Cayley Parametrization Research Articles

Adaptive Localized Cayley Parametrization for Optimization Over Stiefel Manifold and Its Convergence Rate Analysis

Adaptive Localized Cayley Parametrization for Optimization Over Stiefel Manifold and Its Convergence Rate Analysis

Generalized left-localized Cayley parametrization for optimization with orthogonality constraints

We present a reformulation of optimization problems over the Stiefel manifold by using a Cayley-type transform, named the generalized left-localized Cayley transform, for the Stiefel manifold. The reformulated optimization problem is defined over a vector space, whereby we can apply directly powerful computational arts designed for optimization over a vector space. The proposed Cayley-type transform enjoys several key properties which are useful to (i) study relations between the original problem and the proposed problem; (ii) check the conditions to guarantee the global convergence of optimization algorithms. Numerical experiments demonstrate that the proposed algorithm outperforms the standard algorithms designed with a retraction on the Stiefel manifold.

Leading order multi-soft behaviors of tree amplitudes in NLSM

In this paper, we investigate multi-soft behaviors of tree amplitudes in nonlinear sigma model (NLSM). The leading behaviors of amplitudes with odd number of all-adjacent soft pions are zero. We further propose and prove that leading soft factors of amplitudes with even number all-adjacent soft pions can be expressed in terms of products of the leading order Berends-Giele sub-currents in Cayley parametrization. Each subcurrent in the expression contains at most one hard pion. Discussions are generalized to amplitudes containing arbitrary number of nonadjacent soft blocks: the leading behaviors of amplitudes where at least one soft block has odd number of adjacent soft pions are zero; the leading soft factors for amplitudes where all soft blocks containing even number of soft pions are given by products of soft factors for these blocks.

Explicit BCJ numerators of nonlinear simga model

In this paper, we investigate the color-kinematics duality in nonlinear sigma model (NLSM). We present explicit polynomial expressions for the kinematic numerators (BCJ numerators). The calculation is done separately in two parametrization schemes of the theory using Kawai-Lewellen-Tye relation inspired technique, both lead to polynomial numerators. We summarize the calculation in each case into a set of rules that generates BCJ numerators for all multilplicities. In Cayley parametrization we find the numerator is described by a particularly simple formula solely in terms of momentum kernel.

Note on off-shell relations in nonlinear sigma model

In this note, we investigate relations between tree-level off-shell currents in nonlinear sigma model. Under Cayley parametrization, all odd-point currents vanish. We propose and prove a generalized U(1) identity for even-point currents. The off-shell U(1) identity given in [1] is a special case of the generalized identity studied in this note. The on-shell limit of this identity is equivalent with the on-shell KK relation. Thus this relation provides the full off-shell correspondence of tree-level KK relation in nonlinear sigma model.

Amplitude relations in non-linear sigma model

Abstract In this paper, we investigate tree-level scattering amplitude relations in U(N) non-linear sigma model. We use Cayley parametrization. As was shown in the recent works [23,24], both on-shell amplitudes and off-shell currents with odd points have to vanish under Cayley parametrization. We prove the off-shell U(1) identity and fundamental BCJ relation for even-point currents. By taking the on-shell limits of the off-shell relations, we show that the color-ordered tree amplitudes with even points satisfy U(1)-decoupling identity and fundamental BCJ relation, which have the same formations within Yang-Mills theory. We further state that all the on-shell general KK, BCJ relations as well as the minimal-basis expansion are also satisfied by color-ordered tree amplitudes. As a consequence of the relations among color-ordered amplitudes, the total 2 m-point tree amplitudes satisfy DDM form of color decomposition as well as KLT relation.

Quantization of multidimensional cat maps

In this work we study cat maps with many degrees of freedom. Classical cat maps are classified using the Cayley parametrization of symplectic matrices and the closely associated centre and chord generating functions. Particular attention is dedicated to loxodromic and elliptic-hyperbolic behaviour, which are new features of four-dimensional maps: we construct a map that is not Anosov, but is ergodic and mixing. The maps are then quantized using a Weyl representation on the torus and the general condition on the Floquet angles is derived for a particular map to be quantizable. The semiclassical approximation is exact, regardless of the dimensionality or of the nature of the fixed points. We single out the study of the quantum period function (QPF), that is the period of the quantum map as a function of the finite Hilbert space dimension. It is found that the QPF depends basically on the ergodicity and on the existence of degenerate Lyapunov exponents for some power of the cat map.

Improving lattice perturbation theory.

Lepage and Mackenzie have shown that tadpole renormalization and systematic improvement of lattice perturbation theory can lead to much improved numerical results in lattice gauge theory. It is shown that lattice perturbation theory using the Cayley parametrization of unitary matrices gives a simple analytical approach to tadpole renormalization, and that the Cayley parametrization gives lattice gauge potentials gauge transformations close to the continuum form. For example, at the lowest order in perturbation theory, for SU(3) lattice gauge theory, at $\beta=6,$ the `tadpole renormalized' coupling $\tilde g^2 = {4\over 3} g^2,$ to be compared to the non-perturbative numerical value $\tilde g^2 = 1.7 g^2.$

Cayley parametrization of semisimple lie groups and its application to physical laws in a (3+1)-dimensional cubic lattice

The assumption of a discrete space-time is expressed mathematically by restricting the space-time variables to the field of integer numbers, and by restricting to the field of rational numbers the functions describing the laws of motion. This rational character must be preserved under the transformations connecting different systems of reference. The Cayley parametrization of semisimple Lie groups, and in particular of the Lorentz group, satisfies this condition if we require these parameters to take only integer values. The rational points of the most frequently used transcendental functions are obtained with the help of the integer complex and hypercomplex numbers. Some applications are made concerning the laws of motion in special relativity defined over a (3+1)-dimensional cubic lattice.