8,193 research outputs found
Classical Equations for Quantum Systems
The origin of the phenomenological deterministic laws that approximately
govern the quasiclassical domain of familiar experience is considered in the
context of the quantum mechanics of closed systems such as the universe as a
whole. We investigate the requirements for coarse grainings to yield decoherent
sets of histories that are quasiclassical, i.e. such that the individual
histories obey, with high probability, effective classical equations of motion
interrupted continually by small fluctuations and occasionally by large ones.
We discuss these requirements generally but study them specifically for coarse
grainings of the type that follows a distinguished subset of a complete set of
variables while ignoring the rest. More coarse graining is needed to achieve
decoherence than would be suggested by naive arguments based on the uncertainty
principle. Even coarser graining is required in the distinguished variables for
them to have the necessary inertia to approach classical predictability in the
presence of the noise consisting of the fluctuations that typical mechanisms of
decoherence produce. We describe the derivation of phenomenological equations
of motion explicitly for a particular class of models. Probabilities of the
correlations in time that define equations of motion are explicitly considered.
Fully non-linear cases are studied. Methods are exhibited for finding the form
of the phenomenological equations of motion even when these are only distantly
related to those of the fundamental action. The demonstration of the connection
between quantum-mechanical causality and causalty in classical phenomenological
equations of motion is generalized. The connections among decoherence, noise,
dissipation, and the amount of coarse graining necessary to achieve classical
predictability are investigated quantitatively.Comment: 100pages, 1 figur
Newton's method, zeroes of vector fields, and the Riemannian center of mass
We present an iterative technique for finding zeroes of vector fields on
Riemannian manifolds. As a special case we obtain a ``nonlinear averaging
algorithm'' that computes the centroid of a mass distribution supported in a
set of small enough diameter D in a Riemannian manifold M. We estimate the
convergence rate of our general algorithm and the more special Riemannian
averaging algorithm. The algorithm is also used to provide a constructive proof
of Karcher's theorem on the existence and local uniqueness of the center of
mass, under a somewhat stronger requirement than Karcher's on D. Another
corollary of our results is a proof of convergence, for a fairly large open set
of initial conditions, of the ``GPA algorithm'' used in statistics to average
points in a shape-space, and a quantitative explanation of why the GPA
algorithm converges rapidly in practice. We also show that a mass distribution
in M with support Q has a unique center of mass in a (suitably defined) convex
hull of Q.Comment: 43 pages, 1 figur
A theory of hyperfinite sets
We develop an axiomatic set theory -- the Theory of Hyperfinite Sets THS,
which is based on the idea of existence of proper subclasses of big finite
sets. We demonstrate how theorems of classical continuous mathematics can be
transfered to THS, prove consistency of THS and present some applications.Comment: 28 page
Slowly varying control parameters, delayed bifurcations and the stability of spikes in reaction-diffusion systems
We present three examples of delayed bifurcations for spike solutions of
reaction-diffusion systems. The delay effect results as the system passes
slowly from a stable to an unstable regime, and was previously analysed in the
context of ODE's in [P.Mandel, T.Erneux, J.Stat.Phys, 1987]. It was found that
the instability would not be fully realized until the system had entered well
into the unstable regime. The bifurcation is said to have been "delayed"
relative to the threshold value computed directly from a linear stability
analysis. In contrast, we analyze the delay effect in systems of PDE's. In
particular, for spike solutions of singularly perturbed generalized
Gierer-Meinhardt (GM) and Gray-Scott (GS) models, we analyze three examples of
delay resulting from slow passage into regimes of oscillatory and competition
instability. In the first example, for the GM model on the infinite real line,
we analyze the delay resulting from slowly tuning a control parameter through a
Hopf bifurcation. In the second example, we consider a Hopf bifurcation on a
finite one-dimensional domain. In this scenario, as opposed to the extrinsic
tuning of a system parameter through a bifurcation value, we analyze the delay
of a bifurcation triggered by slow intrinsic dynamics of the PDE system. In the
third example, we consider competition instabilities of the GS model triggered
by the extrinsic tuning of a feed rate parameter. In all cases, we find that
the system must pass well into the unstable regime before the onset of
instability is fully observed, indicating delay. We also find that delay has an
important effect on the eventual dynamics of the system in the unstable regime.
We give analytic predictions for the magnitude of the delays as obtained
through analysis of certain explicitly solvable nonlocal eigenvalue problems.
