Volume 44, Issue 7, July 2003
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Bell inequalities in four dimensional phase space and the three marginal theorem
View Description Hide DescriptionWe address the classical and quantum marginal problems, namely the question of simultaneous realizability through a common probability density in phase space of a given set of compatible probability distributions. We consider only distributions authorized by quantum mechanics, i.e., those corresponding to complete commuting sets of observables. For fourdimensional phase space with position variables and momentum variables we establish the two following points: (i) given four compatible probabilities for and there does not always exist a positive phase space density reproducing them as marginals; this settles a long standing conjecture; it is achieved by first deriving Belltype inequalities in phase space which have their own theoretical and experimental interest. (ii) Given instead at most three compatible probabilities, there always exist an associated phase space density the solution is not unique and its general form is worked out. These two points constitute our “three marginal theorem.”

Higher order trace formulas of the Buslaev–Faddeevtype for the halfline Schrödinger operator with longrange potentials
View Description Hide DescriptionWe deal with trace formulas for halfline Schrödinger operators with longrange potentials. We generalize the Buslaev–Faddeev trace formulas to the case of square integrable potentials. The exact relation between the number of the trace formulas and the number of integrable derivatives of the potential is also given. The main results are optimal.

The scattering matrix for the Schrödinger operator with a longrange electromagnetic potential
View Description Hide DescriptionWe consider the Schrödinger operator in the space with longrange electrostatic and magnetic potentials. Using the scheme of smooth perturbations, we give an elementary proof of the existence and completeness of modified wave operators for the pair Our main goal is to study spectral properties of the corresponding scattering matrix We obtain its stationary representation and show that its singular part (up to compact terms) is a pseudodifferential operator with an oscillating amplitude which is an explicit function of and Finally, we deduce from this result that, in general, for each the spectrum of covers the whole unit circle.

General realization of supersymmetric quantum mechanics and its applications
View Description Hide DescriptionBased upon the general supercharges which involve not only generators of the Clifford algebra C(4,0) with positive metric, but also operators of third order, the general form of supersymmetric quantum mechanics (SSQM), which brings out the richer structures, is realized. Then from them, a onedimensional physical realization and a new multidimensional physical realization of SSQM are respectively obtained by solving the constraint conditions. As applications, dynamical superconformal symmetries, which possess both the supersymmetries and the usual dynamical conformal symmetries, are studied in detail by considering two simple superpotentials and and their corresponding superalgebraic structures, which are spanned by eight fermionic generators and six bosonic generators, are established as well.

Exponentially accurate error estimates of quasiclassical eigenvalues. II. Several dimensions
View Description Hide DescriptionWe study the behavior of truncated Rayleigh–Schrödinger series for lowlying eigenvalues of the timeindependent Schrödinger equation, in the semiclassical limit In particular we prove that if the potential energy satisfies certain conditions, there is an optimal truncation of the series for the eigenvalues, in the sense that this truncation is exponentially close to the exact eigenvalue. These results were already discussed for the onedimensional case in a previous article. This time we consider the multidimensional problem, where degeneracy plays a central role.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Dynamical symmetry breaking and the Nambu–Goldstone theorem in the Gaussian wave functional approximation
View Description Hide DescriptionWe analyze the grouptheoretical ramifications of the Nambu–Goldstone (NG) theorem in the selfconsistent relativistic variational Gaussian wave functional approximation to spinless field theories. In an illustrative example we show how the Nambu–Goldstone theorem would work in the symmetric scalar field theory, if the residual symmetry of the vacuum were lesser than e.g., if the vacuum were or symmetric. (This does not imply that any of the “lesser” vacua is actually the absolute energy minimum: stability analysis has not been done.) The requisite number of NG bosons would be or respectively, which may exceed the number of elementary fields in the Lagrangian. We show how the requisite new NG bosons would appear even in channels that do not carry the same quantum numbers as one of “elementary particles” [scalar field quanta, or Castillejo–Dalitz–Dyson (CDD) poles] in the Lagrangian, i.e., in those “flavor” channels that have no CDD poles. The corresponding Nambu–Goldstone bosons are composites (bound states) of pairs of massive elementary (CDD) scalar fieldsexcitations. As a nontrivial example of this method we apply it to the physically more interesting ’t Hooft σ model (an extended bosonic linear σ model with four scalar and four pseudoscalar fields), with spontaneously and explicitly broken chiral symmetry.

A “periodic table” for supersymmetric Mtheory compactifications
View Description Hide DescriptionWe develop a systematic method for classifying supersymmetric orbifold compactifications of theory. By restricting our attention to Abelian orbifolds with low order, in the special cases where elements do not include coordinate shifts, we construct a “periodic table” of such compactifications, organized according to the orbifolding group (order and dimension (up to 7). An intriguing connection between supersymmetric orbifolds and structures is explored.

