252 research outputs found
On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order
We review recently developed new powerful techniques to compute a class of
Feynman diagrams at any loop order, known as sunrise-type diagrams. These
sunrise-type topologies have many important applications in many different
fields of physics and we believe it to be timely to discuss their evaluation
from a unified point of view. The method is based on the analysis of the
diagrams directly in configuration space which, in the case of the sunrise-type
diagrams and diagrams related to them, leads to enormous simplifications as
compared to the traditional evaluation of loops in momentum space. We present
explicit formulae for their analytical evaluation for arbitrary mass
configurations and arbitrary dimensions at any loop order. We discuss several
limiting cases of their kinematical regimes which are e.g. relevant for
applications in HQET and NRQCD.Comment: 100 pages, 16 eps-figures include
The fractional orthogonal derivative
This paper builds on the notion of the so-called orthogonal derivative, where
an n-th order derivative is approximated by an integral involving an orthogonal
polynomial of degree n. This notion was reviewed in great detail in a paper in
J. Approx. Theory (2012) by the author and Koornwinder. Here an approximation
of the Weyl or Riemann-Liouville fractional derivative is considered by
replacing the n-th derivative by its approximation in the formula for the
fractional derivative. In the case of, for instance, Jacobi polynomials an
explicit formula for the kernel of this approximate fractional derivative can
be given. Next we consider the fractional derivative as a filter and compute
the transfer function in the continuous case for the Jacobi polynomials and in
the discrete case for the Hahn polynomials. The transfer function in the Jacobi
case is a confluent hypergeometric function. A different approach is discussed
which starts with this explicit transfer function and then obtains the
approximate fractional derivative by taking the inverse Fourier transform. The
theory is finally illustrated with an application of a fractional
differentiating filter. In particular, graphs are presented of the absolute
value of the modulus of the transfer function. These make clear that for a good
insight in the behavior of a fractional differentiating filter one has to look
for the modulus of its transfer function in a log-log plot, rather than for
plots in the time domain.Comment: 32 pages, 7 figures. The section between formula (4.15) and (4.20) is
correcte
A probabilistic interpretation of a sequence related to Narayana polynomials
A sequence of coefficients appearing in a recurrence for the Narayana
polynomials is generalized. The coefficients are given a probabilistic
interpretation in terms of beta distributed random variables. The recurrence
established by M. Lasalle is then obtained from a classical convolution
identity. Some arithmetical properties of the generalized coefficients are also
established
Comment on ‘Analytical results for a Bessel function times Legendre polynomials class integrals’
A result is obtained, stemming from Gegenbauer, where the products of certain Bessel functions and exponentials are expressed in terms of an infinite series of spherical Bessel functions and products of associated Legendre functions. Closed form solutions for integrals involving Bessel functions times associated Legendre functions times exponentials, recently elucidated by Neves et al(J. Phys. A: Math. Gen. 39 L293), are then shown to result directly from the orthogonality properties of the associated Legendre functions. This result offers greater flexibility in the treatment of classical Heisenberg chains and may do
so in other problems such as occur in electromagnetic diffraction theory
Skew-orthogonal polynomials in the complex plane and their Bergman-like kernels
Non-Hermitian random matrices with symplectic symmetry provide examples for
Pfaffian point processes in the complex plane. These point processes are
characterised by a matrix valued kernel of skew-orthogonal polynomials. We
develop their theory in providing an explicit construction of skew-orthogonal
polynomials in terms of orthogonal polynomials that satisfy a three-term
recurrence relation, for general weight functions in the complex plane. New
examples for symplectic ensembles are provided, based on recent developments in
orthogonal polynomials on planar domains or curves in the complex plane.
Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre
polynomials are derived, from which the conjectured universality of the
elliptic symplectic Ginibre ensemble and its chiral partner follow in the limit
of strong non-Hermiticity at the origin. A Christoffel perturbation of
skew-orthogonal polynomials as it appears in applications to quantum field
theory is provided.Comment: 33 pages; v2: uniqueness of odd polynomials clarified, minor
correction
Group averaging in the (p,q) oscillator representation of SL(2,R)
We investigate refined algebraic quantisation with group averaging in a
finite-dimensional constrained Hamiltonian system that provides a simplified
model of general relativity. The classical theory has gauge group SL(2,R) and a
distinguished o(p,q) observable algebra. The gauge group of the quantum theory
is the double cover of SL(2,R), and its representation on the auxiliary Hilbert
space is isomorphic to the (p,q) oscillator representation. When p>1, q>1 and
p+q == 0 (mod 2), we obtain a physical Hilbert space with a nontrivial
representation of the o(p,q) quantum observable algebra. For p=q=1, the system
provides the first example known to us where group averaging converges to an
indefinite sesquilinear form.Comment: 34 pages. LaTeX with amsfonts, amsmath, amssymb. (References added;
minor typos corrected.
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