The theory is confirmed by numerical solutions of the full PDE systems.Comment: 31 pages, 20 figures, submitted to Physica D: Nonlinear Phenomen
Simple de Sitter Solutions
We present a framework for de Sitter model building in type IIA string
theory, illustrated with specific examples. We find metastable dS minima of the
potential for moduli obtained from a compactification on a product of two Nil
three-manifolds (which have negative scalar curvature) combined with
orientifolds, branes, fractional Chern-Simons forms, and fluxes. As a discrete
quantum number is taken large, the curvature, field strengths, inverse volume,
and four dimensional string coupling become parametrically small, and the de
Sitter Hubble scale can be tuned parametrically smaller than the scales of the
moduli, KK, and winding mode masses. A subtle point in the construction is that
although the curvature remains consistently weak, the circle fibers of the
nilmanifolds become very small in this limit (though this is avoided in
illustrative solutions at modest values of the parameters). In the simplest
version of the construction, the heaviest moduli masses are parametrically of
the same order as the lightest KK and winding masses. However, we provide a
method for separating these marginally overlapping scales, and more generally
the underlying supersymmetry of the model protects against large corrections to
the low-energy moduli potential.Comment: 37 pages, harvmac big, 4 figures. v3: small correction
A No-Go Theorem for Derandomized Parallel Repetition: Beyond Feige-Kilian
In this work we show a barrier towards proving a randomness-efficient
parallel repetition, a promising avenue for achieving many tight
inapproximability results. Feige and Kilian (STOC'95) proved an impossibility
result for randomness-efficient parallel repetition for two prover games with
small degree, i.e., when each prover has only few possibilities for the
question of the other prover. In recent years, there have been indications that
randomness-efficient parallel repetition (also called derandomized parallel
repetition) might be possible for games with large degree, circumventing the
impossibility result of Feige and Kilian. In particular, Dinur and Meir
(CCC'11) construct games with large degree whose repetition can be derandomized
using a theorem of Impagliazzo, Kabanets and Wigderson (SICOMP'12). However,
obtaining derandomized parallel repetition theorems that would yield optimal
inapproximability results has remained elusive.
This paper presents an explanation for the current impasse in progress, by
proving a limitation on derandomized parallel repetition. We formalize two
properties which we call "fortification-friendliness" and "yields robust
embeddings." We show that any proof of derandomized parallel repetition
achieving almost-linear blow-up cannot both (a) be fortification-friendly and
(b) yield robust embeddings. Unlike Feige and Kilian, we do not require the
small degree assumption.
Given that virtually all existing proofs of parallel repetition, including
the derandomized parallel repetition result of Dinur and Meir, share these two
properties, our no-go theorem highlights a major barrier to achieving
almost-linear derandomized parallel repetition
The B-->pi K Puzzle and Supersymmetry
At present, there are discrepancies between the measurements of several
observables in B-->pi K decays and the predictions of the standard model (the
``B-->pi K puzzle''). Although the effect is not yet statistically significant
-- it is at the level of \gsim 3\sigma -- it does hint at the presence of new
physics. In this paper, we explore whether supersymmetry (SUSY) can explain the
B-->pi K puzzle. In particular, we consider the SUSY model of Grossman, Neubert
and Kagan (GNK). We find that it is extremely unlikely that GNK explains the
B-->pi K data. We also find a similar conclusion in many other models of SUSY.
And there are serious criticisms of the two SUSY models that do reproduce the
B-->pi K data. If the B-->pi K puzzle remains, it could pose a problem for SUSY
models.Comment: 14 pages, 2 figures; added reference
Symmetries of Two Higgs Doublet Model and CP violation
We use the invariance of physical picture under a change of Lagrangian, the
reparametrization invariance in the space of Lagrangians and its particular
case -- the rephrasing invariance, for analysis of the two-Higgs-doublet
extension of the SM. We found that some parameters of theory like tan beta are
reparametrization dependent and therefore cannot be fundamental. We use the
Z2-symmetry of the Lagrangian, which prevents a phi_1 phi_2 transitions,
and the different levels of its violation, soft and hard, to describe a
physical content of the model. In general, the broken Z2-symmetry allows for a
CP violation in the physical Higgs sector. We argue that the 2HDM with a soft
breaking of Z2-symmetry is a natural model in the description of EWSB. To
simplify an analysis we choose among different forms of Lagrangian describing
the same physical reality a specific one, in which the vacuum expectation
values of both Higgs fields are real. A possible CP violation in the Higgs
sector is described by using a two-step procedure with the first step identical
to a diagonalization of mass matrix for CP-even fields in the CP conserved
case. We find very simple necessary and sufficient condition for a CP violation
in the Higgs sector. We determine the range of parameters for which CP
violation and Flavor Changing Neutral Current effects are naturally small,what
corresponds to a small dimensionless mass parameter nu= Re m_{12}^2/(2v1v2). We
discuss how for small nu some Higgs bosons can be heavy, with mass up to about
0.6 TeV, without violating of the unitarity constraints. We discuss main
features of the large nu case, which corresponds for nu -> infty to a
decoupling of heavy Higgs bosons.Comment: 27 pages, extended discussion, references added, one figure, Revtex
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