Groupinvariant solutions of relativistic and nonrelativistic models in field theory
View Description Hide DescriptionIn this paper we present a method of constructing explicit classes of solutions of the Chaplygin and Born–Infeld systems of equations in dimensions. The approach adopted here is based on the symmetry reduction method for PDE’s. A systematic use of the subgroup structure of the invariance group is made to generate symmetry variables. We concentrate only on the case when the symmetry variables are invariants of the subgroups having dimension one. The set of symmetry variables enables us to reduce, after some transformation, the original equations to many possible ODE’s. We describe in detail how a composition of Lie subalgebras and singularity analysis applied to these systems provides us with classes of analytic solutions. Several new types of algebraic, rational and solitonlike solutions (among them kinks, bumps and multiple wave solutions) have been obtained in explicit form. We discuss also the existence of several classes of separable solutions admitted by the Chaplygin and Born–Infeld equations. Some physical interpretation of these results in the areas of fluid dynamics and field theory are given.

Exact duality transformations for sigma models and gauge theories
View Description Hide DescriptionWe present an exact duality transformation in the framework of statistical mechanics for various lattice models with nonAbelian global or local symmetries. The transformation applies to sigma models with variables in a compact Lie groupG with global symmetry (the chiralmodel) and with variables in coset spaces and a global Gsymmetry [for example, the nonlinear or models] in any dimension It is also available for lattice gauge theories with local gauge symmetry in dimensions and for the models obtained from minimally coupling a sigma model of the type mentioned above to a gauge theory. The duality transformation maps the strong coupling regime of the original model to the weak coupling regime of the dual model. Transformations are available for the partition function, for expectation values of fundamental variables (correlators and generalized Wilson loops) and for expectation values in the dual model which correspond in the original formulation to certain ratios of partition functions (free energies of dislocations, vortices or monopoles). Whereas the original models are formulated in terms of compact Lie groupsG and H, coset spaces and integrals over them, the configurations of the dual model are given in terms of representations and intertwiners of G and H. They are spin networks and spin foams. The partition function of the dual model describes the group theoretic aspects of the strong coupling expansion in a closed form.

Entropic repulsion for a Gaussian lattice field with certain finite range interaction
View Description Hide DescriptionConsider the centered Gaussian field on with covariance matrix given by where Δ is the discrete Laplacian on and are constants satisfying for and a certain additional condition. We show the probability that all spins are positive in a box of volume decays exponentially at a rate of order and under this hardwall condition, the local sample mean of the field is repelled to a height of order This extends the previously known result for the case that the covariance is given by the Green function of simple random walk on (i.e.,

Foldy–Wouthuysen transformation for relativistic particles in external fields
View Description Hide DescriptionA method of Foldy–Wouthuysen transformation for relativistic spin1/2 particles in external fields is proposed. It permits the determination of the Hamilton operator in the Foldy–Wouthuysen representation with any accuracy. Interactions between a particle having an anomalous magnetic moment and nonstationary electromagnetic and electroweak fields are investigated.

Resolvent convergence in norm for Dirac operator with Aharonov–Bohm field
View Description Hide DescriptionWe consider the Hamiltonian for relativistic particles moving in the Aharonov–Bohm magnetic field in two dimensions. The field has δlike singularity at the origin, and the Hamiltonian is not necessarily essentially selfadjoint. The selfadjoint realization requires one parameter family of boundary conditions at the origin. We approximate the pointlike field by smooth ones and study the problem of norm resolvent convergence to see which boundary condition is physically reasonable among admissible boundary conditions. We also study the effect of perturbations by scalar potentials. Roughly speaking, the obtained result is that the limit selfadjoint realization is different even for small perturbation of scalar potentials according to the values of magnetic fluxes. It changes at halfinteger fluxes. The method is based on the resolvent analysis at low energy on magnetic Schrödinger operators with resonance at zero energy and the resonance plays an important role from a mathematical point of view. However it has been neglected in earlier physical works. The emphasis here is placed on this natural aspect.
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 GENERAL RELATIVITY AND GRAVITATION


Regularity for Lorentz metrics under curvature bounds
View Description Hide DescriptionLet (M, g) be an dimensional space–time, with bounded curvature, with respect to a bounded framing. If (M, g) is vacuum, or satisfies a weak condition on the stressenergy tensor, then it is shown that (M, g) locally admits coordinate systems in which the Lorentz metric g is wellcontrolled in the (space–time) Sobolev space for any This result is essentially optimal. The result allows one to control the regularity of limits of sequences of space–times, with uniformly bounded curvature, and has applications to the structure of boundaries and extensions of space–times.

