The structure of naked singularity in self-similar gravitational collapse: II

Generalizing the results of Joshi and Dwivedi inComm. Math. Phys. 146, 333 (1992), it is pointed out that strong curvature naked singularities could occur in the self-similar gravitational collapse of any form of matter satisfying the weak energy condition for the positivity of mass-energy density.     

Z 3-Graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d³=0, but d²0. The entities dx ⁱ and d² x ^k generate an associative algebra; we shall suppose that the products dx ⁱ dx ^k are independent of dx ^k dx ⁱ , while theternary products will satisfy the relation: dx ⁱ dx ^k dx ^m =jdx ^k dx ^m dx ⁱ =j ²dx ^m dx ^m dx ⁱ dx ^k , complemented by the relation dx ⁱ d² x ^k =jd² x ^k dx ⁱ , withj:=e^2i/3.We shall attribute grade 1 to the differentials dx ⁱ and grade 2 to the second differentials d² x ^k ; under the associative multiplication law the grades add up modulo 3.We show how the notion ofcovariant derivation can be generalized with a 1-formA so thatD:=d+A, and we give the expression in local coordinates of thecurvature 3-form defined as :=d² A+d(A ²)+AdA+A ³.Finally, the introduction of notions of a scalar product and integration of theZ ₃-graded exterior forms enables us to define the variational principle and to derive the differential equations satisfied by the 3-form . The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensorF _ik and its covariant derivativesD _i F _km .     

Symmetries of theq-difference heat equation

The symmetry operators of aq-difference analog of the heat equation in one space dimension are determined. They are seen to generate aq-deformation of the semidirect product of sl(2, ) with the three-dimensional Weyl algebra. It is shown that this algebraic structure is preserved if differentq-analogs of the heat equation are considered. The separation of variables associated to the dilatation symmetry is performed and solutions involving discreteq-Hermite polynomials are obtained.     

Action variables for Toda Lattices

This Letter contains constructions of complex action variables for both the full Kostant-Toda Lattice in sl(n, ) and the generalized nonperiodic tridiagonal Toda lattice associated to an arbitrary complex semisimple Lie algebra g. The main tool is the explicit factorization solution for certain Hamiltonian flows. The Letter also contains a generalization of the standard factorization solution theorem necessary for the analysis of the full Kostant-Toda lattice.     

gl q (n) and quantum monodromy

A generalized Toda lattice based on gl(n) is considered. The Poisson brackets are expressed in terms of a Lax connection, L=–() and a classical r-matrix, {₁,₂}=[r,₁+₂}. The essential point is that the local lattice transfer matrix is taken to be the ordinary exponential, T=e; this assures the intepretation of the local and the global transfer matrices in terms of monodromy, which is not true of the T-matrix used for the sl(n) Toda lattice. To relate this exponential transfer matrix to the more manageable and traditional factorized form, it is necessary to make specific assumptions about the equal time operator product expansions. The simplest possible assumptions lead to an equivalent, factorized expression for T, in terms of operators in (an extension of) the enveloping algebra of gl(n). Restricted to sl(n), and to multiplicity-free representations, these operators satisfy the commutation relations of sl _q (n), which provides a very simple injection of sl _q (n) into the enveloping algebra of sl(n). A deformed coproduct, similar in form to the familiar coproduct on sl _q (n), turns gl(n) into a deformed Hopf algebra gl _q (n). It contains sl _q (n) as a subalgebra, but not as a sub-Hopf algebra.     

Berenstein-Zelevinsky triangles,elementary couplings,and fusion rules

We present a general scheme for describing (N) _k fusion rules in terms of elementary couplings, using Berenstein-Zelevinsky triangles. A fusion coupling is characterized by its corresponding tensor product coupling (i.e. its Berenstein-Zelevinsky triangle) and the threshold level at which it first appears. We show that a closed expression for this threshold level is encoded in the Berenstein-Zelevinsky triangle and an explicit method to calculate it is presented. In this way, a complete solution of (4) _k fusion rules is obtained.Work supported by NSERC (Canada).Work supported by NSERC (Canada) and FCAR (Québec).     

On the linearization of second-order ordinary differential equations

In this Letter, the problem of characterizing all second-order ordinary differential equations y=f(x, y, y) which are locally linearizable by a change of dependent and independent variables (x, y)(X, Y) is considered. Since all second-order linear equations are locally equivalent to y=0, the problem amounts to finding necessary and sufficient conditions for y=f(x, y, y) to be locally equivalent to y=0. It turns out that two apparently different criteria for linearizability have been formulated in the literature: the one found by M. Tresse and later rederived by É. Cartan, and the criterion recently given by Arnol'd [Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983]. It is shown here that these two sets of linearizability conditions are actually equivalent. As a matter of fact, since Arnol'd's criterion is stated without proof in the latter reference, this work can be alternatively considered as a proof of Arnol'd's linearizability conditions based on Cartan-Tresse's. Some further points in connection with the relationship between Arnol'd's and Cartan-Tresse's treatment of the linearization problem are also discussed and illustrated with several examples.     

Asymptotics for Bosonic atoms

It was proved by Benguria and Lieb that for an atom where the electrons do not satisfy the exclusion principle, the critical electron number N _c, i.e., the maximal number of electrons the atom can bind, satisfies lim inf_zN_c/Z 1 + , where Z is the nuclear charge. Here is a positive constant derived from the Hartree model. We complete this result by proving that the correct asymptotics for N _c(Z) is indeed _zN_c/Z = 1 + .This work was done while the author was a graduate student at Princeton University supported by a Danish Research Academy fellowship and U.S. National Science Foundation grant PHY-85-15288-A03.     

