Poisson–Lie Diffeomorphism Groups

We explicitly construct a class of coboundary Poisson–Lie structures on the group of formal diffeomorphisms of ⁿ . Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra W _n of formal vector fields on ⁿ . We conjecture that this class accounts for all such coboundary structures. The natural action of the constructed Poisson–Lie diffeomorphism groups gives rise to large classes of compatible Poisson structures on ⁿ , thus making it a Poisson space. Moreover, the canonical action of the Poisson–Lie groups FDiff( ^m ) × FDiff ⁿ ) gives rise to classes of compatible Poisson structures on the space J ( ^m , ⁿ ) of infinite jets of smooth maps ^{m n} , which makes it also a Poisson space for this action. Poisson modules of generalized densities are also constructed. Initial steps towards a classification of these structures are taken.     

Min-max theory for the Yang-Mills-Higgs equations

In each monopole sector there exist an infinite number of finite energy solutions to the Prasad-Sommerfield limit of the SU(2) Yang-Mills-Higgs equations on ³ whose energy is greater than any finite number.National Science Foundation Postdoctoral Fellow in Mathematics     

Single crystal diffraction patterns of germanium. Part II

The rocking curves of Ge (111), (220), (333) for CuK ₁ radiation were measured by means of the triple-crystal diffractometer. Perfect silicon single crystals, cut parallel to the (111) plane were used in the monochromator part of the triple-crystal diffractometer. The results prove the suitability of such a monochromator for studying diffraction patterns.

---

. II

(rocking curves) (111), (220), (333) CuK ₁ . , (111). .

---

---

In conclusion the authors thank A. Haruý for preparing the germanium single crystals and they are indebted to V. Smutná and A. Irra for the care with which they carried out various tasks.     

The existence of a non-minimal solution to the SU (2) Yang-Mills-Higgs equations on ℝ3. Part I

This paper (Part I) and the sequel (Part II) prove the existence of a smooth, non-trivial, finite action solution to the SU (2) Yang-Mills-Higgs equations on ³ in the Bogomol'nyi-Prasad-Sommerfield limit. The proof uses a simple form of Morse theory known as Ljusternik-nirelman theory. Part I establishes that a form of Lusternik-nirelman theory is applicable to the SU (2) Yang-Mills-Higgs equations. Here, a sufficient condition for the existence of the aforementioned solution is derived. Part II contains the completed existence proof. There it is demonstrated that the sufficient condition of Part I is satisfied by the SU (2) Yang-Mills-Higgs equations.Research is supported in part by the Harvard Society of Fellows and the National Science Foundation under Grant PHY 79-16812     

Dispersive Estimates for Schrödinger Operators in Dimension Two

We prove L¹(²)L(²) for the two-dimensional Schrödinger operator –+V with the decay rate t^–1. We assume that zero energy is neither an eigenvalue nor a resonance. This condition is formulated as in the recent paper by Jensen and Nenciu on threshold expansions for the two-dimensional resolvent.The author was partially supported by the NSF grant DMS-0300081 and a Sloan Fellowship     

The vacuum and lightcone quantization of interaction Hamiltonians

Let denote the conformally invariant neutral free scalar field on ×S ⁿ. The naive lightcone Hamiltonian for a ^p interaction is given by c^p, where C denotes a lightcone in ×S ⁿ, and the Wick power is relative to the free vacuum. We show that this sesquilinear form annihilates the free vacuum if n3 is odd, p>2, and p(n–1)0 mod 4.     

The group of automorphisms of the galilei group

The group of automorphisms of the Galilei groupG: Aut(G) is calculated. It is shown that Aut(G) has the structure of a semi-direct product byG of the group _m ^* ×_m where _m is the group of reals noted multiplicatively and _m ^* <_m is the subgroup of positive reals.     

Common tangent space ℝ 1 4 fromU(2) charges

It is discussed how a common space-time can be constructed from a proposed hiddenU(2) world. Schrödinger's idea to obtain discrete eigenvalues by solving the Maxwell equations for the fieldF on compact spaces without boundaries is modified by orthogonality and identification concepts for the potentialsA. Using residue classes with respect to the metric (Clifford algebra), a common spinor space ⁴=RL and a common Minkowski tangent space ₁ ⁴ are bilinearly constructed from tangent spaces ofU(2) individuals [U(2) manifolds with orthogonal potentials]. The space constructed has the following properties. (1) There are algebraic elements for the identification ofU(2) individuals from ₁ ⁴ as spinors and vectorsA. (2) The transfer of the potentials fromU(2) via ⁴ to ₁ ⁴ is linear. (3) The hiddenU(2) content of the left- and right-handed spaces (L, R) is quite different. The potentials on U(2) individuals are transformed into complex wave functions on the spinor space and into 1-formsA on ₁ ⁴ that can be enlarged to gauge potentials. The construction is discussed from an old point of view of Einstein's, starting with the electric charge as the primary concept for quantum theory. The construction of the tangent space ₁ ⁴ does not depend on a preceding introduction of any points (uncertainty). The identity problem of the interpretation of the quantum theory is discussed in some detail. It is indicated how the algebraic, partiallyad hoc constructions can give a rigid frame for further analytical work.     

