Eduardo A. Notte-Cuello

The squares of the dirac and spin-dirac operators on a riemann-cartan space(time)

In this paper we introduce the Dirac and spin-Dirac operators associated to a connection on Riemann-Cartan space(time) and standard Dirac and spin-Dirac operators associated with a Levi-Civita connection on a Riemannian (Lorentzian) space(time) and calculate the squares of these operators, which play an important role in several topics of modern mathematics, in particular in the study of the geometry of moduli spaces of a class of black holes, the geometry of NS-5 brane solutions of type II supergravity theories and BPS solitons in some string theories. We obtain a generalized Lichnerowicz formula, decompositions of the Dirac and spin-Dirac operators and their squares in terms of the standard Dirac and spin-Dirac operators and using the fact that spinor fields (sections of a spin-Clifford bundle) have representatives in the Clifford bundle we present also a noticeable relation involving the spin-Dirac and the Dirac operators.

Solution to Bethe–Salpeter equation via Mellin–Barnes transform

We consider the Mellin-Barnes (MB) transform of the triangle ladder-like scalar diagram in d = 4 dimensions. It is shown how the multi-fold MB transform of the momentum integral corresponding to an arbitrary number of rungs is reduced to the two-fold MB transform. For this purpose, we use the Belokurov-Usyukina reduction method for four-dimensional scalar integrals in position space. The result is represented in terms of the Euler psi function and its derivatives. We derive new formulas for the MB two-fold integration in the complex planes of two complex variables. We demonstrate that these formulas solve the Bethe-Salpeter equation. We comment on further applications of the solution to the Bethe-Salpeter equation for the vertices in N = 4 supersymmetric Yang-Mills theory. We show that the recursive property of the MB transforms observed in the present work for that kind of diagrams has nothing to do with quantum field theory, the theory of integral transforms, or the theory of polylogarithms in general, but has its origin in a simple recursive property of smooth functions, which may be shown by using basic methods of mathematical analysis

Some Thoughts on Geometries and on the Nature of the Gravitational Field

This paper shows how a gravitational eld generated by a given energy-momentum distribution (for all realistic cases) can be represented by distinct geometrical structures (Lorentzian, teleparallel, and nonnull nonmetricity spacetimes) or that we even can dis-pense all those geometrical structures and simply represent the gravitational eld as a eld in Faradays sense living in Minkowski spacetime. The explicit Lagrangian density for this theory is given, and the eld equations (which are Maxwells like equations) are shown to be equivalent to Einsteins equations. Some examples are worked in detail in order to convince the reader that the geometrical structure of a manifold (modulus some topological constraints) is conventional as already emphasized by Poincare long ago, and thus the realization that there are distinct geometrical representations (and a physical model related to a deformation of the continuum supporting Minkowski spacetime) for any realistic gravitational eld strongly suggests that we must investigate the origin of its physical nature. We hope that this paper will convince readers that this is indeed the case.

The effective lorentzian and teleparallel spacetimes generated by a free electromagnetic field

In this paper we show that a free electromagnetic field living in Minkowski spacetime generates an effective Weitzenbock or an effective Lorentzian spacetime whose properties are determined in details. These results are possible because using the Clifford bundle formalism we found a noticeable result that the energy-momentum densities of a free electromagnetic field are sources of the Hodge duals of exact 2-form fields which satisfy Maxwell like equations.

Two-fold Mellin–Barnes transforms of Usyukina–Davydychev functions

Abstract In our previous paper (Allendes et al., 2013 [10] ), we showed that multi-fold Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions may be reduced to two-fold MB transforms. The MB transforms were written there as polynomials of logarithms of ratios of squares of the external momenta with certain coefficients. We also showed that these coefficients have a combinatoric origin. In this paper, we present an explicit formula for these coefficients. The procedure of recovering the coefficients is based on taking the double-uniform limit in certain series of smooth functions of two variables which is constructed according to a pre-determined iterative way. The result is obtained by using basic methods of mathematical analysis. We observe that the finiteness of the limit of this iterative chain of smooth functions should reflect itself in other mathematical constructions, too, since it is not related in any way to the explicit form of the MB transforms. This finite double-uniform limit is represented in terms of a differential operator with respect to an auxiliary parameter which acts on the integrand of a certain two-fold MB integral. To demonstrate that our result is compatible with original representations of UD functions, we reproduce the integrands of these original integral representations by applying this differential operator to the integrand of the simple integral representation of the scalar triangle four-dimensional integral J ( 1 , 1 , 1 − e ) .

Superposition principle and the problem of additivity of the energies and momenta of distinct electromagnetic fields

In this paper we prove in a rigorous mathematical way (using the Clifford bundle formalism) that the energies and momenta of two distinct and arbitrary free Maxwell fields (of finite energies and momenta) that are superposed are additive and thus that there is no incompatibility between the principle of superposition of fields and the principle of energy-momentum conservation, contrary to some recent claims. Our proof depends on a noticeable formula for the energy-momentum densities, namely, Riesz formula ⋆T a = 1 ⋆ (F θ a ˜ F), which is valid for any electromagnetic field configuration F satisfying Maxwell equation @F= 0.

Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1

Abstract In this paper, we proceed to study properties of Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions. In our previous papers (Allendes et al., 2013 [13] , Kniehl et al., 2013 [14] ), we showed that multi-fold Mellin–Barnes (MB) transforms of Usyukina–Davydychev (UD) functions may be reduced to two-fold MB transforms and that the higher-order UD functions may be obtained in terms of a differential operator by applying it to a slightly modified first UD function. The result is valid in d = 4 dimensions, and its analog in d = 4 − 2 e dimensions exits, too (Gonzalez and Kondrashuk, 2013 [6] ). In Allendes et al. (2013) [13] , the chain of recurrence relations for analytically regularized UD functions was obtained implicitly by comparing the left-hand side and the right-hand side of the diagrammatic relations between the diagrams with different loop orders. In turn, these diagrammatic relations were obtained using the method of loop reduction for the triangle ladder diagrams proposed in 1983 by Belokurov and Usyukina. Here, we reproduce these recurrence relations by calculating explicitly, via Barnes lemmas, the contour integrals produced by the left-hand sides of the diagrammatic relations. In this a way, we explicitly calculate a family of multi-fold contour integrals of certain ratios of Euler gamma functions. We make a conjecture that similar results for the contour integrals are valid for a wider family of smooth functions, which includes the MB transforms of UD functions.

A Clifford Bundle Approach to the Wave Equation of a Spin 1/2 Fermion in the de Sitter Manifold

In this paper we give a Clifford bundle motivated approach to the wave equation of a free spin 1/2 fermion in the de Sitter manifold, a brane with topology

Multi-fold contour integrals of certain ratios of Euler gamma functions from Feynman diagrams: orthogonality of triangles

{M=\mathrm {S0}(4,1)/\mathrm {S0}(3,1)}