Some unexpected proportionalities between components of the spin—other-orbit interaction in the f shell

The reduced matrix elements of the component z _i of the inter-electronic spin-other-orbit interaction have been calculated for all the states of the atomic f shell. Three of the z _i , each of which belongs to the irreducible representation (30) of Racah's group G₂, are found to exhibit matrix elements that are often related to one another in ways that go beyond what the Wigner-Eckart theorem, generalized to G₂, would predict. Examples are presented for matrix elements whose bras belong to the IR (20) of G₂ and whose kets belong to (31). The challenges to current theory are discussed.     

Unitary representations of semi-direct product groups with infinite dimensional Abelian normal subgroup

We generalize Mackey's theory of induced representations to groups which are semidirect products of a locally compact group H by an infinite dimensional Abelian group A. A theorem on inducing in stages is proved and conditions ensuring conservation of the irreducibility of the representation through the inducing process are given. When A is a nuclear space or the Hilbert space A_O of a Gelfand triplet A_O ? A ? A′_O, a cylindrical ergodic measure on the dual space A′ of A (respectively A′_O) is associated to each irreducible representation of G, and it is proved that the representation is induced when the measure is concentrated on an orbit. In the last part, A is supposed to be a Hilbert space and sufficient conditions are given for the preceding ergodic measures to be concentrated on an orbit.     

The horospheric approach to harmonic analysis on a semi-simple Lie group

We extend the Gelfand-Helgason theory of horospheric transformations to the regular representation of an arbitrary connected semi-simple Lie group. Let G=NAK be an Iwasawa decomposition of such a group G, V the subgroup contragredient to N, and M the centralizer of A in K: then the horospheric transform is a mapping of the function space over G onto one over the topological product N×V×M×A; and the Fourier transform over G in the principal most continuous series reduces to an expansion into multihomogeneous functions on A and irreducible representations of M, while the remaining principal series are characterised by associated homogeneity in one or more variables of A. The inversion formula for the Fourier transform is our principal result. The measure required for the most continuous series of a group of real-rank unity is the square of the Plancherel measure for G; and at the double poles of this the residues are precisely the contributions of the remaining principal series. This enables us to write the entire inversion formula for such a group as a single formal contour integral; analogous results hold for groups of higher real-rank, but the measure is slightly less simple.     

On locally continuous cocycles

Cohomology groups of locally continuous cocycles of a topological group G with values in a topological G-module A_Ψ are considered and the role of those of degree 2 in the theory of topological extensions of G by A_Ψ is analyzed. The exactness of long sequence is proven. It is shown that, if G and A_Ψ are Polish and if all topological extensions of G by A_Ψ are fibered, the cohomology groups of Borel cocycles and those of locally continuous Borel cocycles are isomorphic for degree 0,1,2.     

Brueckner G matrix for a planar slab of nuclear matter

The equation for the Brueckner G matrix is investigated for planar-slab geometry. A method for calculating the G matrix for a planar slab of nuclear matter is developed for a separable form of NN interaction. Actually, the separable version of the Paris NN potential is used. The singlet ¹ S ₀ and the triplet ³ S ₁–³ D ₁ channel are considered. The present analysis relies on the mixed momentum-coordinate representation, where use is made of the momentum representation in the slab plane and of the coordinate representation in the orthogonal direction. The full two-particle Hilbert space is broken down into the model subspace, where the two-particle propagator is considered exactly, and the complementary subspace, where the local-potential approximation is used, which was proposed previously for calculating the effective pairing potential. Specific calculations are performed for the case where the model subspace is constructed on the basis of negative-energy single-particle states. The G matrix is parametrically dependent on the total two-particle energy E and the total momentum P _⊥ in the slab plane. Since the G matrix is assumed to be further used to calculate the Landau-Migdal amplitude, the total two-particle energy is fixed at the value E=2μ, where μ is the chemical potential of the system under investigation. The calculations are performed predominantly for P _⊥=0. The role of nonzero values of P _⊥ is assessed. The resulting G matrix is found to depend greatly on μ in the surface region.     

Note on conservation laws based upon traceless energy-momentum tensors in flat space-time

In this note we prove the following theorem. If in a flat space-time with metric g_ij(x) treferred to general coordinates xⁱ a vector ξⁱ(x) satisfies (Tⁱ_jξ^j)_;i=0 (semicolon denotes covariant differentiation) for all energy-momentum tensors of the set $\text{{}\text{T}{}^{\mathrm{ij}}\text{T}{}^{\mathrm{i}}{}_{\mathrm{j;i}}\text{=0;g}{}_{\mathrm{ij}}\text{T}{}^{\mathrm{ij}}\text{=0}$; T^ij = T^ji; T^iju_iu_j > 0 (where u_i is a time-like vector)}, then the vector ξⁱ defines a conformal motion. This theorem, which may be considered as a converse (in flat space-time) to a well-known result of Trautman, is a generalization of a result obtained by J. T. ?opuszański and J. Szczucka-Soko?owska [Reports on Mathematical Physics 11 (1977), 153] in which they assumed the vector ξⁱ was a polynomial in Minkowski coordinates.     

