Search results for "Linear Algebra."
showing 10 items of 552 documents
Bootstrapping pentagon functions
2018
In PRL 116 (2016) no.6, 062001, the space of planar pentagon functions that describes all two-loop on-shell five-particle scattering amplitudes was introduced. In the present paper we present a natural extension of this space to non-planar pentagon functions. This provides the basis for our pentagon bootstrap program. We classify the relevant functions up to weight four, which is relevant for two-loop scattering amplitudes. We constrain the first entry of the symbol of the functions using information on branch cuts. Drawing on an analogy from the planar case, we introduce a conjectural second-entry condition on the symbol. We then show that the information on the function space, when comple…
Modular transformations of elliptic Feynman integrals
2021
We investigate the behaviour of elliptic Feynman integrals under modular transformations. This has a practical motivation: Through a suitable modular transformation we can achieve that the nome squared is a small quantity, leading to fast numerical evaluations. Contrary to the case of multiple polylogarithms, where it is sufficient to consider just variable transformations for the numerical evaluations of multiple polylogarithms, it is more natural in the elliptic case to consider a combination of a variable transformation (i.e. a modular transformation) together with a redefinition of the master integrals. Thus we combine a coordinate transformation on the base manifold with a basis transf…
Construction of the ground state in nonrelativistic QED by continuous flows
2006
AbstractFor a nonrelativistic hydrogen atom minimally coupled to the quantized radiation field we construct the ground state projection Pgs by a continuous approximation scheme as an alternative to the iteration scheme recently used by Fröhlich, Pizzo, and the first author [V. Bach, J. Fröhlich, A. Pizzo, Infrared-finite algorithms in QED: The groundstate of an atom interacting with the quantized radiation field, Comm. Math. Phys. (2006), doi: 10.1007/s00220-005-1478-3]. That is, we construct Pgs=limt→∞Pt as the limit of a continuously differentiable family (Pt)t⩾0 of ground state projections of infrared regularized Hamiltonians Ht. Using the ODE solved by this family of projections, we sho…
Physical interpretation of laser phase dynamics
1990
The basic features characterizing the dynamical evolution of the phase of a detuned-laser field under an unstable regime are physically interpreted in terms of dispersive and dynamical effects. A general method for obtaining any attractor projection containing the phase information is established, which provides evidence for the heteroclinic character of the attractor in the presence of cavity detuning for any emission regime.
gjfactor of an electron bound in a hydrogenlike ion
2000
We present a detailed theoretical evaluation for the ${g}_{j}$ factor of a bound electron in hydrogenlike ions up to $Z=94.$ All quantum electrodynamical corrections of order $(\ensuremath{\alpha}/\ensuremath{\pi})$ are evaluated in detail and various other contributions to the ${g}_{j}$ factor are computed and listed for 61 Z. A comparison with all existing experiments is carried out and excellent agreement is found. The present uncertainty in our calculations is discussed. It is not possible to improve this precision with only minor effort since two-photon bound-state QED terms are uncalculated up to now.
Molecular correlations in a supercooled liquid
1998
We present static and dynamic properties of molecular correlation functions S_{lmn,l'm'n'}(q,t) in a simulated supercooled liquid of water molecules, as a preliminary effort in the direction of solving the molecular mode coupling theory (MMCT) equations for supercooled molecular liquids. The temperature and time dependence of various molecular correlation functions, calculated from 250 ns long molecular dynamics simulations, show the characteristic patterns predicted by MMCT and shed light on the driving mechanism responsible for the slowing down of the molecular dynamics. We also discuss the symmetry properties of the molecular correlation functions which can be predicted on the basis of t…
The zero-point energy for rotation
1978
The Gaussian overlap approach (GOA) becomes inappropriate for describing the rotation of weakly deformed systems. A modification is proposed which allows to maintain the GOA for small deformations. The zero-point energy subtraction, derived from it, provides a simple and reliable approximation for angular momentum projection. It becomes obvious, however, that the projection complicates the equations which determine the motion along the deformation path. These effects are studied in some simple models and the results are condensed into a simple interpolation formula for the total zero-point energy.
A model study of Hartree-Fock and Linear Response in coordinate space
1979
A fast procedure for spherical Hartree-Fock is obtained by coordinate space representation and a modification of gradient iteration. Along similar lines, the corresponding Linear Response equations are derived and solved, in order to achieve a fully consistent treatment. The Linear Response equations are applied to a change in particle numbers, i.e. to the description of isotopic differences. In a model study we look for their physical and numerical properties, i.e. linearity of the response, numerical stability and consistency requirements for the Hartree-Fock basis.
A CRITICAL VIEW ON THE PERTURBATIVE RG METHOD
2012
The perturbative renormalization group (RG) treatment of the Ginzburg–Landau model is reconsidered based on the Feynman diagram technique. We derive RG flow equations, exactly calculating all vertices appearing in the perturbative RG transformation of the φ4 model up to the ε3 order of the ε-expansion. The Fourier-transformed two-point correlation function G(k) has been considered. Although the ε-expansion of X(k) = 1/G(k) is well defined on the critical surface, we have revealed an inconsistency with the exact rescaling of X(k), represented as an expansion in powers of k at k →0. This new result can serve as a basis to challenge the correctness of the ε-expansion-based perturbative RG met…
η–η′ mixing in the flavor basis and large N
2010
Abstract The mass matrix for η – η ′ is derived in the flavor basis at O ( p 4 ) of the chiral Lagrangian using the large N approximation. Under certain assumptions, the mixing angle ϕ = 41.4 ° and the decay constants ratio f K / f π = 1.15 are calculated in agreement with the data. It appears that the FKS scheme arises as a special limit of the chiral Lagrangian. Their mass matrix is obtained without the hypothesis on the mixing pattern of the decay constants.