausal contribution. The causal contribution has a lightcone expansion which is closely related to the lightcone expansion of the Green’s functions; it describes the singular behavior of the Dirac sea in terms of nested line integrals along the light cone. The noncausal contribution, on the other hand, is, to every order in perturbation theory, a smooth function in position space.

Invariant measures on polarized submanifolds in group quantization
View Description Hide DescriptionWe provide an explicit construction of quasiinvariant measures on polarized coadjoint orbits of a Lie groupG. The use of specific (trivial) central extensions of G by the multiplicative group allows us to restore the strict invariance of the measures and, accordingly, the unitarity of the quantization of coadjoint orbits. As an example, the representations of are recovered.

Optimal ensemble length of mixed separable states
View Description Hide DescriptionThe optimal (pure state) ensemble length of a separable state is the minimum number of (pure) states needed in convex combination to construct We study the set of all separable states with optimal (pure state) ensemble length equal to or fewer. Lower bounds on are found below which these sets have measure 0 in the set of separable states. In the bipartite case and the multiparticle case where one of the particles has significantly more quantum numbers than the rest the lower bounds are sharp. A consequence of our results is that for all twoparticle systems, except possibly those with one qubit or those with a ninedimensional Hilbert space, and for all systems with more than two particles the optimal pure state ensemble length for a randomly picked separable state is with probability 1 greater than the state’s rank. In bipartite systems with probabilty 1 it is greater than 1/4 the rank raised to the 3/2 power and in a system of qubits with probability 1 it is greater than i.e., almost the square of the rank.

Why two qubits are special
View Description Hide DescriptionWe analyze some special properties of a system of two qubits, and in particular of the socalled Bell basis for this system, and discuss the possibility of extending these properties to higher dimensional systems. We give a general construction for orthonormal bases of maximally entangled vectors, which works in any dimension, and is based on Latin squares and complex Hadamard matrices. However, for none of these bases the special properties of the operation of complex conjugation in Bell basis hold, namely that maximally entangled vectors have uptoaphase real coefficients and that factorizable unitaries have real matrix elements.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


On the representation of inhomogeneous linear forcefree fields
View Description Hide DescriptionIt is shown that there is a false assumption hidden in the description of a relaxed state with inhomogeneous boundary conditions as the vector sum of a potential field, satisfying the boundary conditions, and a sum of eigenfunctions of the associated eigenvalue problem expanded by certain coefficients. In particular, although the Jensen and Chu formula (1984) can provide the correct expansion coefficients, it contains an implicit paradox in its derivation according to a general vector theorem. The same paradox led Chu et al. (1999) to be concerned about a contradiction obtained by taking the curl of their magnetic field expansion which, if permitted, becomes inconsistent with a current normal to the surface. The assumption that the curl can be commuted across an infinite sum of terms is the mechanism leading to these, apparently paradoxical, conclusions. Two mechanisms for resolving this apparent paradox are possible, one of which will be described in some detail below and the other discussed further in a forthcoming, more theoretical paper (Laurence et al., 2000). The decomposition of the magnetic field above is valid with convergence in the mean squared sense, but a decomposition of the current needs to be reinterpreted in terms of negative Sobolev spaces. To avoid this, and remain in a more easily managable and familiar setting, we derive the expansion coefficients in a way that involves the commuting of the inverse curl (as opposed to the curl) and the series. The resulting series converges in a mean square sense. When this is done the calculation can conform to the general vector theorem and a new gaugeinvariant expression for the coefficients is obtained. However the consequence of the noncommutability is nullified in the Jensen and Chu formula, in both simply and multiply connected domains, by the important extra requirement of a boundary condition on the vector potential eigenfunctions; this excludes magnetic field eigenfunctions that carry flux, but there remains a complete set for the expansion and all flux is carried by the potential field. The two formulas are then identical. On a different issue, it is shown that if the general expansion is taken over a halfspace, by combining positive and negative eigenvalue terms, then the coefficients are anisotropic, that is they are tensors except when evaluated at the first eigenvalue. A specific example is presented to illustrate the situation and to validate the new method of deriving the coefficients.

