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.     

Low-x evolution equations in M?bius representation

The M?bius form of the BFKL kernel in the next-to-leading order (NLO) in theories containing fermions and scalars in arbitrary representations of the colour group is presented. The ambiguity of the NLO kernels permits to get agreement between the BFKL approach and the colour dipole model and to find the quasi-conformal representation of the BFKL kernel.     

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.     

Dynamics of atoms in strong laser fields I: A quasi analytical model in momentum space based on a Sturmian expansion of the interacting nonlocal Coulomb potential

In this study we present a model that we have formulated in the momentum space to describe atoms interacting with intense laser fields. As a further step, it follows our recent theoretical approach in which the kernel of the reciprocal-space time-dependent Schrödinger equation (TDSE) is replaced by a finite sum of separable potentials, each of them supporting one bound state of atomic hydrogen (Tetchou Nganso et al. 2013). The key point of the model is that the nonlocal interacting Coulomb potential is expanded in a Coulomb Sturmian basis set derived itself from a Sturmian representation of Bessel functions of the first kind in the position space. As a result, this decomposition allows a simple spectral treatment of the TDSE in the momentum space. In order to illustrate the credibility of the model, we have considered the test case of atomic hydrogen driven by a linearly polarized laser pulse, and have evaluated analytically matrix elements of the atomic Hamiltonian and dipole coupling interaction. For various regimes of the laser parameters used in computations our results are in very good agreement with data obtained from other time-dependent calculations.     

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.     

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.     

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 of honeycomb lattices on Klein bottle,Möbius strip and cylinder

Dimer statistics is a central problem in statistical physics. In this paper the enumerations of close-packed dimers of honeycomb lattices on Klein bottle, Möbius strip and cylinder are considered. By establishing a Pfaffian orientation or a crossing orientation, and then computing the determinants of the skew-symmetric matrices of the resulting orientation graphs, we obtain explicit expressions of the number of close-packed dimers of the Klein-bottle polyhex, the Möbius polyhex and the cylindrical polyhex.     

Distance geometry and geometric algebra

As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.⁽¹⁾ In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.⁽¹⁸⁾ Möbius invention of homogeneous coordinates was one of the most far-reaching ideas in the history of mathematics: comparable to Leibnitz's invention of differentials.H.S.M. Coxeter, Introduction to Geometry, p. 221... the inhabitants of a hyperbolic world would also study horospherical geometry, which is the same as Euclidean geometry!H.S.M. Coxeter, Introduction to Geometry, p. 304     

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.     

Small x resummation and the Odderon

We present a general argument which suggests that the Bartels–Lipatov–Vacca Odderon intercept should be equal to one to all orders in the perturbation theory. The argument is based on the validity of the so-called omega-expansion in the high energy limit. It can be further supported by the analogous pattern observed in the case of the anomalous dimensions which is a consequence of the momentum sum rule. In addition, we conjecture that the BFKL kernel should satisfy the transverse momentum sum rule. Finally, it is shown that the higher order kinematical effects do not change the BLV Odderon intercept.     

A Hilbert space approach to the Lippmann-Schwinger equation in momentum representation

We study the Lippmann-Schwinger equation for the quantum mechanical two-body scattering problem. We propose a Hilbert space approach in momentum-angular momentum representation. Imposing a Hölder integrability condition on the potential, the kernel of the integral equation becomes a compact operator in an adequate Hilbert space H⁰. We show that expansion into orthogonal polynomials becomes very simple, and we give an application to the three-particle problem.     

Diffractive vector meson production at large momentum transfer

The diffractive process γp→V+X (whereV is a vector meson andX results from the dissociation of the proton) is studied. In particular, we consider the region of large momentum transfer (i.e. |t| ?Λ _QCD ² ) and large centre-of-mass (CM) energy,s. The asymptotic (s→∞,s/|t|?1) behaviour is derived from the BFKL equation and compared to that which is obtained in the Born approximation (two-gluon exchange). We also calculate the corrections to the Born graphs by iterating the BFKL kernel numerically. Improved convergence of the BFKL series is found by summing the logarithms which occur when an exchanged gluon goes nearly on shell. Importantly, we find evidence that the asymptotic solution to the BFKL equation is inappropriate over most of the HERA range and we provide more realistic predictions for the cross section. The predicted cross section is not too small and can be measured at HERA, up to momentum transfers |t|≈10 GeV².     

Cluster variation method and Möbius inversion formula

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

Form of the memory function for a degenerate electron gas with elastic scatterers

The high frequency response of an electron gas with elastic scatterers can be described by a Drude equation with a memory term, which is an integral over a kernel times the square of the scattering potential. In the momentum (and frequency or time) representation, this kernel is discontinuous at the diameter of the Fermi sphere.In the time representation it has the causality property. As a function of space it shows Friedel oscillations.     

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.     

Eight-dimensional spinor representation of the Poincaré group

Finite-dimensional matrix representations of the Poincaré group are discussed with particular emphasis on the eight-dimensional spinor representation. It is speculated that the complex eight-dimensional representation space might be interpreted as a more fundamental entity than Minkowski space, being in a sense a square root of the latter. One can model the usual position, momentum, and angular momentum variables of a particle of nonzero rest mass and arbitrary spin by real bilinear forms in the 8-spinor components, and obtain their correct equations of motion by subjecting the spinor to a Schrödinger-like evolution equation.     

Color dipole BFKL-Regge factorization and high-energy photon-photon scattering

Based on the color dipole representation, we investigate consequences for the γ*γ*, γ *γ scattering of the finding by Fadin, Kuraev, and Lipatov that the incorporation of asymptotic freedom into the BFKL equation makes the QCD pomeron a series of isolated poles in the angular momentum plane. The emerging color dipole BFKL-Regge factorization allows us to relate in a model-independent way the contributions of each BFKL pole to the γ *γ*, γ*γ scattteirng and the deep inelastic scattering on protons. Numerical predictions based on our early work on the color dipole BFKL phenomenology of the deep inelastic scattering on protons gives a good agreement with the recent experimental data from OPAL and L3 experiments at LEP200. We discuss the role of nonperturbative dynamics and predict a pronounced effect of the Regge-factorization breaking due to large unfactorizable nonperturbative corrections to the perturbative vacuum exchange. We comment on the salient features of the BFKL-Regge expansion for the γ*γ*, γ*γ scattering including the issue of the decoupling of subleading BFKL poles and the soft plus rightmost hard BFKL pole dominance.