Dynamical groups and spherical potentials in Classical Mechanics

The one particle problem in a spherical potential is examined in Classical Mechanics from a group theorical point of view. The constants of motion are classified according to their behaviour under the rotation groupSO(3), i.e. according to the irreducible representationsD _j ofSO(3) (section 1).The Lie algebras ofSO(4) andSO(3) are explicitly built in terms of Poisson brackets for an arbitrary potential, from global considerations. The Kepler and the 3 dimensional oscillator problems are shown to play particular roles with respect to these groups (sections 2 and 3).In the last section, the Kepler problem is analyzed with the aid of theSO(4) group instead of the Lie algebra. It is proved that the transformations generated by the angular momentum and the Runge-Lenz vector form indeed a group of canonical transformations isomorphic toSO(4). Consequences with respect to the quantization problem are examined.     

Torsion,Scalar Field,Mass and FRW Cosmology

In the Einstein–Cartan space U₄, an axial vector torsion together with a scalar field connected to a local scale factor have been considered. By combining two particular terms from the SO(4, 1) Pontryagin density and then modifying it in a SO(3, 1) invariant way, we get a Lagrangian density with Lagrange multipliers. Then under FRW-cosmological background, where the scalar field is connected to the source of gravitation, the Euler–Lagrange equations ultimately give the constancy of the gravitational constant together with only three kinds of energy densities representing mass, radiation and cosmological constant. The gravitational constant has been found to be linked with the geometrical Nieh-Yan density.     

Natural halotrichites—an EDX and Raman spectroscopic study

Raman spectroscopy has been used to characterise four natural halotrichites: halotrichite FeSO₄.Al₂(SO₄)₃. 22H₂O, apjohnite MnSO₄.Al₂(SO₄)₃.22H₂O, pickingerite MgSO₄.Al₂(SO₄)₃.22H₂O and wupatkiite CoSO₄.Al₂(SO₄)₃.22H₂O. A comparison of the Raman spectra is made with the spectra of the equivalent synthetic pseudo‐alums. Energy dispersive X‐ray analysis (EDX) was used to determine the exact composition of the minerals. The Raman spectrum of apjohnite and halotrichite display intense symmetric bands at ∼985 cm⁻¹ assigned to the ν₁(SO₄)²⁻ symmetric stretching mode. For pickingerite and wupatkiite, an intense band at ∼995 cm⁻¹ is observed. A second band is observed for these minerals at 976 cm⁻¹ attributed to a water librational mode The series of bands for apjohnite at 1104, 1078 and 1054 cm⁻¹, for halotrichite at 1106, 1072 and 1049 cm⁻¹, for pickingerite at 1106, 1070 and 1049 cm⁻¹ and for wupatkiite at 1106, 1075 and 1049 cm⁻¹ are attributed to the ν₃(SO₄)²⁻ antisymmetric stretching modes of ν₃(B_g) SO₄. Raman bands at around 474, 460 and 423 cm⁻¹ are attributed to the ν₂(A_g) SO₄ mode. The band at 618 cm⁻¹ is assigned to the ν₄(B_g) SO₄ mode. The splitting of the ν₂, ν₃ and ν₄ modes is attributed to the reduction of symmetry of the SO₄ and it is proposed that the sulphate coordinates to water in the hydrated aluminium in bidentate chelation. Copyright © 2007 John Wiley & Sons, Ltd.     

Liouville field theory revisited: Zero modes andS-matrix element

A new analysis of the intrinsic structure of Liouville field theory (LFT) is presented. We prove that LFT displays a zero mode if its Laplacian is defined in terms of the square of the corresponding Dirac operator. Further, by interpreting the spacetime asSO(2,1)/SO(1, 1) (analogous toSO(3, 2)/SO(3, 1)) we present, arguments which support the nontrivial approximativeS-matrix element we derived in [2]. Some connected questions are also discussed.     

Master Analytic Representations and unified representation theory of certain orthogonal and pseudo-orthogonal groups

The representation theory of the groupsSO(5),SO(4, 1),SO(6) andSO(5, 1) is studied using the method of Master Analytic Representations (MAR). It is shown that a single analytic expression for the matrix elements of the generators ofSO(n+1) andSO(n, 1) in anSO(n) basis yields all the unitary representations (forn=4,5); and that the compact and non-compact groups have essentially the same analytic representation. Once the MAR of a group is worked out, the search for the unitary irreducible representations is reduced to a purely arithmetic operation. The utmost care has been exercised to conduct the discussions at an elementary level: knowledge of simple angular momentum theory is the only prerequisite.Work supported in part by the National Science Foundation.Work supported in part by the U.S. Atomic Energy Commission.     

