Connection between complete and Möbius forms of gauge invariant operators

We study the connection between complete representations of gauge invariant operators and their Möbius representations acting in a limited space of functions. The possibility to restore the complete representations from Möbius forms in the coordinate space is proven and a method of restoration is worked out. The operators for transition from the standard BFKL kernel to the quasi-conformal one are found both in Möbius and total representations.     

Supersymmetry and the Möbius inversion function

We show that the Möbius inversion function of number theory can be interpreted as the operator (–1) ^F in quantum field theory. Consequently, we are able to provide physical interpretations for various properties of the Möbius inversion function. These include a physical understanding of the Möbius Inversion Formula and of a result that is equivalent to the prime number theorem. Supersymmetry and the Witten index play a central rôle in these constructions.     

On the Uniqueness of Diffeomorphism Symmetry in Conformal Field Theory

A Möbius covariant net of von Neumann algebras on S¹ is diffeomorphism covariant if its Möbius symmetry extends to diffeomorphism symmetry. We prove that in case the net is either a Virasoro net or any at least 4-regular net such an extension is unique: the local algebras together with the Möbius symmetry (equivalently: the local algebras together with the vacuum vector) completely determine it. We draw the two following conclusions for such theories. (1) The value of the central charge c is an invariant and hence the Virasoro nets for different values of c are not isomorphic as Möbius covariant nets. (2) A vacuum preserving internal symmetry always commutes with the diffeomorphism symmetries. We further use our result to give a large class of new examples of nets (even strongly additive ones), which are not diffeomorphism covariant; i.e. which do not admit an extension of the symmetry to Diff⁺(S¹).Supported in part by the Italian MIUR and GNAMPA-INDAM.     

Dimer statistics on the Möbius strip and the Klein bottle

Closed-form expressions are obtained for the generating function of close-packed dimers on a 2M×2N simple quartic lattice embedded on a Möbius strip and a Klein bottle. Finite-size corrections are also analyzed and compared with those under cylindrical and free boundary conditions. Particularly, it is found that, for large lattices of the same size and with a square symmetry, the number of dimer configurations on a Möbius strip is 70.2% of that on a cylinder. We also establish two identities relating dimer generating functions for Möbius strips and cylinders.     

Cluster variation method and Möbius inversion formula

The cluster variation method is reformulated with the aid of the Möbius inversion formula.     

Optical Möbius strips in three-dimensional ellipse fields: I. Lines of circular polarization

The major and minor axes of the polarization ellipses that surround singular lines of circular polarization in three-dimensional optical ellipse fields are shown to be organized into Möbius strips (twisted ribbons). These strips can have either one or three half-twists, and can be either right- or left-handed. The normals to the surrounding ellipses generate cone-like structures. Two special projections, and eight new indices are developed to characterize the rather complex structures of the Möbius strips and cones. These eight indices, together with the two well-known indices used until now to characterize singular lines of circular polarization, could, if independent, generate 16,384 different index combinations. Geometric constraints and 15 selection rules are discussed that reduce the number of combinations to 1676. Of these 1150 have been observed in 10⁶ independent realizations of a simulated random ellipse field. Statistical probabilities are presented for the most important index combinations. It is argued that it is presently feasible to perform experimental measurements of the Möbius strips and cones described here theoretically.     

Optical Möbius strips in three dimensional ellipse fields: II. Lines of linear polarization

The minor axes of, and the normals to, the polarization ellipses that surround singular lines of linear polarization in three dimensional optical ellipse fields are shown to be organized into Möbius strips (technically twisted ribbons) and into structures we call “rippled rings” (r-rings). The Möbius strips have two full twists, and can be either right- or left-handed. The major axes of the surrounding ellipses generate cone-like structures. Three orthogonal projections that give rise to 15 indices are used to characterize the different structures These indices, if independent, could generate 839,808 geometrically and topologically distinct lines; selection rules are presented that reduce the number of lines to 8248, some 5562 of which have been observed in a computer simulation. Analytical expressions are presented for 11 of the 15 indices in terms of wavefield parameters; four indices proved to be intractable. Statistical probabilities are presented for the most important index combinations in random fields. It is argued that it is presently feasible to perform experimental measurements of the Möbius strips, r-rings, and cones described here theoretically.     

Difference between standard and quasi-conformal BFKL kernels

As it was recently shown, the colour singlet BFKL kernel, taken in Möbius representation in the space of impact parameters, can be written in quasi-conformal shape, which is unbelievably simple compared with the conventional form of the BFKL kernel in momentum space. It was also proved that the total kernel is completely defined by its Möbius representation. In this paper we calculated the difference between standard and quasi-conformal BFKL kernels in momentum space and discovered that it is rather simple. Therefore we come to the conclusion that the simplicity of the quasi-conformal kernel is caused mainly by using the impact parameter space.     

Arbitrary-state solutions of the Dirac equation for a Möbius square potential using the Nikiforov-Uvarov method

We inquire into spin and pseudospin symmetries of the Dirac equation under a Möbius square-type potential using the Nikiforov-Uvarov method to calculate the bound state solutions. We numerically discuss the problem and include various explanatory figures.     

A note on the cluster variation method

Kikuchi's cluster variation method (CVM) is reformulated as the truncation of a Möbius inversion. An attempt is made to explicate and simplify the various approaches to the CVM. This formulation makes apparent the connection of the method with other types of cluster approximation. An illustration of the procedure is provided.     

