107 research outputs found
No Drama Quantum Theory?
This work builds on the following result of a previous article
(quant-ph/0509044): the matter field can be naturally eliminated from the
equations of the scalar electrodynamics (the Klein-Gordon-Maxwell
electrodynamics) in the unitary gauge. The resulting equations describe
independent dynamics of the electromagnetic field (they form a closed system of
partial differential equations). An improved derivation of this surprising
result is offered in the current work. It is also shown that for this system of
equations, a generalized Carleman linearization (Carleman embedding) procedure
generates a system of linear equations in the Hilbert space, which looks like a
second-quantized theory and is equivalent to the original nonlinear system on
the set of solutions of the latter. Thus, the relevant local realistic model
can be embedded into a quantum field theory. This model is equivalent to a
well-established model - the scalar electrodynamics, so it correctly describes
a large body of experimental data. Although it does not describe the electronic
spin and possibly some other experimental facts, it may be of great interest as
a "no drama quantum theory", as simple (in principle) as classical
electrodynamics. Possible issues with the Bell theorem are discussed.Comment: 4 page
A simple piston problem in one dimension
We study a heavy piston that separates finitely many ideal gas particles
moving inside a one-dimensional gas chamber. Using averaging techniques, we
prove precise rates of convergence of the actual motions of the piston to its
averaged behavior. The convergence is uniform over all initial conditions in a
compact set. The results extend earlier work by Sinai and Neishtadt, who
determined that the averaged behavior is periodic oscillation. In addition, we
investigate the piston system when the particle interactions have been
smoothed. The convergence to the averaged behavior again takes place uniformly,
both over initial conditions and over the amount of smoothing.Comment: Accepted by Nonlinearity. 27 pages, 2 figure
Normal origamis of Mumford curves
An origami (also known as square-tiled surface) is a Riemann surface covering
a torus with at most one branch point. Lifting two generators of the
fundamental group of the punctured torus decomposes the surface into finitely
many unit squares. By varying the complex structure of the torus one obtains
easily accessible examples of Teichm\"uller curves in the moduli space of
Riemann surfaces. The p-adic analogues of Riemann surfaces are Mumford curves.
A p-adic origami is defined as a covering of Mumford curves with at most one
branch point, where the bottom curve has genus one. A classification of all
normal non-trivial p-adic origamis is presented and used to calculate some
invariants. These can be used to describe p-adic origamis in terms of glueing
squares.Comment: 21 pages, to appear in manuscripta mathematica (Springer
An Adiabatic Theorem without a Gap Condition
The basic adiabatic theorems of classical and quantum mechanics are
over-viewed and an adiabatic theorem in quantum mechanics without a gap
condition is described.Comment: Talk at QMath 7, Prague, 1998. 10 pages, 7 figure
On the connection between the Nekhoroshev theorem and Arnold Diffusion
The analytical techniques of the Nekhoroshev theorem are used to provide
estimates on the coefficient of Arnold diffusion along a particular resonance
in the Hamiltonian model of Froeschl\'{e} et al. (2000). A resonant normal form
is constructed by a computer program and the size of its remainder
at the optimal order of normalization is calculated as a function
of the small parameter . We find that the diffusion coefficient
scales as , while the size of the optimal remainder
scales as in the range
. A comparison is made with the numerical
results of Lega et al. (2003) in the same model.Comment: Accepted in Celestial Mechanics and Dynamical Astronom
Normal Form and Nekhoroshev stability for nearly-integrable Hamiltonian systems with unconditionally slow aperiodic time dependence
The aim of this paper is to extend the results of Giorgilli and Zehnder for
aperiodic time dependent systems to a case of general nearly-integrable convex
analytic Hamiltonians. The existence of a normal form and then a stability
result are shown in the case of a slow aperiodic time dependence that, under
some smallness conditions, is independent on the size of the perturbation.Comment: Corrected typo in the title and statement of Lemma 3.
Square-tiled cyclic covers
A cyclic cover of the complex projective line branched at four appropriate
points has a natural structure of a square-tiled surface. We describe the
combinatorics of such a square-tiled surface, the geometry of the corresponding
Teichm\"uller curve, and compute the Lyapunov exponents of the determinant
bundle over the Teichm\"uller curve with respect to the geodesic flow. This
paper includes a new example (announced by G. Forni and C. Matheus in
\cite{Forni:Matheus}) of a Teichm\"uller curve of a square-tiled cyclic cover
in a stratum of Abelian differentials in genus four with a maximally degenerate
Kontsevich--Zorich spectrum (the only known example found previously by Forni
in genus three also corresponds to a square-tiled cyclic cover
\cite{ForniSurvey}).
We present several new examples of Teichm\"uller curves in strata of
holomorphic and meromorphic quadratic differentials with maximally degenerate
Kontsevich--Zorich spectrum. Presumably, these examples cover all possible
Teichm\"uller curves with maximally degenerate spectrum. We prove that this is
indeed the case within the class of square-tiled cyclic covers.Comment: 34 pages, 6 figures. Final version incorporating referees comments.
In particular, a gap in the previous version was corrected. This file uses
the journal's class file (jmd.cls), so that it is very similar to published
versio
Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio
We study the exponentially small splitting of invariant manifolds of
whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable
Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a
torus whose frequency ratio is the silver number . We show
that the Poincar\'e-Melnikov method can be applied to establish the existence
of 4 transverse homoclinic orbits to the whiskered torus, and provide
asymptotic estimates for the tranversality of the splitting whose dependence on
the perturbation parameter satisfies a periodicity property. We
also prove the continuation of the transversality of the homoclinic orbits for
all the sufficiently small values of , generalizing the results
previously known for the golden number.Comment: 17 pages, 2 figure
Gravitational Ionization: A Chaotic Net in the Kepler System
The long term nonlinear dynamics of a Keplerian binary system under the
combined influences of gravitational radiation damping and external tidal
perturbations is analyzed. Gravitational radiation reaction leads the binary
system towards eventual collapse, while the external periodic perturbations
could lead to the ionization of the system via Arnold diffusion. When these two
opposing tendencies nearly balance each other, interesting chaotic behavior
occurs that is briefly studied in this paper. It is possible to show that
periodic orbits can exist in this system for sufficiently small damping.
Moreover, we employ the method of averaging to investigate the phenomenon of
capture into resonance.Comment: REVTEX Style, Submitte
Time-averaging for weakly nonlinear CGL equations with arbitrary potentials
Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the
form: under the periodic boundary conditions, where and
is a smooth function. Let be
the -basis formed by eigenfunctions of the operator . For a
complex function , write it as and
set . Then for any solution of the linear
equation we have . In this work it is
proved that if equation with a sufficiently smooth real potential
is well posed on time-intervals , then for any its
solution , the limiting behavior of the curve
on time intervals of order , as
, can be uniquely characterized by a solution of a certain
well-posed effective equation:
where is a resonant averaging of the nonlinearity . We
also prove a similar results for the stochastically perturbed equation, when a
white in time and smooth in random force of order is added
to the right-hand side of the equation.
The approach of this work is rather general. In particular, it applies to
equations in bounded domains in under Dirichlet boundary conditions
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