Holonomy Groups Coming from F-Theory Compactification

We study holonomy groups coming from F-theory compactifications. We focus mainly on SO(8) as 12−4=8 and subgroups SU(4), Spin(7), G ₂ and SU(3) suitable for descent from F-theory, M-theory and Superstring theories. We consider the relation of these groups with the octonions, which is striking and reinforces their role in higher dimensions and dualities. These holonomy groups are related in various mathematical forms, which we exhibit.     

The Spectrum of D = 11 Supergravity via Harmonic Expansions on S4 X S7

We derive the complete particle spectrum of the d = 11 supergravity compactified on AdS X S⁷, by analytically continuing AdS to S⁴, and using the Salam-Strathdee method of harmonic expansion on S⁴ X S⁷. The spectrum is arranged into supermultiplets labelled by an integer l = 0, 1, 2,. The massless supermultiplet corresponds to l = 0. For the l'th supermultiplet we find a relation, involving the eigenvalues of the second order Casimir operators of SO(3, 2) and SO(8), given by 2C₂[SO(3, 2)] + C₂[SO(8)] = 3/2(l + 2) (l + 4).     

Spectroscopic properties of Er3+ ions in cadmium and alkali cadmium borosulphate glasses

Spectroscopic properties of Er³⁺:CBS (CdSO₄+B₂O₃ and R₂SO₄+CdSO₄+B₂O₃, R₂SO₄=Li₂SO₄.H₂O, Na₂SO₄, K₂SO₄ and Gd₂(SO₄)₃.8H₂O) glasses are reported. The assigned energy level data of Er³⁺(4f ¹¹) in these glasses are analysed in terms of a parametrized model Hamiltonian. The standard deviations of the data fits are between 39 and 47 cm⁻¹ so that the energy level schemes of the Er³⁺(4f ¹¹) ions in borosulphate (CBS) glasses are reasonably well reproduced. Radiative properties for the fluorescent levels of Er³⁺:CBS glasses are determined by using the Judd-Ofelt theory. The potential laser transitions are identified with the help of predicted radiative properties which are compared and discussed with similar results.     

A spectrum generating algebra for meson resonances

The algebraSO(6,1) is considered as a unification ofSO(6), which is isomorphic toSU(4) SU(3), and the de Sitter algebraSO(4,1). The latter replaces the Poincaré algebra as the algebra of the group of motions of physical space-time. A representation ofSO(6,1) is constructed, which, on restriction toSU(3), decomposes into the direct sum of allSU(3) representations, each occurring just once in the decomposition. The expectation values of the mass-squared operator, when evaluated in the octet, give accurate mass formulae for the octets of 1^– and 2⁺ meson resonances.     

Dynamical and symmetry groups of the SU(2) Kepler problem

It is well known that the MIC–Kepler problem, an extension of the three-dimensional Kepler problems, admits the same dynamical and symmetry groups as the Kepler problem. This paper aims to study dynamical and symmetry groups of the SU(2) Kepler problem, where the SU(2) Kepler problem is defined to be the dynamical system reduced from the eight-dimensional conformal Kepler problem through an SU(2) symmetry and turns out to be an extension of the five-dimensional Kepler problem. It is shown that the SU(2) Kepler problem admits a dynamical group SO^*(8) and that the phase space of the SU(2) Kepler problem is symplectomorphic with a co-adjoint orbit of SO^*(8), on which the Kirillov–Kostant–Souriau form is defined. It is further shown that the subgroups, SU(4), SU^*(4), and Sp(2)×_SR⁵, of SO^*(8) provide the symmetry groups, SU(4)/Z₂SO(6), SU^*(4)/Z₂SO₀(1,5), and (Sp(2)×_SR⁵)/Z₂SO(5)×_SR⁵, of the SU(2) Kepler problem with negative, positive, and zero energies, respectively, where ×_S denotes a semi-direct product. Furthermore, constants of motion for the SU(2) Kepler problem are found together with their Poisson brackets. The symmetry Lie algebra formed by constants of motion is shown to be isomorphic with so(6)su(4), so(1,5)su^*(4), or so(5)_SR⁵sp(2)_SR⁵, depending on whether the energy is negative, positive, or zero, where _S denotes a semi-direct sum. These Lie algebras are subalgebras of so^*(8)so(2,6).     

