Natural origin of SO3-spinors in an eight-dimensional Kaluza theory exhibiting an additional U1×SU2 internal symmetry

The higher-dimensional spaceM ₄×S ₁×S ₃ (M ₄ = Minkowski space) is assumed. The symmetry group ofS ₃ isSO ₄ SU ₂×SU₂ Gravitational waves produce a symmetry breaking so thatSO ₃, rotations inM ₄ are linked with one of theSU ₂. Certain components of the metrical tensor ofM ₄×S ₁×S ₃, appear then as spinor fields.     

Conformal regularization of the Kepler Problem

The manifoldM of null rays through the origin of ^2,n+1 is diffeomorphic toS ¹×S ⁿ , and it is a homogeneous space of SO(2,n+1). This group therefore acts onT*M, which we show to be the generating manifold of the extended phase space of the regularized Kepler Problem. A local canonical chart inT*M is found such that the restriction to the subbundle of the null nonvanishing covectors is given byp ₀+H(q,p)=0, whereH(q,p) is the Hamiltonian of the Kepler Problem. By means of this construction, we get some results that clarify and complete the previous approaches to the problem.     

Exact results for the two-dimensional,two-component plasma atΓ=2 in doubly periodic boundary conditions

The two-dimensional, two-component plasma is considered in doubly periodic boundary conditions with the positive and negative charges confined to separate interlacing rectangular lattices. It is shown that at the special coupling=2, on a lattice of 2M ₁×2M ₂ sites, the grand partition function can be written as a double integral over a product of determinants of dimension 2M ₂×2M ₂. On the basis of a conjecture regarding the zero distribution of the grand partition function, the large-M ₂ behavior of the determinant is given and the pressure evaluated exactly.     

Construction of position and momentum operators and Hamiltonians by embedding and submersion

Extrinsic position and momentum operators are defined on a manifold M via an embedding i: M→(R _n , g _n ) and, via a submersion, π: R _n →M. A Hamiltonian , describing free dynamics on M, is constructed as an intrinsic property of M by the embedding i and an associated Riemannian geodesic submersion.     

Isospectral Hamiltonian flows in finite and infinite dimensions

The approach to isospectral Hamiltonian flow introduced in part I is further developed to include integration of flows with singular spectral curves. The flow on finite dimensional Ad^*-invariant Poisson submanifolds of the dual of the positive part of the loop algebra is obtained through a generalization of the standard method of linearization on the Jacobi variety of the invariant spectral curveS. These curves are embedded in the total space of a line bundleTP ₁(C), allowing an explicit analysis of singularities arising from the structure of the image of a moment map from the space of rank-r deformations of a fixedN×N matrixA. It is shown how the linear flow of line bundles over a suitably desingularized curve may be used to determine both the flow of matricial polynomialsL() and the Hamiltonian flow in the spaceM _N,r×M_N,r in terms of -functions. The resulting flows are proved to be completely integrable. The reductions to subalgebras developed in part I are shown to correspond to invariance of the spectral curves and line bundles under certain linear or anti-linear involutions. The integration of two examples from part I is given to illustrate the method: the Rosochatius system, and the CNLS (coupled non-linear Schrödinger) equation.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by U.S. Army grant DAA L03-87-K-0110     

Effect of heavy quark symmetry on the mass difference ofB-system in minimal left right symmetric model

An estimation of the mass difference of system with heavy quark symmetry formalism is presented. The effective Hamiltonian describing the transition (whereh=b forB _d ⁰ -system) is considered in a manifest left right symmetric (MLRS) model along with contribution from neutral Higgs boson. We use the spin and flavor symmetry for heavy quarks to obtain the transition matrix element 〈B _d ⁰ |ℋ_eff(x)| _d ⁰ 〉 in terms of Isgur-Wise function. Assuming thatB _d ⁰ and states are at rest, we find that Isgur-Wise function turns out to be unity. However using the experimental values of ΔM _K and as input, we find thatM _R=835 GeV andM _H⩾2·9 TeV.     

Calabi-Yau manifolds of some special forms

Let M=M ₁×...×M _m be a product of Kähler C-spaces with second Betti numbers b ₂(M _i )=1 (1im). The work establishes that the complete intersections X of M produce a finite number of N-dimensional Calabi-Yau manifolds. Moreover, if b ₄(M _i )=1, then the complete intersections with vanishing first Pontrjagin classes are finitely many, as well.On the other hand, we consider hypersurfaces of weighted projective spaces and give an explicit formula for their Euler characteristics. As in the previous case, it turns out that only a finite number of these are Calabi-Yau manifolds.     

Hamilton Spaces of Order k ≥ 1

A suitable dual for the k-acceleration bundle(T ^k M, ^k ,M) is the fiberedbundle (T ^k–1 M× _M T*M). The mentioned bundle carries a canonicalpresymplectic structure and k canonical Poisson structures. By means of thisdual we define the notion of Hamilton spaces of orderk, whose total spaceconsists of points x of the configuration spaceM, accelerations of order 1,...,k – 1, y ⁽¹⁾,...,y ^(k–1), and momenta p. Some remarkable Hamiltonian systemsare pointed out. There exists a Legendre mapping from the Lagrange spaces oforder k to the Hamilton space of order k.     

