104 research outputs found
Homology class of a Lagrangian Klein bottle
It is shown that an embedded Lagrangian Klein bottle represents a non-trivial
mod 2 homology class in a compact symplectic four-manifold with
. (In versions 1 and 2, the last assumption was missing.
A counterexample to this general claim and the first proof of the corrected
result have been found by Vsevolod Shevchishin.) As a corollary one obtains
that the Klein bottle does not admit a Lagrangian embedding into the standard
symplectic four-space.Comment: Version 3 - completely rewritten to correct a mistake; Version 4 -
minor edits, added references; AMSLaTeX, 6 page
On the Use of Minimum Volume Ellipsoids and Symplectic Capacities for Studying Classical Uncertainties for Joint Position-Momentum Measurements
We study the minimum volume ellipsoid estimator associates to a cloud of
points in phase space. Using as a natural measure of uncertainty the symplectic
capacity of the covariance ellipsoid we find that classical uncertainties obey
relations similar to those found in non-standard quantum mechanics
Quantum Blobs
Quantum blobs are the smallest phase space units of phase space compatible
with the uncertainty principle of quantum mechanics and having the symplectic
group as group of symmetries. Quantum blobs are in a bijective correspondence
with the squeezed coherent states from standard quantum mechanics, of which
they are a phase space picture. This allows us to propose a substitute for
phase space in quantum mechanics. We study the relationship between quantum
blobs with a certain class of level sets defined by Fermi for the purpose of
representing geometrically quantum states.Comment: Prepublication. Dedicated to Basil Hile
On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
Courant theorem provides an upper bound for the number of nodal domains of
eigenfunctions of a wide class of Laplacian-type operators. In particular, it
holds for generic eigenfunctions of quantum graph. The theorem stipulates that,
after ordering the eigenvalues as a non decreasing sequence, the number of
nodal domains of the -th eigenfunction satisfies . Here,
we provide a new interpretation for the Courant nodal deficiency in the case of quantum graphs. It equals the Morse index --- at a
critical point --- of an energy functional on a suitably defined space of graph
partitions. Thus, the nodal deficiency assumes a previously unknown and
profound meaning --- it is the number of unstable directions in the vicinity of
the critical point corresponding to the -th eigenfunction. To demonstrate
this connection, the space of graph partitions and the energy functional are
defined and the corresponding critical partitions are studied in detail.Comment: 22 pages, 6 figure
Symplectic geometry of quantum noise
We discuss a quantum counterpart, in the sense of the Berezin-Toeplitz
quantization, of certain constraints on Poisson brackets coming from "hard"
symplectic geometry. It turns out that they can be interpreted in terms of the
quantum noise of observables and their joint measurements in operational
quantum mechanics. Our findings include various geometric mechanisms of quantum
noise production and a noise-localization uncertainty relation. The methods
involve Floer theory and Poisson bracket invariants originated in function
theory on symplectic manifolds.Comment: Revised version, 57 pages, 3 figures. Incorporates arXiv:1203.234
A beginner's introduction to Fukaya categories
The goal of these notes is to give a short introduction to Fukaya categories
and some of their applications. The first half of the text is devoted to a
brief review of Lagrangian Floer (co)homology and product structures. Then we
introduce the Fukaya category (informally and without a lot of the necessary
technical detail), and briefly discuss algebraic concepts such as exact
triangles and generators. Finally, we mention wrapped Fukaya categories and
outline a few applications to symplectic topology, mirror symmetry and
low-dimensional topology. This text is based on a series of lectures given at a
Summer School on Contact and Symplectic Topology at Universit\'e de Nantes in
June 2011.Comment: 42 pages, 13 figure
Imprints of the Quantum World in Classical Mechanics
The imprints left by quantum mechanics in classical (Hamiltonian) mechanics
are much more numerous than is usually believed. We show Using no physical
hypotheses) that the Schroedinger equation for a nonrelativistic system of
spinless particles is a classical equation which is equivalent to Hamilton's
equations.Comment: Paper submitted to Foundations of Physic
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