Möbius groups over general fields using clifford algebras associated with spheres

The space of 2-by-2 Hermitian matrices is isometric to Minkowski space. This is commonly used to exhibit the groupSL(2, ) as a twofold cover of the identity component of the Lorentz group. That these Hermitian matrices also represent equations of circles in the Euclidean plane leads to the groupPSL(2, ) as the Möbius group of the Euclidean plane. Clifford algebras naturally arise in the construction of covers of the orthogonal group by spin groups. By considering in addition the Clifford algebra of the space of equations of spheres, we are able to extend these ideas to the Möbius group of finite-dimensional vector spaces over general fields.     

A Monte Carlo algorithm for lattice ribbons

A lattice ribbon is a connected sequence of plaquettes subject to certain selfavoidance conditions. The ribbon can be closed to form an object which is topologically either a cylinder or a Möbius band, depending on whether its surface is orientable or nonorientable. We describe a grand canonical Monte Carlo algorithm for generating a sample of these ribbons, prove that the associated Markov chain is ergodic, and present and discuss numerical results about the dimensions and entanglement complexity of the ribbons.     

Three-body interaction and phonon dispersion relations in aluminium

Based on an ab initio cohesive energy calculation and a model of three-body interaction, the pair potential can be calculated using the Möbius inversion theorem in the theory of numbers. Then the atomic force constants and the phonon dispersion for A1 are evaluated both with and without three-body interaction. Compared with experiments, the results show that taking the three-body interaction into account considerably improves the dispersions. Contrary to previous work, the method for calculating the atomic force constants and phonon dispersions presented here is simple, with only two adjustable parameters.     

A comment on discrete Kalb–Ramond field on orientifold and rank reduction

We show that the rank reduction of the gauge group on orientifolds in presence of non-vanishing discrete Kalb–Ramond field can be explained by the presence of an induced field strength in a non-trivial bundle on the branes. This field strength is also necessary for the tadpole cancellation and the number of branes is left unchanged by the presence of the discrete Kalb–Ramond background. We moreover give a derivation of the normalization of the Möbius amplitude in presence of a non-vanishing Kalb–Ramond field from first principles.     

A Mössbauer spectroscopy study of antiferromagnetic iron arachidate Langmuir-Blodgett films

Langmuir-Blodgett multilayers of ferric arachidate (abbreviated: FeA) on silicon wafers have been investigated by means of conversion electron Mössbauer spectroscopy (CEMS) and absorption Mössbauer spectroscopy at different temperatures between room temperature and 4.2 K, without and with external magnetic fields up to 5 T. The films show a quasicrystalline structure with ac-axis orientation perpendicular to the plane and a random distribution of orientations in the plane of the films. All Mössbauer spectra show Fe³⁺. At low temperatures, antiferromagnetic ordering has been observed. The lines in both the room temperature and the low-temperature spectra are significantly broadened, which is due to a distribution of electric field gradients and hyperfine fields. The reason for this is a slightly disordered iron environment. The orientation and the distribution function of the internal magnetic fields have been determined.     

A method to predict the orientation relationship,interface planes and morphology between a crystalline precipitate and matrix: part II – application

A model based on near coincidence of diffraction intensity-weighted reciprocal lattice spots was used to study the orientation relationships between a precipitate and matrix in various alloys. The model was used to calculate the orientation relationship and interface orientations between phases including body-centred cubic, body-centred tetragonal, face-centred cubic and hexagonal close-packed crystals. Comparison of calculated results with those reported from various experimental observations demonstrate that in most cases the model can predict the orientation relationship between two phases with an accuracy of a few degrees or better. Calculation of the interface orientation was found to be very sensitive to the exact orientation relationship and therefore, in some cases, showed significant deviation from experimental observations.     

Geometric phases in twisted strips

We compute the optical phase shifts between the left and the right-circularly polarized light after it traverses non-planar cyclic paths described by the boundary curves of twisted closed strips of various widths. While the phase shift increases monotonically with the width of the strip for multi-twisted closed strips, it has a minimum at a certain critical width for the Möbius. In our detailed study of various twisted strips, this intriguing property singles out the Möbius geometry. Possible applications are pointed out.     

Finitized Conformal Spectra of the Ising Model on the Klein Bottle and Möbius Strip

We study the conformal spectra of the critical square lattice Ising model on the Klein bottle and Möbius strip using Yang–Baxter techniques and the solution of functional equations. In particular, we obtain expressions for the finitized conformal partition functions in terms of finitized Virasoro characters. This demonstrates that Yang–Baxter techniques and functional equations can be used to study the conformal spectra of more general exactly solvable lattice models in these topologies. The results rely on certain properties of the eigenvalues which are confirmed numerically.     

Mössbauer diffraction measurements on polycrystalline bcc iron

In the early days of Mössbauer spectroscopy, it was demonstrated that coherent scattering of photons emitted by a Mössbauer source can be observed. In spite of the fact that scattering experiments could give information not accessible by the absorption method, they are not widely used. The reasons for this are, on the one hand, experimental difficulties, and on the other hand, the form of the samples, which should be large single crystals in most cases. In this work, we present a Mössbauer diffraction measurement on polycrystalline bcc iron where these problems have been overcome. We demonstrate that using the kinematical theory of gamma-ray diffraction, useful information on the relative orientation of the crystallographic axis to the hyperfine field directions can be subtracted.     

Nuclear orientation of125I in n-Si and p-Si using defect induced electric field gradients

Defect induced electric field gradients were generated by ion implantation of¹²⁵I in highly B-doped p-Si and P-doped n-Si. The resulting low temperature nuclear orientation at the parent¹²⁵I state was detected by¹²⁵Te Mössbauer spectroscopy. From the asymmetry of the Mössbauer spectrum, the sign of the quadrupole interaction can be derived. This sign is found to be the same for n-Si as for p-Si.