Raman spectroscopic study of the sulfite‐bearing minerals scotlandite,hannebachite and orschallite: implications for the desulfation of soils

The structures of the naturally occurring sulfite‐bearing minerals scotlandite, hannebachite and orschallite have been studied by Raman spectroscopy. Raman bands are observed for scotlandite PbSO₃ at 935, 880, 622 and 474 cm⁻¹ and are assigned to the (SO₃)²⁻ν₁(A₁), ν₃(E), ν₂(A₁) and ν₄(E) vibrational modes, respectively. For hannebachite (CaSO₃)₂·H₂O these bands are observed at 1005, 969 and 655 cm⁻¹ with multiple bands for the ν₄(E) mode at 444, 492 and 520 cm⁻¹. The Raman spectrum of hannebachite is very different from that of the compound CaSO₃·2H₂O. It is proposed, on the basis of Raman spectroscopy, that in the mineral hannebachite, the sulfite anion bonds to Ca through the sulfur atom. The Raman spectrum of the mineral orschallite Ca₃[SO₄](SO₃)₂·12H₂O is complex resulting from the overlap of sulfate and sulfite bands. Raman bands at 1005 cm⁻¹, 1096 and 1215 cm⁻¹ are assigned to the (SO₄)²⁻ν₁ symmetric and ν₃ asymmetric stretching modes. The two Raman bands at 971 and 984 cm⁻¹ are attributed to the (SO₃)²⁻ν₃(E) and ν₁(A₁) stretching vibrations. The formation of sulfite compounds in nature offers a potential mechanism for the removal of sulfates and sulfites from soils. Copyright © 2008 John Wiley & Sons, Ltd.     

Hamilton's Turns for the Lorentz Group

Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by “turns,” which are equivalence classes of directed great circle arcs on the unit sphere S ², in such a manner that the rule for composition of group elements takes the form of the familiar parallelogram law for the Euclidean translation group. It is only recently that this construction has been generalized to the simplest noncompact group SU(1, 1)=Sp(2, R)=SL(2, R), the double cover of SO(2, 1). The present work develops a theory of turns for SL(2, C), the double and universal cover of SO(3, 1) and SO(3, C), rendering a geometric representation in the spirit of Hamilton available for all low dimensional semisimple Lie groups of interest in physics. The geometric construction is illustrated through application to polar decomposition, and to the composition of Lorentz boosts and the resulting Wigner or Thomas rotation. PACS numbers: 02.20.-a     

Synthesis and thermoluminescence characteristics of γ-irradiated K3Ca2(SO4)3F:Eu or Ce fluoride

New halophosphor K₃Ca₂(SO₄)₃F activated by Eu and Ce has been synthesized by a co-precipitation method and characterized according to its thermoluminescence. The formation of traps in rare earth doped K₃Ca₂(SO₄)₃F and the effects of γ-radiation dose on the glow curve are discussed. The glow curve of K₃Ca₂(SO₄)₃F:Ce shows a prominent single peak at 150°C, whereas K₃Ca₂(SO₄)₃F:Eu and K₃Ca₂(SO₄)₃F:Ce,Eu at 142°C and 192°C, respectively. A single glow peak indicates that there is only one set of trap being activated within the particular temperature range. The presented phosphors are also studied because of its fading, reusability and trapping parameters. There was just 2% fading during a period of 10 days, indicating no serious fading problem. Trapping parameters such as order of kinetics (b), activation energy (E) and frequency factor (S) were calculated by using Chen's half-width method. The observations presented in this paper are good for lamp phosphors as well as solid-state dosimeter.     

Projective spacetime

It is suggested that the world is locally projectively flat rather than Euclidean. From this postulate it is shown that an (N+1)-particle system has the global geometry of the symmetric spaceSO(4,N+1)/SO(4)×SO(N+1). A complex representation also exists, with structureSU(2,N+1)/S[U(2)×U(N+1)]. Several aspects of these geometrics are developed. Physical states are taken to be eigenfunctions of the Laplace-Beltrami operators. The theory may provide a rational basis for comprehending the groupsSO(4, 2),SU(2)×U(1),SU(3), etc., of current interest.     

SO q (n+1,n−1) as a real form of SO q (2n,ℂ)

Quantum pseudo-orthogonal groups SO _q (n+1,n–1) are defined as real forms of quantum orthogonal groups SO _q (n+1,n–1) by means of a suitable antilinear involution. In particular, the casen=2 gives a quantized Lorentz group.     

Complex geometry,unification, and quantum gravity. I. The geometry of elementary particles

The Poincaré group is replaced byU(3, 2), the pseudounitary extension of the de Sitter groupSO(3, 2), as internal and space-time symmetries are combined in a geometric setting which invalidates the no-go theorems. A new model of elementary particles as vertical vectors on the principal fiber bundleU(3, 2) U(3, 2)/U(3, 1)×U(1) is introduced and their interactions via Lie bracket analyzed. The model accounts for the four known superselection rules: spin, electric charge, baryon number, and lepton number.     

