204 research outputs found
Differential Calculi on Associative Algebras and Integrable Systems
After an introduction to some aspects of bidifferential calculus on
associative algebras, we focus on the notion of a "symmetry" of a generalized
zero curvature equation and derive Backlund and (forward, backward and binary)
Darboux transformations from it. We also recall a matrix version of the binary
Darboux transformation and, inspired by the so-called Cauchy matrix approach,
present an infinite system of equations solved by it. Finally, we sketch recent
work on a deformation of the matrix binary Darboux transformation in
bidifferential calculus, leading to a treatment of integrable equations with
sources.Comment: 19 pages, to appear in "Algebraic Structures and Applications", S.
Silvestrov et al (eds.), Springer Proceedings in Mathematics & Statistics,
202
Soliton equations and the zero curvature condition in noncommutative geometry
Familiar nonlinear and in particular soliton equations arise as zero
curvature conditions for GL(1,R) connections with noncommutative differential
calculi. The Burgers equation is formulated in this way and the Cole-Hopf
transformation for it attains the interpretation of a transformation of the
connection to a pure gauge in this mathematical framework. The KdV, modified
KdV equation and the Miura transformation are obtained jointly in a similar
setting and a rather straightforward generalization leads to the KP and a
modified KP equation.
Furthermore, a differential calculus associated with the Boussinesq equation
is derived from the KP calculus.Comment: Latex, 10 page
Dynamical Evolution in Noncommutative Discrete Phase Space and the Derivation of Classical Kinetic Equations
By considering a lattice model of extended phase space, and using techniques
of noncommutative differential geometry, we are led to: (a) the conception of
vector fields as generators of motion and transition probability distributions
on the lattice; (b) the emergence of the time direction on the basis of the
encoding of probabilities in the lattice structure; (c) the general
prescription for the observables' evolution in analogy with classical dynamics.
We show that, in the limit of a continuous description, these results lead to
the time evolution of observables in terms of (the adjoint of) generalized
Fokker-Planck equations having: (1) a diffusion coefficient given by the limit
of the correlation matrix of the lattice coordinates with respect to the
probability distribution associated with the generator of motion; (2) a drift
term given by the microscopic average of the dynamical equations in the present
context. These results are applied to 1D and 2D problems. Specifically, we
derive: (I) The equations of diffusion, Smoluchowski and Fokker-Planck in
velocity space, thus indicating the way random walk models are incorporated in
the present context; (II) Kramers' equation, by further assuming that, motion
is deterministic in coordinate spaceComment: LaTeX2e, 40 pages, 1 Postscript figure, uses package epsfi
Integrability and chemical potential in the (3+1)-dimensional Skyrme model
Using a remarkable mapping from the original (3+1)dimensional Skyrme model to
the Sine-Gordon model, we construct the first analytic examples of Skyrmions as
well as of Skyrmions--anti-Skyrmions bound states within a finite box in 3+1
dimensional flat space-time. An analytic upper bound on the number of these
Skyrmions--anti-Skyrmions bound states is derived. We compute the critical
isospin chemical potential beyond which these Skyrmions cease to exist. With
these tools, we also construct topologically protected time-crystals:
time-periodic configurations whose time-dependence is protected by their
non-trivial winding number. These are striking realizations of the ideas of
Shapere and Wilczek. The critical isospin chemical potential for these
time-crystals is determined.Comment: 15 pages; 1 figure; a discussion on the closeness to the topological
bound as well as some clarifying comments on the semi-classical quantization
have been included. Relevant references have been added. Version accepted for
publication on Physics Letters
Pythagoras' Theorem on a 2D-Lattice from a "Natural" Dirac Operator and Connes' Distance Formula
One of the key ingredients of A. Connes' noncommutative geometry is a
generalized Dirac operator which induces a metric(Connes' distance) on the
state space. We generalize such a Dirac operator devised by A. Dimakis et al,
whose Connes' distance recovers the linear distance on a 1D lattice, into 2D
lattice. This Dirac operator being "naturally" defined has the so-called "local
eigenvalue property" and induces Euclidean distance on this 2D lattice. This
kind of Dirac operator can be generalized into any higher dimensional lattices.Comment: Latex 11pages, no figure
Noncommutative Geometry of Finite Groups
A finite set can be supplied with a group structure which can then be used to
select (classes of) differential calculi on it via the notions of left-, right-
and bicovariance. A corresponding framework has been developed by Woronowicz,
more generally for Hopf algebras including quantum groups. A differential
calculus is regarded as the most basic structure needed for the introduction of
further geometric notions like linear connections and, moreover, for the
formulation of field theories and dynamics on finite sets. Associated with each
bicovariant first order differential calculus on a finite group is a braid
operator which plays an important role for the construction of distinguished
geometric structures. For a covariant calculus, there are notions of invariance
for linear connections and tensors. All these concepts are explored for finite
groups and illustrated with examples. Some results are formulated more
generally for arbitrary associative (Hopf) algebras. In particular, the problem
of extension of a connection on a bimodule (over an associative algebra) to
tensor products is investigated, leading to the class of `extensible
connections'. It is shown that invariance properties of an extensible
connection on a bimodule over a Hopf algebra are carried over to the extension.
Furthermore, an invariance property of a connection is also shared by a `dual
connection' which exists on the dual bimodule (as defined in this work).Comment: 34 pages, Late
Bi-differential calculi and integrable models
The existence of an infinite set of conserved currents in completely
integrable classical models, including chiral and Toda models as well as the KP
and self-dual Yang-Mills equations, is traced back to a simple construction of
an infinite chain of closed (respectively, covariantly constant) 1-forms in a
(gauged) bi-differential calculus. The latter consists of a differential
algebra on which two differential maps act. In a gauged bi-differential
calculus these maps are extended to flat covariant derivatives.Comment: 24 pages, 2 figures, uses amssymb.sty and diagrams.sty, substantial
extensions of examples (relative to first version
Non-commutative Geometry and Kinetic Theory of Open Systems
The basic mathematical assumptions for autonomous linear kinetic equations
for a classical system are formulated, leading to the conclusion that if they
are differential equations on its phase space , they are at most of the 2nd
order. For open systems interacting with a bath at canonical equilibrium they
have a particular form of an equation of a generalized Fokker-Planck type. We
show that it is possible to obtain them as Liouville equations of Hamiltonian
dynamics on with a particular non-commutative differential structure,
provided certain geometric in character, conditions are fulfilled. To this end,
symplectic geometry on is developped in this context, and an outline of the
required tensor analysis and differential geometry is given. Certain questions
for the possible mathematical interpretation of this structure are also
discussed.Comment: 22 pages, LaTe
Abelian Toda field theories on the noncommutative plane
Generalizations of GL(n) abelian Toda and abelian affine
Toda field theories to the noncommutative plane are constructed. Our proposal
relies on the noncommutative extension of a zero-curvature condition satisfied
by algebra-valued gauge potentials dependent on the fields. This condition can
be expressed as noncommutative Leznov-Saveliev equations which make possible to
define the noncommutative generalizations as systems of second order
differential equations, with an infinite chain of conserved currents. The
actions corresponding to these field theories are also provided. The special
cases of GL(2) Liouville and sinh/sine-Gordon are
explicitly studied. It is also shown that from the noncommutative
(anti-)self-dual Yang-Mills equations in four dimensions it is possible to
obtain by dimensional reduction the equations of motion of the two-dimensional
models constructed. This fact supports the validity of the noncommutative
version of the Ward conjecture. The relation of our proposal to previous
versions of some specific Toda field theories reported in the literature is
presented as well.Comment: v3 30 pages, changes in the text, new sections included and
references adde
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