Nonextendible positive maps

Positive maps of ordered vector spaces into the algebra of all bounded operators acting on a Hilbert space are considered. A special class of so called nonextendible maps is introduced and investigated. This class is much smaller than the class of extreme maps.Any positive map can be obtained from a nonextendible one by restriction.In theC*-algebra case, the nonextendibility of a normalized positive map is related to the properties of the expression (a ²)–(a)². In particular Jordan representations are non-extendible.2-positive nonextendible maps are representations. Similar result holds for copositive maps. For abelianC*-algebras, notion of nonextendible map and that of representation coincide.The nonextendible positive maps of the Jordan algebraM _2s of all 2×2 symmetric matrices and of the full 2×2 matrix algebra are especially investigated. Any nonextendible normalized positive map ofM _2s is a Jordan representation.M ₂ admits nonextendible normalized positive maps not being Jordan representations. A large class of examples is given.On leave of absence from: Department of Mathematical Methods in Physics, University of Warsaw, Hoa 74, 00-682 Warsaw, Poland     

X-Band Pulsed ENDOR Study ofFe-Substituted Sodalite— The Effect of the Zero-Field Splitting

X-band (∼9.3 GHz) pulsed ENDOR measurements were carried out on⁵⁷Fe-substituted sodalite (FeSOD) which contains only one type of Fe(III) (S=) located at a framework site. The ENDOR spectrum recorded atg= 2 shows three doublets corresponding to the sixM_Smanifolds. The assignment of these signals was confirmed by hyperfine-selective and triple ENDOR experiments. The components of each of the doublets had different intensities, reflecting the different populations of the EPR energy levels at the measurement temperature, 1.8 K. ENDOR spectra were recorded at magnetic fields within the EPR powder pattern, and the field dependence observed showed an anisotropic behavior, unexpected from the isotropic character of the⁵⁷Fe(III) hyperfine coupling. This dependence was attributed to the high-order effects of the zero-field splitting (ZFS) interaction on the ENDOR frequencies. Three different theoretical approaches were used to account for the dependence of the ENDOR spectrum on the ZFS interaction. The first involves the exact diagonalization of the total spin Hamiltonian, the second uses third-order perturbation approximations, and the third employs an effective nuclear Hamiltonian for each of theM_Smanifolds. The simulations showed that the ENDOR signals of theM_S= ±5/2 (ν_±5/2) manifold are the least sensitive to the magnitude of the ZFS parameterDand are therefore the most appropriate for the determination ofa_iso. It is shown that at X band anda_isovalues of about 30 MHz, the perturbation approach is valid up toDvalues of 500 MHz if all three doublets are concerned. However, if only the ν_±5/2doublet is considered, then this approach is valid forD< 1000 MHz. The third approach was found inappropriate fora_isovalues of ∼30 MHz. Using the method of exact diagonalization together with orientation selectivity, the trends observed in the experimental spectra could be reproduced. The ENDOR spectra of the⁵⁷Fe-substituted zeolites ZSM5, L, and mazzite showed broad and ill-defined peaks since the ZFS of Fe(III) in these zeolites is significantly larger than that of FeSOD. Because this broadening is a high-order effect, it can be significantly reduced at higher spectrometer frequencies.     

Hidden symmetry breaking and the Haldane phase inS=1 quantum spin chains

We study the phase diagram ofS=1 antiferromagnetic chains with particular emphasis on the Haldane phase. The hidden symmetry breaking measured by the string order parameter of den Nijs and Rommelse can be transformed into an explicit breaking of aZ ₂×Z ₂ symmetry by a nonlocal unitary transformation of the chain. For a particular class of Hamiltonians which includes the usual Heisenberg Hamiltonian, we prove that the usual Néel order parameter is always less than or equal to the string order parameter. We give a general treatment of rigorous perturbation theory for the ground state of quantum spin systems which are small perturbations of diagonal Hamiltonians. We then extend this rigorous perturbation theory to a class of diagonally dominant Hamiltonians. Using this theory we prove the existence of the Haldane phase in an open subset of the parameter space of a particular class of Hamiltonians by showing that the string order parameter does not vanish and the hiddenZ ₂×Z ₂ symmetry is completely broken. While this open subset does not include the usual Heisenberg Hamiltonian, it does include models other than VBS models.     

