Dynamical equations for the quantum product on a symplectic space in affine coordinates

We derive a system of dynamical equations for an associative noncommutative product of functions on a symplectic space. This system is explicitly solved in semiclassical approximation.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 42–52.Original Russian Text Copyright © 2005 by O. N. Grigorev, M. V. Karasev.This revised version was published online in April 2005 with a corrected issue number.     

Dynamical equations for the quantum product on a symplectic space in affine coordinates

We derive a system of dynamical equations for an associative noncommutative product of functions on a symplectic space. This system is explicitly solved in semiclassical approximation.     

Classification of K3-surfaces with involution and maximal symplectic symmetry

K3-surfaces with antisymplectic involution and compatible symplectic actions of finite groups are considered. In this situation actions of large finite groups of symplectic transformations are shown to arise via double covers of Del Pezzo surfaces, and a complete classification of K3-surfaces with maximal symplectic symmetry is obtained.     

Regular reduction of controlled Hamiltonian system with symplectic structure and symmetry

In this paper, our goal is to study the regular reduction theory of regular controlled Hamiltonian (RCH) systems with symplectic structure and symmetry, and this reduction is an extension of regular symplectic reduction theory of Hamiltonian systems under regular controlled Hamiltonian equivalence conditions. Thus, in order to describe uniformly RCH systems defined on a cotangent bundle and on the regular reduced spaces, we first define a kind of RCH systems on a symplectic fiber bundle. Then we introduce regular point and regular orbit reducible RCH systems with symmetry by using momentum map and the associated reduced symplectic forms. Moreover, we give regular point and regular orbit reduction theorems for RCH systems to explain the relationships between RpCH-equivalence, RoCH-equivalence for reducible RCH systems with symmetry and RCH-equivalence for associated reduced RCH systems. Finally, as an application we regard rigid body and heavy top as well as them with internal rotors as the regular point reducible RCH systems on the rotation group SO(3) and on the Euclidean group SE(3), as well as on their generalizations, respectively, and discuss their RCH-equivalence. We also describe the RCH system and RCH-equivalence from the viewpoint of port Hamiltonian system with a symplectic structure.     

Dynamical symmetry breaking in hyperbolic 4D spacetime and in extra dimensions

We study the dynamical symmetry breaking in quark matter within two different models. First, we consider the effect of gravitational catalysis of chiral and color symmetries breaking in strong gravitational field of ultrastatic hyperbolic spacetime ℝ ⊗ H ³ in the framework of an extended Nambu-Jona-Lasinio model. Second, we discuss the dynamical fermion mass generation in the flat 4-dimensional brane situated in the 5D spacetime with one extra dimension compactified on a circle. In the model, bulk fermions interact with fermions on the brane in the presence of a constant abelian gauge field A ₅ in the bulk. The influence of the A ₅-gauge field on the symmetry breaking is considered both when this field is a background parameter and a dynamical variable.     

Infinitely strong coupling limit in quantum chromodynamics,and symplectic symmetry of quark bound states

The symplectic groupSp(2N _fN_c) is proposed as the internal symmetry group of quark bound states; hereN _f andN _c designate respectively the number of quark flavors and that of color degrees of freedom of the quark. The effective Lagrangian of quark bound states in a quark—gluon system is constructed in the infinitely strong coupling limit. Several consequences of theSp(2N _fN_c)-symmetry in the caseN _f=3 are examined.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 131, pp. 47–71, 1983.The authors are grateful to I. Ya. Aref'eva, M. K. Volkov, G. V. Efimov, L. N. Lipatov, A. A. Migdal, V. N. Pervushin, and V. N. Popov for discussions and critical remarks.     

A molecular dynamics model for symplectic intergrators

Mechanical systems in the very large scale like in celestial mechanics or in the very small scale like in the molecular dynamics can be modelled without dissipation. The resulting Hamiltonian systems possess conservation properties, which are characterized with the term symplecticness, Numerical integration schemes should preserve the symplecticness. Different methods are introduced and their performance is studied for constant and variable step size. As test examples two systems from molecular dynamics are introduced.     

Dynamical phase transition in the simplest molecular chain model

We consider the dynamics of the simplest chain of a large number N of particles. In the double scaling limit, we find the partition of the parameter space into two domains: for one domain, the supremum over the time interval (0,∞) of the relative extension of the chain tends to 1 as N → ∞, and for the other domain, to infinity.