360 research outputs found
Quantum models of classical mechanics: maximum entropy packets
In a previous paper, a project of constructing quantum models of classical
properties has been started. The present paper concludes the project by turning
to classical mechanics. The quantum states that maximize entropy for given
averages and variances of coordinates and momenta are called ME packets. They
generalize the Gaussian wave packets. A non-trivial extension of the
partition-function method of probability calculus to quantum mechanics is
given. Non-commutativity of quantum variables limits its usefulness. Still, the
general form of the state operators of ME packets is obtained with its help.
The diagonal representation of the operators is found. A general way of
calculating averages that can replace the partition function method is
described. Classical mechanics is reinterpreted as a statistical theory.
Classical trajectories are replaced by classical ME packets. Quantum states
approximate classical ones if the product of the coordinate and momentum
variances is much larger than Planck constant. Thus, ME packets with large
variances follow their classical counterparts better than Gaussian wave
packets.Comment: 26 pages, no figure. Introduction and the section on classical limit
are extended, new references added. Definitive version accepted by Found.
Phy
Infinitesimals without Logic
We introduce the ring of Fermat reals, an extension of the real field
containing nilpotent infinitesimals. The construction takes inspiration from
Smooth Infinitesimal Analysis (SIA), but provides a powerful theory of actual
infinitesimals without any need of a background in mathematical logic. In
particular, on the contrary with respect to SIA, which admits models only in
intuitionistic logic, the theory of Fermat reals is consistent with classical
logic. We face the problem to decide if the product of powers of nilpotent
infinitesimals is zero or not, the identity principle for polynomials, the
definition and properties of the total order relation. The construction is
highly constructive, and every Fermat real admits a clear and order preserving
geometrical representation. Using nilpotent infinitesimals, every smooth
functions becomes a polynomial because in Taylor's formulas the rest is now
zero. Finally, we present several applications to informal classical
calculations used in Physics: now all these calculations become rigorous and,
at the same time, formally equal to the informal ones. In particular, an
interesting rigorous deduction of the wave equation is given, that clarifies
how to formalize the approximations tied with Hook's law using this language of
nilpotent infinitesimals.Comment: The first part of the preprint is taken directly form arXiv:0907.1872
The second part is new and contains a list of example
Macroscopic limit of a solvable dynamical model
The interaction between an ultrarelativistic particle and a linear array made
up of two-level systems (^^ ^^ AgBr" molecules) is studied by making use of
a modified version of the Coleman-Hepp Hamiltonian. Energy-exchange processes
between the particle and the molecules are properly taken into account, and the
evolution of the total system is calculated exactly both when the array is
initially in the ground state and in a thermal state. In the macroscopic limit
(), the system remains solvable and leads to interesting
connections with the Jaynes-Cummings model, that describes the interaction of a
particle with a maser. The visibility of the interference pattern produced by
the two branch waves of the particle is computed, and the conditions under
which the spin array in the limit behaves as a ^^ ^^
detector" are investigated. The behavior of the visibility yields good insights
into the issue of quantum measurements: It is found that, in the
thermodynamical limit, a superselection-rule space appears in the description
of the (macroscopic) apparatus. In general, an initial thermal state of the ^^
^^ detector" provokes a more substantial loss of quantum coherence than an
initial ground state. It is argued that a system decoheres more as the
temperature of the detector increases. The problem of ^^ ^^ imperfect
measurements" is also shortly discussed.Comment: 30 pages, report BA-TH/93-13
Efficient and robust entanglement generation in a many-particle system with resonant dipole-dipole interactions
We propose and discuss a scheme for robust and efficient generation of
many-particle entanglement in an ensemble of Rydberg atoms with resonant
dipole-dipole interactions. It is shown that in the limit of complete dipole
blocking, the system is isomorphic to a multimode Jaynes-Cummings model. While
dark-state population transfer is not capable of creating entanglement, other
adiabatic processes are identified that lead to complex, maximally entangled
states, such as the N-particle analog of the GHZ state in a few steps. The
process is robust, works for even and odd particle numbers and the
characteristic time for entanglement generation scales with N^a, with a being
less than unity.Comment: 4 figure
Gauged motion in general relativity and in Kaluza-Klein theories
In a recent paper [1] a new generalization of the Killing motion, the {\it
gauged motion}, has been introduced for stationary spacetimes where it was
shown that the physical symmetries of such spacetimes are well described
through this new symmetry. In this article after a more detailed study in the
stationary case we present the definition of gauged motion for general
spacetimes. The definition is based on the gauged Lie derivative induced by a
threading family of observers and the relevant reparametrization invariance. We
also extend the gauged motion to the case of Kaluza-Klein theories.Comment: 42 pages, revised version, typos correction along with some minor
changes, Revtex forma
Accelerating electromagnetic magic field from the C-metric
Various aspects of the C-metric representing two rotating charged black holes
accelerated in opposite directions are summarized and its limits are
considered. A particular attention is paid to the special-relativistic limit in
which the electromagnetic field becomes the "magic field" of two oppositely
accelerated rotating charged relativistic discs. When the acceleration vanishes