Time inversion symmetry and transport properties of solids

Identification of the time inversion operator θ with the magnetic moment inversion operator A in finding the symmetry restrictions on the form of transport tensors in solids is shown to be incorrect.     

Generalizing some properties of short range classical solutions to Yang-Mills theory

We extend a theorem which states that for classical solutions of Yang-Mills theory, the field G_μν has to decrease at least as fast as the source S_μ at spatial infinity, provided G_μν decreases exponentially [G_μν ～ exp(?Mr)]. This generalization encompasses all decreases G_μν ～ exp(?Mr^η) with η > 0, r→∞. This is done by assuming an integral representation for A_μ, the vector potential, and imposing some regularity conditions on A_μ, valid as r→∞.     

空间羣算符的矩阵元

本文首先通过对波失星直接乘积的分析指出,如果 ^*K″∈^*K?^*K′,则波矢峯G_k″可以有下面四种情况:A)G_k同G_k′,是G_k″的子峯,B)G_k=G_k′=G_k″,C)G_k″=G_s, D)G_s是G_k″的子峯。根据上述四种情况,分别考虑积分A=μ″^i″(k″)|f_μⁱ(k)|f_μ′^i′(k′)>的简约。并且证明,对所有四种情况,积分A的值均可由短阵U的矩阵元统一地表达出来。矩阵U是使可约表示Г′简约的转换矩阵,对四种情况,可约表示Г′分别是:A)(Г_kⁱ?Г_k′^i′)^(?),B)Гⁱ?Г_k′^i′,C)Г_k^i(s)?Г_k′^i′(s),D)(Г_k^i(s)?Г_k′^i′(s))^(?)。最后,利用准投影算符的方法,对矩阵U进行了计算,导出空间峯Wigner-Eckart定理的表式。     

On Majorana Representations of A6 and A7

The Majorana representations of groups were introduced in Ivanov (The Monster Group and Majorana Involutions, 2009) by axiomatising some properties of the 2A-axial vectors of the 196 884-dimensional Monster algebra, inspired by the sensational classification of such representations for the dihedral groups achieved by Sakuma (Int Math Res Notes, 2007). This classification took place in the heart of the theory of Vertex Operator Algebras and expanded earlier results by Miyamoto (J Alg 268:653–671, 2003). Every subgroup G of the Monster which is generated by its intersection with the conjugacy class of 2A-involutions possesses the (possibly unfaithful) Majorana representation obtained by restricting to G the action of the Monster on its algebra. This representation of G is said to be based on an embedding of G in the Monster. So far the Majorana representations have been classified for the groups G isomorphic to the symmetric group S ₄ of degree 4 (Ivanov et al. in J Alg 324:2432–2463, 2010), the alternating group A ₅ of degree 5 (Ivanov AA, Seress á in Majorana Representations of A ₅, 2010), and the general linear group GL ₃(2) in dimension 3 over the field of two elements (Ivanov AA, Shpectorov S in Majorana Representations of L ₃(2), 2010). All these representations are based on embeddings in the Monster of either the group G itself or of its direct product with a cyclic group of order 2. The dimensions and shapes of these representations are given in the following table:     

A phenomenological theory of possible sequences of ferrotoroidal phase transitions in boracites

A phenomenological theory of the sequence of two second-order phase transitions with close temperatures is considered; such transitions occur in the Ni-Br boracite. The thermodynamic potential is written as a function of polarization P _i, magnetization M _i, and toroidal moment T _i vectors and fields E _i and H _i; T _i is treated as an order parameter. It is assumed that only one coefficient of T _i ² passes through zero as T decreases. The possibility of a sequence of two proper ferrotoroidal phase transitions along the T ₁ and T ₂ components is demonstrated. Spontaneous T _i, P _i, and M _i vector values and equations for susceptibility tensors (dielectric χ _ij = dP _i/dE _j, magnetic k _ij = dM _i/dH _j, and magnetoelectric α_ij = dP _i/dH _j = dM _j/dE _i) were obtained for three phases. Some of these values have well-defined anomalies in the vicinity of transitions. All possible sequences of ferrotoroidal phase transitions in boracites are considered. Depending on two potential coefficient values, these sequences may consist of one, two, or three such transitions.     

The Tc-SSR correlation in A3B(A15-type) compounds

The 28A₃B(A15-type) compounds, for which both the phase equilibrium diagram and the superconducting critical temperature, T_c, are known, can be divided into two groups. The first group consists of compounds whose SSR (solid solution range) is either zero or extends only to the A-rich side of the stoichiometric composition. The second group consists of the compounds whose SSR extends to both sides or only to the B-rich side. The first group of compounds generally has high T_c's and follow the T_c-SSR correlation proposed earlier.Both the grouping and the T_c-SSR correlation are directly relatable to “A-chain integrity” and thus lend further support to the thesis that “A-chain integrity” is of fundamental importance in the superconductivity of these compounds.     