Flavored surface defects in 4d $$\mathcal{N}=1$$ N = 1 SCFTs

We discuss supersymmetric surface defects in compactifications of six-dimensional minimal conformal matter of types SU(3) and SO(8) to four dimensions. The relevant field theories in four dimensions are \(\mathcal{N}=1\) quiver gauge theories with SU(3) and SU(4) gauge groups, respectively. The defects are engineered by giving space-time-dependent vacuum expectation values to baryonic operators. We find evidence that in the case of SU(3) minimal conformal matter, the defects carry SU(2) flavor symmetry which is not a symmetry of the four-dimensional model. The simplest case of a model in this class is SU(3) SQCD with nine flavors, and thus the results suggest that this admits natural surface defects with SU(2) flavor symmetry. We analyze the defects using the superconformal index and derive analytic difference operators introducing the defects into the index computation. The duality properties of the four-dimensional theories imply that the index of the models is a kernel function for such difference operators. In turn, checking the kernel property constitutes an independent check of the dualities and the dictionary between six- dimensional compactifications and four-dimensional models.

     

Raman and infrared spectral studies of [C(NH2)3]2MII(H2O)4(SO4)2, MII = Mn,Cd and VO

The Raman and FTIR spectra of three metal guanidinium sulfates, [C(NH₂)₃]₂M^II(H₂O)₄(SO₄)₂, (M^II = Mn, Cd and VO), are recorded. The observed spectral bands are assigned in terms of the fundamental modes of vibration of the guanidinium ions, sulfate groups and water molecules. The appearance of the sulfate tetrahedra's ν₁ and ν₂ modes in the IR spectra and the partial lifting of the ν₄ mode in the Raman spectra indicate the distortion of the SO₄²⁻ tetrahedra in the structure, so that its symmetry is lowered from T_d to C₁. The geometry of the sulfate group in guanidinium vanadyl sulfate does not deviate much from that of the average sulfate group. The distortion of the SO₄ tetrahedra is stronger in GuCds than in GuMnS. The CN₃ group in the guanidinium ion is planar (D_3h point group) in GuCdS and GuMnS, whereas it is lowered in the vanadyl compound. Furthermore, the spectral analyses show the presence of weak hydrogen bonds in the structures. Copyright © 2007 John Wiley & Sons, Ltd.     

Principles of Quaternionic Vacuum Thermodynamics and a Unified Gravistrong Interaction Model

The conceptions of the preceding paper [1] are completed by the discussion of first outlines of quaternionically originated vacuum thermodynamics. Thereby the algebraic Planck scale discontinuum statistically and approximately is extended by steps to affinely unconnected and to affinely connected Riemannian spacetime continuum. There results the concept of an irreversible particle realization process out of vacuum, as well as a vacuum foundation of the gauge principle and of the long-range coupling constants. The version of the Electroweak Standard Model derived from the simple-regular SO(3,3) representations is extended to a unified interaction model of gravitation and strong forces (gravistrong interaction model) derived from the double-regular SO(3,3) representations. An SO(3,3)-originated connection between the linear graviton field and the non-linear gravitation field is proposed.     

The SO2 Stretching Vibrations in Some Metal Saccharinates: Spectra-Structure Correlations

The infrared spectra of a series of metal saccharinates (those of Na, Mg, Mn Fe, Co; Ni, Zn, Cd, Pb as well as mercury(II) saccharinate and chloromercury(II) saccharinate) were studied in the region of the antisymmetric and symmetric SO₂ stretching vibrations [v_as(SO₂) and v_s(SO₂) hereafter]. The results concerning the number of bands attributable to the abovementioned vibrations, their frequencies and intensities were correlated with the crystallographically determined type of metal-to-ligand bonding, on the one hand, and the number of the structurally different SO₂ groups, on the other.

It was shown that, irrespective on whether the sulphonyl oxygen atoms do or do not participate in bonding to the metal atom and/or in hydrogen bonding and irrespective on the type of the metal-to-saccharin bond (from covalent to purely ionic), the frequencies of both v_as(SO₂) and v_s(SO₂) modes are lower for the metal saccharinates than for saccharin itself, the effect being more pronounced in the case of the v_as(SO₂) bands.

As expected, the appearance of the spectra in the SO₂ stretching regions is rather strongly dependent on the values of the O-S-O angles. Thus, two rather different types of O-S-O angles exist in the structure of the covalently bonded saccharinate of mercury and of the ionic lead saccharinate (the values of these angles are 118.7 and 111.8 ° and 120.4 and 111.8 ° respectively) and two pairs of v_as(SO₂) and v_s(SO₂) bands are present in the spectra of these two compounds. On the other hand, the existence of only one type of SO₂ groups in the saccharinates of Mn and ClHg or the presence of SO₂ groups with very close O-S-O angles in the saccharinates of Na and Mg is manifested by the appearance of only a single pair of intense v_as(SO₂) and v_s(SO₂) bands in their spectra.