589 research outputs found
Geometrical Description of Quantum Mechanics - Transformations and Dynamics
In this paper we review a proposed geometrical formulation of quantum
mechanics. We argue that this geometrization makes available mathematical
methods from classical mechanics to the quantum frame work. We apply this
formulation to the study of separability and entanglement for states of
composite quantum systems.Comment: 22 pages, to be published in Physica Script
Dynamics and Lax-Phillips scattering for generalized Lamb models
This paper treats the dynamics and scattering of a model of coupled
oscillating systems, a finite dimensional one and a wave field on the half
line. The coupling is realized producing the family of selfadjoint extensions
of the suitably restricted self-adjoint operator describing the uncoupled
dynamics. The spectral theory of the family is studied and the associated
quadratic forms constructed. The dynamics turns out to be Hamiltonian and the
Hamiltonian is described, including the case in which the finite dimensional
systems comprises nonlinear oscillators; in this case the dynamics is shown to
exist as well. In the linear case the system is equivalent, on a dense
subspace, to a wave equation on the half line with higher order boundary
conditions, described by a differential polynomial explicitely
related to the model parameters. In terms of such structure the Lax-Phillips
scattering of the system is studied. In particular we determine the incoming
and outgoing translation representations, the scattering operator, which turns
out to be unitarily equivalent to the multiplication operator given by the
rational function , and the Lax-Phillips semigroup,
which describes the evolution of the states which are neither incoming in the
past nor outgoing in the future
On perturbations of Dirac operators with variable magnetic field of constant direction
We carry out the spectral analysis of matrix valued perturbations of
3-dimensional Dirac operators with variable magnetic field of constant
direction. Under suitable assumptions on the magnetic field and on the
pertubations, we obtain a limiting absorption principle, we prove the absence
of singular continuous spectrum in certain intervals and state properties of
the point spectrum. Various situations, for example when the magnetic field is
constant, periodic or diverging at infinity, are covered. The importance of an
internal-type operator (a 2-dimensional Dirac operator) is also revealed in our
study. The proofs rely on commutator methods.Comment: 12 page
Bosonization method for second super quantization
A bosonic-fermionic correspondence allows an analytic definition of
functional super derivative, in particular, and a bosonic functional calculus,
in general, on Bargmann- Gelfand triples for the second super quantization. A
Feynman integral for the super transformation matrix elements in terms of
bosonic anti-normal Berezin symbols is rigorously constructed.Comment: In memoriam of F. A. Berezin, accepted in Journal of Nonlinear
Mathematical Physics, 15 page
Essential self-adjointness in one-loop quantum cosmology
The quantization of closed cosmologies makes it necessary to study squared
Dirac operators on closed intervals and the corresponding quantum amplitudes.
This paper proves self-adjointness of these second-order elliptic operators.Comment: 14 pages, plain Tex. An Erratum has been added to the end, which
corrects section
Spectral properties on a circle with a singularity
We investigate the spectral and symmetry properties of a quantum particle
moving on a circle with a pointlike singularity (or point interaction). We find
that, within the U(2) family of the quantum mechanically allowed distinct
singularities, a U(1) equivalence (of duality-type) exists, and accordingly the
space of distinct spectra is U(1) x [SU(2)/U(1)], topologically a filled torus.
We explore the relationship of special subfamilies of the U(2) family to
corresponding symmetries, and identify the singularities that admit an N = 2
supersymmetry. Subfamilies that are distinguished in the spectral properties or
the WKB exactness are also pointed out. The spectral and symmetry properties
are also studied in the context of the circle with two singularities, which
provides a useful scheme to discuss the symmetry properties on a general basis.Comment: TeX, 26 pages. v2: one reference added and two update
Trading quantum for classical resources in quantum data compression
We study the visible compression of a source E of pure quantum signal states,
or, more formally, the minimal resources per signal required to represent
arbitrarily long strings of signals with arbitrarily high fidelity, when the
compressor is given the identity of the input state sequence as classical
information. According to the quantum source coding theorem, the optimal
quantum rate is the von Neumann entropy S(E) qubits per signal.