Exterior differential systems, Janet–Riquier theory and the Riemann–Lanczos problems in two, three, and four dimensions
View Description Hide DescriptionWe discuss the Riemann–Lanczos problems in two, three, and four dimensions using the theory of exterior differential systems and Janet–Riquier theory. We show that the Riemann–Lanczos problem in two dimensions is always a system in involution. For each of the two possible signatures we give the general solution in both instances and show that the occurrence of characteristic coordinates need not affect the result. In three dimensions, the Riemann–Lanczos problem is not in involution as an identity occurs. This does not prevent the existence of singular solutions and we give an example for the reduced Gödel space–time. A prolongation of this problem, whereby an integrability condition is added, leads to a prolonged system in involution. The Riemann–Lanczos problem in four dimensions is not in involution and needs to be prolongated as Bampi and Caviglia suggested. But singular solutions of it can be found and we give examples for the Gödel, Kasner, and Debever–Hubaut space–times.

The Weyl–Lanczos equations and the Lanczos wave equation in four dimensions as systems in involution
View Description Hide DescriptionThe Weyl–Lanczos equations in four dimensions form a system in involution. We compute its Cartan characters explicitly and use Janet–Riquier theory to confirm the results in the case of all space–times with a diagonal metric tensor and for the plane wave limit of space–times. We write the Lanczos wave equation as an exterior differential system and, with assistance from Janet–Riquier theory, we compute its Cartan characters and find that it forms a system in involution. We compare these Cartan characters with those of the WeylLanczos equations. All results hold for the real analytic case.

A space–time in toroidal coordinates
View Description Hide DescriptionWe present an exact solution of Einstein’s field equations in toroidal coordinates. The solution has three regions: an interior with a string equation of state; an Israel boundary layer; and an exterior with constant isotropic pressure and constant density, locally isometric to anti–de Sitter space–time. The exterior can be a cosmological vacuum with negative cosmological constant. The size and mass of the toroidal loop depend on the size of Λ.
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 DYNAMICAL SYSTEMS


Prolongation methods and Cartan characters for the threedimensional Riemann–Lanczos problem
View Description Hide DescriptionThe Riemann–Lanczos problem in two dimensions is in involution and its general solution is known. The threedimensional Riemann–Lanczos problem is not in involution and needs to be prolongated. We write the threedimensional Riemann–Lanczos problem as an exterior differential system and prolong it to second order. Using algebraic computing we find that we have to add an integrability condition to make it a system in involution. We also suggest a prolongation of the threedimensional Riemann–Lanczos problem in the same way as Bampi and Caviglia did for four dimensions. After supplementing this threedimensional system with an integrability condition, it becomes involutive with Cartan characters (17,8,2) or (20,10,3) if no cyclic conditions are imposed. We present the relevant sections of REDUCE computer codes by means of which these results were obtained.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


The algebraic entropy of classical mechanics
View Description Hide DescriptionWe describe the “Lie algebra of classical mechanics,” modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. It is a polynomially graded Lie algebra, a class we introduce. We describe these Lie algebras, give an algorithm to calculate the dimensions of the homogeneous subspaces of the Lie algebra of classical mechanics, and determine the value of its entropy It is 1.825 423 774 201 08… , a fundamental constant associated with classical mechanics.
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 METHODS OF MATHEMATICAL PHYSICS


Towards the complete classification of homogeneous twocomponent integrable equations
View Description Hide DescriptionIn this article we suggest an improved method for classifying general twocomponent integrable evolution equations, homogeneous in a given weighting scheme. This method relies on linear changes of variables and on an appropriate splitting of the solution space. To illustrate the method, we implement the classification of coupled KdVtype equations. We show that there are five nontriangular systems possessing higher order generalized symmetries. One of these systems is previously unknown. This seems to be the first classification of coupled integrable equations homogeneous in a given weighting scheme, without any restrictions on the form of the main matrix.

Isospectrality of spherical MHD dynamo operators: Pseudohermiticity and a nogo theorem
View Description Hide DescriptionThe isospectrality problem is studied for the operator of the spherical hydromagnetic dynamo. It is shown that this operator is formally pseudoHermitian symmetric) and lives in a Krein space. Based on the symmetry, an operator intertwining Ansatz with firstorder differential intertwining operators is tested for its compatibility with the structure of the dynamo operator matrix. An intrinsic structural inconsistency is obtained in the set of associated matrix Riccati equations. This inconsistency is interpreted as a nogo theorem which forbids the construction of isospectral dynamo operator classes with the help of firstorder differential intertwining operators.