Covariant field theory on frame bundles of fibered manifolds
View Description Hide DescriptionWe show that covariant field theory for sections of π : lifts in a natural way to the bundle of vertically adapted linear frames Our analysis is based on the fact that is a principal fiber bundle over the bundle of 1jets On the canonical soldering 1forms play the role of the contact structure of A lifted LagrangianL: is used to construct modified soldering 1forms, which we refer to as the Cartan–Hamilton–Poincaré 1forms. These 1forms on pass to the quotient to define the standard Cartan–Hamilton–Poincaré form on We derive generalized Hamilton–Jacobi and Hamilton equations on and show that the Hamilton–Jacobi and canonical equations of Carathéodory–Rund and de Donder–Weyl are obtained as special cases.

Symmetries in covariant classical mechanics
View Description Hide DescriptionIn the framework of covariant classical mechanics (i.e., general relativistic classical mechanics on a space–time with absolute time), developed by Jadczyk and Modugno, we analyze systematically the relationship between symmetries of geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplectic structure on space–time and of its horizontal potentials are also symmetries of spacelike metric, gravitational and electromagnetic fields, Euler–Lagrange morphism and Lagrangians. Then, we provide a definition for a covariant momentum map associated with a group of cosymplectic symmetries by means of a covariant lift of functions of phase space. In the case of holonomic symmetries, we see that any covariant momentum map takes values in the quantizable functions in the sense of Jadczyk and Modugno, i.e., functions quadratic in velocities with leading coefficient proportional to the spacelike metric. Finally, we illustrate the results by some examples.

 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Ornstein–Uhlenbeck–Cauchy process
View Description Hide DescriptionWe combine earlier investigations of linear systems subject to Lévy fluctuations with recent attempts to give meaning to socalled Lévy flights in external force fields. We give a complete construction of the Ornstein–Uhlenbeck–Cauchy process as a fully computable paradigm example of Doob’s stable noisesupported Ornstein–Uhlenbeck process. Despite the nonexistence of all moments, we determine local characteristics (forward drift) of the process, generators of forward and backward dynamics, and relevant (pseudodifferential) evolution equations. The induced nonstationary spatial process is proved to be Markovian and quite apart from its inherent discontinuity defines an associated velocity process in a probabilistic sense.

Generalized lowenergy expansion formula for Green’s function of the Fokker–Planck equation
View Description Hide DescriptionThe frequency expansion formula for the onedimensional Fokker–Planck equation derived in a previous paper is generalized to an expansion formula for arbitrary powers of the Green’s function. Based on this expansion, the smallfrequency behavior of the Green’s function is studied for the cases where the potential tends to infinity at both and

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


A class of Liouvilleintegrable Hamiltonian systems with two degrees of freedom
View Description Hide DescriptionA class of twodimensional Liouvilleintegrable Hamiltonian systems is studied. Separability of the corresponding Hamilton–Jacobi equation for these systems is shown to be equivalent to other fundamental properties of Hamiltonian systems, such as the existence of the Lax and biHamiltonian representations of certain fixed types. Applications to physical models, including the Calogero–Moser model, an integrable case of the HénonHeiles potential and the nonperiodic Toda lattice are presented.

