20 research outputs found
A Maxwell Like Formulation of Gravitational Theory in Minkowski Spacetime
In this paper using the Clifford bundle formalism a Lagrangian theory of the
Yang-Mills type (with a gauge fixing term and an auto interacting term) for the
gravitational field in Minkowski spacetime is presented. It is shown how two
simple hypothesis permit the interpretation of the formalism in terms of
effective Lorentzian or teleparallel geometries. In the case of a Lorentzian
geometry interpretation of the theory the field equations are shown to be
equivalent to Einstein's equations.Comment: This is a version of a paper published in Int. J. Mod. Phs. D 16(6),
1027-1041 (2007) where some misprints and typos have been corrected, some
references have been updated, a footnote has been added and some few
sentences have been rewritten to better explain the role of the (plastic)
deformation tensor
Explicit calculation of multi-fold contour integrals of certain ratios of Euler gamma functions. Part 1
In this paper we proceed to study properties of Mellin-Barnes (MB) transforms
of Usyukina-Davydychev (UD) functions. In our previous papers [Nuclear Physics
B 870 (2013) 243], [Nuclear Physics B 876 (2013) 322] 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 were
obtained in terms of a differential operator by applying it to a slightly
modified first UD function. The result is valid in dimensions and its
analog in dimensions exits too [Theoretical and Mathematical
Physics 177 (2013) 1515]. In [Nuclear Physics B 870 (2013) 243] the chain of
recurrent 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 due to the method of loop
reduction for the triangle ladder diagrams proposed in 1983 by Belokurov and
Usyukina. Here we reproduce these recurrent relations by calculating explicitly
via Barnes lemmas the contour integrals produced by the left hand sides of the
diagrammatic relations. In such 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.Comment: 7 pages, 1 figure, minor changes in the text; accepted for
publication in Nuclear Physics
New four-dimensional integrals by Mellin-Barnes transform
This paper is devoted to the calculation by Mellin-Barnes transform of a
especial class of integrals. It contains double integrals in the position space
in d = 4-2e dimensions, where e is parameter of dimensional regularization.
These integrals contribute to the effective action of the N = 4 supersymmetric
Yang-Mills theory. The integrand is a fraction in which the numerator is a
logarithm of ratio of spacetime intervals, and the denominator is the product
of powers of spacetime intervals. According to the method developed in the
previous papers, in order to make use of the uniqueness technique for one of
two integrations, we shift exponents in powers in the denominator of integrands
by some multiples of e. As the next step, the second integration in the
position space is done by Mellin-Barnes transform. For normalizing procedure,
we reproduce first the known result obtained earlier by Gegenbauer polynomial
technique. Then, we make another shift of exponents in powers in the
denominator to create the logarithm in the numerator as the derivative with
respect to the shift parameter delta. We show that the technique of work with
the contour of the integral modified in this way by using Mellin-Barnes
transform repeats the technique of work with the contour of the integral
without such a modification. In particular, all the operations with a shift of
contour of integration over complex variables of two-fold Mellin-Barnes
transform are the same as before the delta modification of indices, and even
the poles of residues coincide. This confirms the observation made in the
previous papers that in the position space all the Green function of N = 4
supersymmetric Yang-Mills theory can be expressed in terms of UD functions.Comment: Talk at El Congreso de Matematica Capricornio, COMCA 2009,
Antofagasta, Chile and at DMFA seminar, UCSC, Concepcion, Chile, 24 pages;
revised version, Introduction is modified, Conclusion is added, five
Appendices are added, Appendix E is ne
Solution to Bethe-Salpeter equation via Mellin-Barnes transform
We consider Mellin-Barnes transform of 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 Belokurov-Usyukina reduction
method for four-dimensional scalar integrals in the position space. The result
is represented in terms of Euler psi-function and its derivatives. We derive
new formulas for the MB two-fold integration in complex planes of two complex
variables. We demonstrate that these formulas solve 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, theory of
integral transforms, or with theory of polylogarithms in general, but has an
origin in a simple recursive property for smooth functions which can be shown
by using basic methods of mathematical analysis.Comment: 33 pages, 9 figures, revised version, all the factors and symbols of
integration are written explcitly, symbols of proportionality are removed,
section 2 is extended, section 6.3 is shortene
Hodge operator and asymmetric fluid in unbounded domains
ABSTRACT
A system of equations modeling the stationary flow of an incompressible asymmetric fluid is studied for bounded domains of an arbitrary form. Based on the methods of Clifford analysis, we write the system of asymmetric fluid in the hypercomplex formulation and represent its solution in Clifford operator terms. We have significantly used Clifford algebra, and in particular the Hodge operator of the Clifford algebra to demonstrate the existence and uniqueness of the strong solution for arbitrary unbounded domains.