Quantum Mechanical Virial Theorem in Systems with Translational and Rotational Symmetry

Generalized virial theorem for quantum mechanical nonrelativistic and relativistic systems with translational and rotational symmetry is derived in the form of the commutator between the generator of dilations G and the Hamiltonian H. If the conditions of translational and rotational symmetry together with the additional conditions of the theorem are satisfied, the matrix elements of the commutator [G,H] are equal to zero on the subspace of the Hilbert space. Normalized simultaneous eigenvectors of the particular set of commuting operators which contains H, J ², J _z and additional operators form an orthonormal basis in this subspace. It is expected that the theorem is relevant for a large number of quantum mechanical N-particle systems with translational and rotational symmetry.     

Antiferromagnetic resonance spectrum in LaMnO3: Interrelation of the orbital structure and the magnetic properties

The relation between the orbital ordering and magnetic structure of the crystal LaMnO₃ is investigated. The dependence of the exchange parameters on the angle Φ of the orbital structure is determined. When the isotropic exchange interaction and the single-ion anisotropy, which depends on the angle Φ and the rotational distortions, are introduced into the spin Hamiltonian, a four-sublattice structure (A _X, F _Y, G _Z) is obtained with orientation of the magnetic moments of the sublattices near the long axis of the orthorhombic cell of the crystal in the basal plane of the crystal (A _X ? G _Z, F _Y). The effect of a magnetic field on the magnetic structure and the antiferromagnetic resonance spectrum are investigated taking account of the nonequivalent, anisotropic, orbitally-dependent g tensors. The spin-flop and spin-flip transition fields are calculated.     

EPR studies of Cu2+ in anhydrous sodium oxalate

Electron paramagnetic resonance spectra of Cu²⁺ doped in single crystal of anhydrous sodium oxalate grown by slow evaporation from saturated aqueous solutions have been investigated. EPR measurements gave the following values for the spin Hamiltonian parameters: g_xx = g_yy = 2.0741, g_zz = 2.3253, A_xx = A_yy = ?14.9G and A_zz = ?147.7 G. The principal axes of the hyperfine coupling and g tensors are spatially coincident. Using the optical absorption energy values given for a similar complex we have estimated the values of the bonding parameters and orbital-reduction factors for the system under investigation.     

The role of the neutron electric form factor ind(e,e′ N)N including polarization observables

The influence of the neutron electric form factor on various observables in two-body break-up of deuterons by electrons such as differential cross section, beam, target and beam-target asymmetries and outgoing nucleon polarization as well is investigated for different kinematic regions. The electron-deuteron vector asymmetryA _ed ^V and the outgoing nucleon polarization component P′_x(n) are the most promising observables in and off the quasi-free region for a determination ofG _En. Also the single polarization observablesA _d ^T and P_y ⁰(p) and the double polarization observable P′_z(n) show significant influences fromG _En.     

The renormalized variational theory: Its structure,interpretation, and relation to other methods

We apply the Ritz variational principle to a renormalized form of the Iwamoto-Yamada cluster expansion, restricting our discussion to infinite systems. The structure of the resulting theory is governed by the renormalization which keeps track of the normalization denominator in the expectation value. The single-particle potential for hole states (u_i) is introduced as a Lagrange multiplier in the variational principle, and the self-consistent choice of u_i guarantees that the renormalization factors are determined correctly. The importance of the renormalization is illustrated by a discussion of the two-body approximation to our theory. The general formalism is evaluated in more detail for the representation Ψ_T = exp(S)Φ of the trial wave function. Very fundamental considerations show that the theory is especially adapted to that choice of Ψ_T. In addition, if we use that choice of Ψ_T the self-consistent single-particle energies are directly related to experiment, and the theory is almost identical to renormalized Brueckner theory. Thus we are able to clarify many aspects of the latter. We also discuss the relation to the theory of Coester and Kümmel.     

Oscillatory free induction decay in angular distribution of nuclear radiation

A theoretical consideration of the free induction decay (FID) signal formation in the angular distribution of radiation, emitted by oriented nuclei after excitation by a long radiofrequency resonance pulse t_p ? T¹₂, is carried out. The well-known theorem on the oscillatory FID signal is expanded for the case of second rank tensors.     

C*-algebras and automorphism groups

Let (A,G, α) be aC*-dynamical system withG a topological group. Let π be a representation ofA. We will show that there exists a quasiequivalent representation \(\hat \pi \) to π which is a covariant representation, if and only if the folium of π is invariant under the action ofG and this action is strongly continuous.