We develop a refinement of this theorem in order to analyze the situation in
which the states are coded into classical and quantum bits that are quantified
separately. This leads to a trade--off curve Q(R), where Q(R) qubits per signal
is the optimal quantum rate for a given classical rate of R bits per signal.
Our main result is an explicit characterization of this trade--off function
by a simple formula in terms of only single signal, perfect fidelity encodings
of the source. We give a thorough discussion of many further mathematical
properties of our formula, including an analysis of its behavior for group
covariant sources and a generalization to sources with continuously
parameterized states. We also show that our result leads to a number of
corollaries characterizing the trade--off between information gain and state
disturbance for quantum sources. In addition, we indicate how our techniques
also provide a solution to the so--called remote state preparation problem.
Finally, we develop a probability--free version of our main result which may be
interpreted as an answer to the question: ``How many classical bits does a
qubit cost?'' This theorem provides a type of dual to Holevo's theorem, insofar
as the latter characterizes the cost of coding classical bits into qubits.Comment: 51 pages, 7 figure
Qualitative Properties of the Dirac Equation in a Central Potential
The Dirac equation for a massive spin-1/2 field in a central potential V in
three dimensions is studied without fixing a priori the functional form of V.
The second-order equations for the radial parts of the spinor wave function are
shown to involve a squared Dirac operator for the free case, whose essential
self-adjointness is proved by using the Weyl limit point-limit circle
criterion, and a `perturbation' resulting from the potential. One then finds
that a potential of Coulomb type in the Dirac equation leads to a potential
term in the above second-order equations which is not even infinitesimally
form-bounded with respect to the free operator. Moreover, the conditions
ensuring essential self-adjointness of the second-order operators in the
interacting case are changed with respect to the free case, i.e. they are
expressed by a majorization involving the parameter in the Coulomb potential
and the angular momentum quantum number. The same methods are applied to the
analysis of coupled eigenvalue equations when the anomalous magnetic moment of
the electron is not neglected.Comment: 22 pages, plain Tex. In the final version, a section has been added,
and the presentation has been improve
Homoclinic chaos in the dynamics of a general Bianchi IX model
The dynamics of a general Bianchi IX model with three scale factors is
examined. The matter content of the model is assumed to be comoving dust plus a
positive cosmological constant. The model presents a critical point of
saddle-center-center type in the finite region of phase space. This critical
point engenders in the phase space dynamics the topology of stable and unstable
four dimensional tubes , where is a saddle direction and
is the manifold of unstable periodic orbits in the center-center sector.
A general characteristic of the dynamical flow is an oscillatory mode about
orbits of an invariant plane of the dynamics which contains the critical point
and a Friedmann-Robertson-Walker (FRW) singularity. We show that a pair of
tubes (one stable, one unstable) emerging from the neighborhood of the critical
point towards the FRW singularity have homoclinic transversal crossings. The
homoclinic intersection manifold has topology and is constituted
of homoclinic orbits which are bi-asymptotic to the center-center
manifold. This is an invariant signature of chaos in the model, and produces
chaotic sets in phase space. The model also presents an asymptotic DeSitter
attractor at infinity and initial conditions sets are shown to have fractal
basin boundaries connected to the escape into the DeSitter configuration
(escape into inflation), characterizing the critical point as a chaotic
scatterer.Comment: 11 pages, 6 ps figures. Accepted for publication in Phys. Rev.
Sound archaeology: terminology, Palaeolithic cave art and the soundscape
This article is focused on the ways that terminology describing the study of music and sound within archaeology has changed over time, and how this reflects developing methodologies, exploring the expectations and issues raised by the use of differing kinds of language to define and describe such work. It begins with a discussion of music archaeology, addressing the problems of using the term âmusicâ in an archaeological context. It continues with an examination of archaeoacoustics and acoustics, and an emphasis on sound rather than music. This leads on to a study of sound archaeology and soundscapes, pointing out that it is important to consider the complete acoustic ecology of an archaeological site, in order to identify its affordances, those possibilities offered by invariant acoustic properties. Using a case study from northern Spain, the paper suggests that all of these methodological approaches have merit, and that a project benefits from their integration
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