The general four-dimensional spherically symmetric Finsler space

Finsler geometries give natural generalisations of Riemannian geometries, and hence possible natural extensions of general relativity. In this latter (gravitational) context, it is of particular interest to find the general spherically symmetric Finsler metric on ⁴. In this paper, we derive this metric. The general solution is given in two alternative forms, the second of which allows easy comparison with Riemannian-like metrics. We also derive the general axially symmetric Finsler metric on ⁴.     

Hyper-Kähler metrics and a generalization of the Bogomolny equations

We generalize the Bogomolny equations to field equations on ^{3 n} and describe a twistor correspondence. We consider a general hyper-Kähler metric in dimension 4n with an action of the torusT ⁿ compatible with the hyper-Kähler structure. We prove that such a metric can be described in terms of theT ⁿ -solution of the field equations coming from the twistor space of the metric.     

The existence of a non-minimal solution to the SU(2) Yang-Mills-Higgs equations on ℝ3: Part II

This paper proves that there exists a finite action solution to the SU(2) Yang-Mills-Higgs equations on ³ in the Bogomol'nyi-Prasad-Sommerfield limit which is not a solution to the first order Bogonol'nyi equations. The existence is established using Ljusternik-nirelman theory on non-contractible loops in the configuration space.Research is supported in part by the Harvard Society of Fellows and the National Science Foundation under Grant PHY79-16812     

A direct variational approach to Skyrme's model for meson fields

The solutions of Skyrme's variational problem describe the structure of mesons in a field of weak energy. The problem consists in minimizing the corresponding energy among the functions from ³ toS ³ which have a fixed degree without making any symmetry assumptions. We prove the existence of minima and study their properties.     

On the isomorphism of a ‘quantum logic’ with the logic of the projections in a hilbert space

A topology is introduced in a logic using the set of pure states of . It is shown that , equipped with this topology, under suitable conditions, determines the division ring , or 2e. With the continuity of the antiautomorphism of the division ring added, it is shown that these conditions are necessary and sufficient for the projective logic to be isomorphic with the projective logic of the projections in a Hilbert space over , or 2e.P. Cotta-Ramusino gratefully acknowledges a fellowship of the Consiglio Nazionale delle Ricerche.     

Elliptic-type solutions to the scale invariant Yang-Mills and sigma-model hierarchies

We outline the construction of non-self-dual elliptic solutions by relating the spherically symmetric subsystem of the (scale invariant) Yang-Mills and sigmamodel hierarchies to the hierarchies of ⁴ and Sine-Gordon models in one dimension respectively. The construction is carried out explicitly for the usual Yang-Mills model on ₄, and the first two sigma-models on ₂ and ₄. The solution to the first member of the Yang-Mills hierarchy is related to elliptic solutions found previously.     

Geometry ofSU(2) gauge fields

We studySU(2) Yang-Mills theory onS ³× from the canonical view-point. We use topological and differential geometric techniques, identifying the true configuration space as the base-space of a principal bundle with the gauge-group as structure group.     

Hamiltonian collapsing of irrational Lagrangian submanifolds with small first Betti number

Among the main symplectic invariants of a closed Lagrange submanifoldL of the cotangent of ⁿ is the tubular radiusR(L) defined as the smallest tube D(r) × ^n–1 of ⁿ T^{* n} in whichL can be pushed by an Hamiltonian diffeotopy of ⁿ. We show here, using pseudo-holomorphic techniques, that such a submanifold cannot collapse if the first Betti number ofL is smaller than 3 and if the Maslov class ofL does not vanish; in other words,R(L) is then strictly positive and one can actually give an explicit lower bound in terms of the Liouville and Maslov classes ofL.Partially supported by Research Grants NSERC OGP0092913 and FCAR EQ3518     

A note on the vacuum structure of an SU(2) Yang-Mills theory

We discuss different compactifications of the spacial part ³ of Minkowski space and give classifications of the vacuum structure for a Yang-Mills theory.Partially supported by DFG and by Simon Fraser UniversityPartially supported by National Research Council of Canada grant 751-010     

On the correlation for Kac-like models in the convex case

The aim of this paper is to stu the behavior asm tends to of a family of measures exp[- ^(m)(x)]dx ^(m) on ^m , where ^(m) is a potential on ^m which is a perturbation in a suitable sense of the harmonic potential _j x _j ² .     

Bianchi identities,electromagnetic waves,and charge conservation in theP(4) theory of gravitation and electromagnetism

The Bianchi identities for theP(4)=O(1, 3) ^4* theory of gravitation and electromagnetism are decomposed into the standardO(1, 3) Riemannian Bianchi identity plus an additional ^4* component. When combined with the Einstein-Maxwell affine field equations the ^4* components of theP(4) Bianchi identities imply conservation of magnetic charge and the wave equation for the Maxwell field strength tensor. These results are analyzed in light of the special geometrical postulates of theP(4) theory. We show that our development is the analog of the manner in which the Riemannian Bianchi identities, when combined with Einstein's field equations, imply conservation of stress-energy-momentum and the wave equation for the LanczosH-tensor.     

Semi-global solutions of Einstein equations,minkowskian near past infinity

On a universe homeomorphic toV ^T =]– ,T[x³, we prove the existence of solutions of Einstein equations, minkowskian near past infinity, if the sources are small enough for some norms. We prove that some of these solutions verify at least the positivity condition (Weak energy condition) on some domains homeomorphic toV ^T .