Quantum Out-States Holographically Induced by Asymptotic Flatness: Invariance under Spacetime Symmetries, Energy Positivity and Hadamard Property

This paper continues the analysis of the quantum states introduced in previous works and determined by the universal asymptotic structure of four-dimensional asymptotically flat vacuum spacetimes at null infinity M. It is now focused on the quantum state λ _M , of a massless conformally coupled scalar field propagating in M. λ _M is “holographically” induced in the bulk by the universal BMS-invariant state λ defined on the future null infinity of M. It is done by means of the correspondence between observables in the bulk and those on the boundary at future null infinity discussed in previous papers. This induction is possible when some requirements are fulfilled, in particular whenever the spacetime M and the associated unphysical one, M͂, are globally hyperbolic and M admits future time infinity i ⁺. λ _M coincides with Minkowski vacuum if M is Minkowski spacetime. It is now proved that, in the general case of a curved spacetime M, the state λ _M enjoys the following further remarkable properties:

An instanton-invariant for 3-manifolds

To an oriented closed 3-dimensional manifoldM withH ₁(M, )=0, we assign a ₈-graded homology groupI _*(M) whose Euler characteristic is twice Casson's invariant. The definition uses a construction on the space of instantons onM×.     

Experimental limits onM t andM H — present and future

Existing data from precision tests of the electroweak standard model at LEP, the -colliders andvN-scattering experiments constrain the mass of the top quark toM _t =144 _–26–21 ⁺²³⁺¹⁹ GeV, where the central value and first error refer to a Higgs mass value of 300 GeV. The second error shows the variation ofM _t for Higgs mass values spanning the range 50M _H(GeV)<1000. the=" present=" apparent=" sensitivity=" of=" the=" data=" to=" the=" value=" of=" the=" higgs=" mass=" is=" investigated.=" future=" data=" from=" lep=" will=" significantly=" tighten=" the=" bounds=">M _t . Limits onM _H however will remain of order M _H/M _H2 without an independent measurement ofM _t .     

Black holes associated with galaxies

A small and a large black hole are naturally associated with a galaxy of total massM and spherical halo radiusR. Also of massM, the large black hole is a spatial contraction of the galaxy down to its Schwarzschild radius,r r, with=2GM/c ²R, whereG/c ²=4.78×10^–17 kpc/M is Newton's gravitational constant divided by the speed of light squared. The small black hole is ther r contraction of the large hole, i.e., the iterated double contraction of the galaxy itself, with the resulting massm=M=2GM ²/c²R. In the case of the Milky Way (M=7.0×10¹¹ M andR=15 kpc) the latter equation for the small black hole mass yieldsm=3.1×10⁶ M , which is close to the observed value for the mass of the black hole at the center of the Milky Way. Black holes of the small type may evolve to the large by mass accretion, perhaps during a quasar phase. Vast regions of the universe may in fact be populated by large black holes—missing mass—which validates the cosmological principle and effects the closure of the universe.     

Magnetic properties of Fe and Ni films evaporated in poor vacuum

To examine the pressure dependence of magnetic characteristics of Fe and Ni films, the films were prepared in vacuum ranging from 10^–5 to 10^–2 torr.Saturation magnetizationM _s ^* , perpendicular anisotropyK ^* and coercive forceH _c for Fe and Ni films deposited at around 10^–5 torr were in good agreement with the values obtained by others. When the film thickness was less than 500 ,M _s ^* for Fe films increased with pressure, while it decreased monotonically for Ni films. At pressures between 2×10^–3 and 10^–2 torr,M _s ^* decreased rapidly for Fe but it increased slightly for Ni. This interesting behaviour was most marked with film thickness of about 500 . Corresponding to the change inM _s ^* , bothK ^* andH _c also changed with deposition pressure.The result should be explained in terms of the presence of Fe₄N, Fe₈N and Ni₃N, as well as the macroscopic and microscopic structures of films.     

A natural framework for the minimal supersymmetric gauge theories

We show that the sequence of Jordan algebras M _inf3 ^sup1 , M _inf3 ^sup2 , M _inf3 ^sup4 , and M _inf3 ^sup8 , whose elements are in the 3×3 Hermitean matrices over , , , and O, respectively, provide an elegant and natural framework in which to describe supersymmetric gauge theories. The four minimal supersymmetric gauge theories are in a one-to-one correspondence with these four Jordan algebras and, hence, with the four division algebras.     

Critical Stability of Three-Body Relativistic Bound States with Zero-Range Interaction

For the zero-range interaction providing a given mass M₂ of the two-body bound state, the mass M₃ of the relativistic three-body system is calculated. We have found that the three-body system exists only when M₂ is greater than a critical value M_c ( 1.43 m for bosons and 1.35 m for fermions, m being the constituent mass). For M₂=M_c the mass M₃ turns into zero and for M₂ < M_c there is no solution with real value of M₃.     

Sufficient subalgebras and the relative entropy of states of a von Neumann algebra

A subalgebraM ₀ of a von Neumann algebraM is called weakly sufficient with respect to a pair (,) of states if the relative entropy of and coincides with the relative entropy of their restrictions toM ₀. The main result says thatM ₀ is weakly sufficient for (,) if and only ifM ₀ contains the Radon-Nikodym cocycle [D,D] _t . Other conditions are formulated in terms of generalized conditional expectations and the relative Hamiltonian.