 RELATIVITY AND GRAVITATION


Equivalence classes of perturbations in cosmologies of Bianchi types I and V: Propagation and constraint equations
View Description Hide DescriptionThis is the third in a series of papers [J. Math. Phys. 40, 3978 (1999); 40, 3995 (1999)], the overall objective of which is the demonstration that a set of 26 gaugeinvariant variables, denoted collectively by D and referred to as the complete set of basic variables, can be used to describe the equivalence classes of perturbations in a Bianchi type I or type V universe filled with a nonbarotropic perfect fluid. The object here is the derivation of a full system of propagation and constraint equations for these basic variables. We show that the constraint equations, which involve only the spatial derivatives of D, are preserved in time along the unperturbed fluid flow lines, i.e., that the time derivative of each constraint equation is identically satisfied as a consequence of the other equations that hold. Let us put things another way. What we prove is the statement that if the constraints in our system are satisfied at one time and the evolution equations are satisfied at all times, then the constraints are satisfied at all times. A further important point is simply this. When the linearized field equations of Einstein’s gravity theory are reexpressed in a manifestly gaugeinvariant form, an open set of equations is obtained for D since there are too many unknowns. Thus this set must be suitably closed by means of accurate “closure” relations. In order to find them, we observe that the definition of basic gaugeinvariant variables gives rise to additional geometrical identities from which an exact method of closure can be determined. Our formalism turns out to be especially appropriate for handling the linearized perturbations in a Bianchi type V universe model where the standard approaches conceptually break down.

Isotropization of twocomponent fluids
View Description Hide DescriptionWe consider the problem of latetime isotropization in spatially homogeneous but anisotropic cosmological models when the source of the gravitational field consists of two noninteracting perfect fluids—one tilted and one nontilted. In particular, we study irrotational Bianchi type V models. By introducing appropriate dimensionless variables, a full global understanding of the state space of the gravitational field equations becomes possible. The issue of isotropization can then be addressed in a simple fashion. We also discuss implications for the cosmic “nohair” theorem for Bianchi models when part of the source is a tilted fluid.

Applications of harmonic morphisms to gravity
View Description Hide DescriptionWe introduce the notion of gravity coupled to a horizontally conformal submersion as a modification of the wellknown concept of gravity coupled to a harmonic map, thus obtaining a coupled gravity system with a more geometric flavor. By using integral techniques we determine the necessary conditions for coupling and cosmological constants. Finally, in the context of higher dimensional gravitation theory, we show that harmonic morphisms provide a natural ansatz to trigger spontaneous splitting and reduction of the gravity system coupled to a harmonic map on dimensional space–times.

Solutions of the spherically symmetric SU(2) Einstein–Yang–Mills equations defined in the far field
View Description Hide DescriptionIt is shown analytically that every static, spherically symmetric solution to the Einstein–Yang–Mills equations with SU(2) gauge group that is defined in the far field has finite ADM mass. Moreover, there can be at most two horizons for such solutions. The three types of solutions possible, Bartnik–McKinnon particlelike solutions, Reissner–Nordströmtype solutions, and black hole solutions having only one horizon are distinguished by the behavior of the metric coefficients at the origin.

 MISCELLANEOUS TOPICS IN MATHEMATICAL METHODS


Spectral decomposition and resolvent kernel for a magnetic Laplacian in
View Description Hide DescriptionSpectral decomposition and resolvent kernel for a magnetic Laplacian in are given. As an application we obtain the corresponding Schrödinger propagator and wave kernel.

Statistical Lorentzian geometry and the closeness of Lorentzian manifolds
View Description Hide DescriptionI introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume element, some information on the topology and metric is encoded in the partial order that the causal structure induces among those points; one can then define closeness between Lorentzian geometries by comparing the sets of probabilities they give for obtaining the same posets. If the density of points is finite, one gets a pseudodistance, which only compares the manifolds down to a finite volume scale, as illustrated here by a fully worked out example of two twodimensional manifolds of different topology; if the density is allowed to become infinite, a true distance can be defined on the space of all Lorentzian geometries. The introductory and concluding sections include some remarks on the motivation for this definition and its applications to quantum gravity.

On crossed product of algebras
View Description Hide DescriptionThe concept of a crossed tensor product of algebras is studied from a few points of view. Some related constructions are considered. Crossed enveloping algebras and their representations are discussed. Applications to the noncommutative geometry and particle systems with generalized statistics are indicated.

Differential geometry of
View Description Hide DescriptionWe introduce a construction of the differential calculus on the quantum supergroup We obtain two differential calculi, respectively, associated with the left and the right Cartan–Maurer oneforms. We also obtain the quantum superalgebra of Although all of the structures we obtain are derived without an R matrix, they nevertheless can be expressed using an R matrix.
