4,152 research outputs found
Strange Attractors in Dissipative Nambu Mechanics : Classical and Quantum Aspects
We extend the framework of Nambu-Hamiltonian Mechanics to include dissipation
in phase space. We demonstrate that it accommodates the phase space
dynamics of low dimensional dissipative systems such as the much studied Lorenz
and R\"{o}ssler Strange attractors, as well as the more recent constructions of
Chen and Leipnik-Newton. The rotational, volume preserving part of the flow
preserves in time a family of two intersecting surfaces, the so called {\em
Nambu Hamiltonians}. They foliate the entire phase space and are, in turn,
deformed in time by Dissipation which represents their irrotational part of the
flow. It is given by the gradient of a scalar function and is responsible for
the emergence of the Strange Attractors.
Based on our recent work on Quantum Nambu Mechanics, we provide an explicit
quantization of the Lorenz attractor through the introduction of
Non-commutative phase space coordinates as Hermitian matrices in
. They satisfy the commutation relations induced by one of the two
Nambu Hamiltonians, the second one generating a unique time evolution.
Dissipation is incorporated quantum mechanically in a self-consistent way
having the correct classical limit without the introduction of external degrees
of freedom. Due to its volume phase space contraction it violates the quantum
commutation relations. We demonstrate that the Heisenberg-Nambu evolution
equations for the Quantum Lorenz system give rise to an attracting ellipsoid in
the dimensional phase space.Comment: 35 pages, 4 figures, LaTe
The Berry Phase and Monopoles in Non-Abelian Gauge Theories
We consider the quantum mechanical notion of the geometrical (Berry) phase in
SU(2) gauge theory, both in the continuum and on the lattice. It is shown that
in the coherent state basis eigenvalues of the Wilson loop operator naturally
decompose into the geometrical and dynamical phase factors. Moreover, for each
Wilson loop there is a unique choice of U(1) gauge rotations which do not
change the value of the Berry phase. Determining this U(1) locally in terms of
infinitesimal Wilson loops we define monopole-like defects and study their
properties in numerical simulations on the lattice. The construction is gauge
dependent, as is common for all known definitions of monopoles. We argue that
for physical applications the use of the Lorenz gauge is most appropriate. And,
indeed, the constructed monopoles have the correct continuum limit in this
gauge. Physical consequences are briefly discussed.Comment: 18 pp., Latex2e, 4 figures, psfig.st
A longitudinal gauge degree of freedom and the Pais Uhlenbeck field
We show that a longitudinal gauge degree of freedom for a vector field is
equivalent to a Pais-Uhlenbeck scalar field. With the help of this equivalence,
we can determine natural interactions of this field with scalars and fermions.
Since the theory has a global U(1) symmetry, we have the usual conserved
current of the charged fields, thanks to which the dynamics of the scalar field
is not modified by the interactions. We use this fact to consistently quantize
the theory even in the presence of interactions. We argue that such a degree of
freedom can only be excited by gravitational effects like the inflationary era
of the early universe and may play the role of dark energy in the form of an
effective cosmological constant whose value is linked to the inflation scale.Comment: 20 pages, no figures. Minor changes and comments added to match the
accepted version in JHE
Energetics, skeletal dynamics and long-term predictions in Kolmogorov-Lorenz systems
We study a particular return map for a class of low dimensional chaotic
models called Kolmogorov Lorenz systems, which received an elegant general
Hamiltonian description and includes also the famous Lorenz63 case, from the
viewpoint of energy and Casimir balance. In particular it is considered in
detail a subclass of these models, precisely those obtained from the Lorenz63
by a small perturbation on the standard parameters, which includes for example
the forced Lorenz case in Ref.[6]. The paper is divided into two parts. In the
first part the extremes of the mentioned state functions are considered, which
define an invariant manifold, used to construct an appropriate Poincare surface
for our return map. From the experimental observation of the simple orbital
motion around the two unstable fixed points, together with the circumstance
that these orbits are classified by their energy or Casimir maximum, we
construct a conceptually simple skeletal dynamics valid within our sub class,
reproducing quite well the Lorenz map for Casimir. This energetic approach
sheds some light on the physical mechanism underlying regime transitions. The
second part of the paper is devoted to the investigation of a new type of
maximum energy based long term predictions, by which the knowledge of a
particular maximum energy shell amounts to the knowledge of the future
(qualitative) behaviour of the system. It is shown that, in this respect, a
local analysis of predictability is not appropriate for a complete
characterization of this behaviour. A perspective on the possible extensions of
this type of predictability analysis to more realistic cases in (geo)fluid
dynamics is discussed at the end of the paper.Comment: 21 pages, 14 figure
Regularization as quantization in reducible representations of CCR
A covariant quantization scheme employing reducible representations of
canonical commutation relations with positive-definite metric and Hermitian
four-potentials is tested on the example of quantum electrodynamic fields
produced by a classical current. The scheme implies a modified but very
physically looking Hamiltonian. We solve Heisenberg equations of motion and
compute photon statistics. Poisson statistics naturally occurs and no infrared
divergence is found even for pointlike sources. Classical fields produced by
classical sources can be obtained if one computes coherent-state averages of
Heisenberg-picture operators. It is shown that the new form of representation
automatically smears out pointlike currents. We discuss in detail Poincar\'e
covariance of the theory and the role of Bogoliubov transformations for the
issue of gauge invariance. The representation we employ is parametrized by a
number that is related to R\'enyi's . It is shown that the ``Shannon
limit" plays here a role of correspondence principle with the
standard regularized formalism.Comment: minor extensions, version submitted for publicatio
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