the usual electromagnetic magic field of the Kerr-Newman black hole with
gravitational constant set to zero arises. Properties of the accelerated discs
and the fields produced are studied and illustrated graphically. The charges at
the rim of the accelerated discs move along spiral trajectories with the speed
of light. If the magic field has some deeper connection with the field of the
Dirac electron, as is sometimes conjectured because of the same gyromagnetic
ratio, the "accelerating magic field" represents the electromagnetic field of a
uniformly accelerated spinning electron. It generalizes the classical Born's
solution for two uniformly accelerated monopole charges.Comment: 22 pages, 5 figure
Magnetic Fields, Relativistic Particles, and Shock Waves in Cluster Outskirts
It is only now, with low-frequency radio telescopes, long exposures with
high-resolution X-ray satellites and gamma-ray telescopes, that we are
beginning to learn about the physics in the periphery of galaxy clusters. In
the coming years, Sunyaev-Zeldovich telescopes are going to deliver further
great insights into the plasma physics of these special regions in the
Universe. The last years have already shown tremendous progress with detections
of shocks, estimates of magnetic field strengths and constraints on the
particle acceleration efficiency. X-ray observations have revealed shock fronts
in cluster outskirts which have allowed inferences about the microphysical
structure of shocks fronts in such extreme environments. The best indications
for magnetic fields and relativistic particles in cluster outskirts come from
observations of so-called radio relics, which are megaparsec-sized regions of
radio emission from the edges of galaxy clusters. As these are difficult to
detect due to their low surface brightness, only few of these objects are
known. But they have provided unprecedented evidence for the acceleration of
relativistic particles at shock fronts and the existence of muG strength fields
as far out as the virial radius of clusters. In this review we summarise the
observational and theoretical state of our knowledge of magnetic fields,
relativistic particles and shocks in cluster outskirts.Comment: 34 pages, to be published in Space Science Review
Nash Equilibria in Discrete Routing Games with Convex Latency Functions
In a discrete routing game, each of n selfish users employs a mixed strategy to ship her (unsplittable) traffic over m parallel links. The (expected) latency on a link is determined by an arbitrary non-decreasing, non-constant and convex latency function φ. In a Nash equilibrium, each user alone is minimizing her (Expected) Individual Cost, which is the (expected) latency on the link she chooses. To evaluate Nash equilibria, we formulate Social Cost as the sum of the users ’ (Expected) Individual Costs. The Price of Anarchy is the worst-case ratio of Social Cost for a Nash equilibrium over the least possible Social Cost. A Nash equilibrium is pure if each user deterministically chooses a single link; a Nash equilibrium is fully mixed if each user chooses each link with non-zero probability. We obtain: For the case of identical users, the Social Cost of any Nash equilibrium is no more than the Social Cost of the fully mixed Nash equilibrium, which may exist only uniquely. Moreover, instances admitting a fully mixed Nash equilibrium enjoy an efficient characterization. For the case of identical users, we derive two upper bounds on the Price of Anarchy: For the case of identical links with a monomial latency function φ(x) = x d, the Price of Anarchy is the Bell number of order d + 1. For pure Nash equilibria, a generic upper bound from the Wardrop model can be transfered to discrete routing games. For polynomial latency functions with non-negative coefficients and degree d, this yields an upper bound of d + 1. For th
DT/T beyond linear theory
The major contribution to the anisotropy of the temperature of the Cosmic
Microwave Background (CMB) radiation is believed to come from the interaction
of linear density perturbations with the radiation previous to the decoupling
time. Assuming a standard thermal history for the gas after recombination, only
the gravitational field produced by the linear density perturbations present on
a universe can generate anisotropies at low z (these
anisotropies would manifest on large angular scales). However, secondary
anisotropies are inevitably produced during the nonlinear evolution of matter
at late times even in a universe with a standard thermal history. Two effects
associated to this nonlinear phase can give rise to new anisotropies: the
time-varying gravitational potential of nonlinear structures (Rees-Sciama RS
effect) and the inverse Compton scattering of the microwave photons with hot
electrons in clusters of galaxies (Sunyaev-Zeldovich SZ effect). These two
effects can produce distinct imprints on the CMB temperature anisotropy. We
discuss the amplitude of the anisotropies expected and the relevant angular
scales in different cosmological scenarios. Future sensitive experiments will
be able to probe the CMB anisotropies beyong the first order primary
contribution.Comment: plain tex, 16 pages, 3 figures. Proceedings of the Laredo Advance
School on Astrophysics "The universe at high-z, large-scale structure and the
cosmic microwave background". To be publised by Springer-Verla
Ten Misconceptions from the History of Analysis and Their Debunking
The widespread idea that infinitesimals were "eliminated" by the "great
triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an
uninterrupted chain of work on infinitesimal-enriched number systems. The
elimination claim is an oversimplification created by triumvirate followers,
who tend to view the history of analysis as a pre-ordained march toward the
radiant future of Weierstrassian epsilontics. In the present text, we document
distortions of the history of analysis stemming from the triumvirate ideology
of ontological minimalism, which identified the continuum with a single number
system. Such anachronistic distortions characterize the received interpretation
of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note:
text overlap with arXiv:1108.2885 and arXiv:1110.545
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