Gelfand-Zetlin basis for Uq(gl(N+1)) modules

The Gelfand-Zetlin basis of U_q(gl(N+1)) modules is constructed via the lowering operator method.     

The two-parameter deformation of the supergroup GL(1|1), its differential calculus and its Lie algebra

We construct a right-invariant differential calculus on the quantum supergroup GL _q,s (1|1) and we show that its quantum Lie algebra with comultiplication is isomorphic to that which we obtain using the Reshetikhin-Takhtajan-Faddeev approach.     

A generalized consistency condition for massless fields

Using a generalized consistency condition (gcc), we construct couplings between massless scalar fields and the spin 2 gravitational field. Specifically, we consider all possible third-order interaction terms for scalar fields and a and use the gcc to single out one. We also find three generalized current identities associated with the massless gauge fields.     

On Dijkgraaf-Witten-type invariants

We explicitly construct a series of lattice models based upon the gauge group Z _p which have the property of subdivision invariance, when the coupling parameter is quantized and the field configurations are restricted to satisfy a type of mod-p flatness condition. The simplest model of this type yields the Dijkgraaf-Witten invariant of a 3-manifold and is based upon a single link, or 1-simplex, field. Depending upon the manifold's dimension, other models may have more than one species of field variable, and these may be based on higher-dimensional simplices.Supported by Stichting voor Fundamenteel Onderzoek der Materie (FOM).     

Darboux coordinates and isospectral Hamiltonian flows for the massive thirring model

The two-dimensional massive Thirring model is described as the integrability condition of a pair of commuting completely integrable isospectral Hamiltonian flows in the dual (2)^+* of the positive part (2)⁺ of the twisted loop algebra (2). Action-angle coordinates corresponding to the spectral invariants are derived on rational coadjoint orbits and a linearization of the flows obtained in the Jacobi variety of the underlying invariant spectral curve through a Liouville generating function for canonical coordinates.Research supported in part by the Natural Sciences Engineering Research Council of Canada and the Fonds FCAR du Québec.     

The commutator method for form-relatively compact perturbations

We present a variant of the conjugate operator method which can be used when the group generated by the conjugate operator leaves invariant only the form domain of the Hamiltonian. As an example, we get detailed spectral properties and a large class of locally smooth operators for two-body Schrödinger Hamiltonians with form-relatively compact potentials.     

Algebraic quantization

The different correspondences (or orderings) used in quantum mechanics and the associated deformations, are both seen from an algebraic viewpoint. The deformations which are compatible with the diagonal map (the ₀-deformations) are introduced and connected to the formal groups. A very straighforward example of a ₀-deformation (the multiplicative deformation) appears in the normal quantization of the harmonic oscillator.     

Harness Processes and Non-Homogeneous Crystals

We consider the Harmonic crystal, a measure on with Hamiltonian H(x)=∑ _i,j J _i,j (x(i)−x(j))²+h∑ _i (x(i)−d(i))², where x, d are configurations, x(i), d(i)∈ℝ, i,j∈ℤ ^d . The configuration d is given and considered as observations. The ‘couplings’ J _i,j are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite temperature and to find the unique ground state configuration m corresponding to the Hamiltonian.     

Remarks on Quantization of Classical r-Matrices

Abstract

If a classical r-matrix r is skewsymmetric, its quantization R can lose the skewsymmetry property. Even when R is skewsymmetric, it may not be unique.     

Realization of a unique time evolution unitary operator in Klein Gordon theory

The scattering theory for the Klein Gordon equation, with time-dependent potential and in a non-static space-time, is considered. Using the Klein Gordon equation formulated in the Hubert spaceL ²(R ³) and the Einstein’s relativistic equation in the spaceL ²(R ³, dx) and establishing the equivalence of the vacuum states of their linearized forms in the Hubert spaceL ²(R ³) with the help of unique symmetric symplectic operator, the time evolution unitary operatorU(t) has been fixed for the Klein Gordon equation, incorporating either the positive or negative frequencies, in the infinite dimensional Hubert spaceL ²(R ³).     

Role of rational surfaces on fluctuations and transport in the plasma edge of the TJ-II stellarator

It has been shown that transport barriers in toroidal magnetically confined plasmas tend to be linked to regions of unique magnetic topology such as the location of a minimum in the safety factor, rational surfaces or the boundary between closed and open flux surfaces. In the absence ofE×B sheared flows, fluctuations are expected to show maximum amplitude near rational surfaces, and plasma confinement might tend to deteriorate. On the other hand, if the generation ofE×B sheared flows were linked to low order rational surfaces, these would be beneficial for confinement. Experimental evidence ofE×B sheared flows linked to rational surfaces has been obtained in the plasma edge region of the TJ-II stellarator. Presented at the Workshop on the Role of Electric Fields in Plasma Confinement and Exhaust, Budapest, 18–19 June, 2000.     

On invariant and covariant Schwartz distributions in the case of a compact linear group

A proof is given for the representations of invariant and covariant (Schwartz) distributions onR ⁿ , which are often used in theoretical physics. We express invariant distributions as distributions of standard polynomial invariants and decompose covariant distributions in standard polynomial covariants. Our consideration is restricted to compact groups acting linearly onR ⁿ . The representation for invariant distributions is obtained provided the standard invariants form an algebraically independent generating set in the ring of invariant polynomials. As for the standard covariants we assume that in the class of covariant polynomials they provide a unique decomposition into a sum of the standard covariants multiplied with invariant polynomials.