15 research outputs found
On obtaining classical mechanics from quantum mechanics
Constructing a classical mechanical system associated with a given quantum
mechanical one, entails construction of a classical phase space and a
corresponding Hamiltonian function from the available quantum structures and a
notion of coarser observations. The Hilbert space of any quantum mechanical
system naturally has the structure of an infinite dimensional symplectic
manifold (`quantum phase space'). There is also a systematic, quotienting
procedure which imparts a bundle structure to the quantum phase space and
extracts a classical phase space as the base space. This works straight
forwardly when the Hilbert space carries weakly continuous representation of
the Heisenberg group and recovers the linear classical phase space
. We report on how the procedure also allows
extraction of non-linear classical phase spaces and illustrate it for Hilbert
spaces being finite dimensional (spin-j systems), infinite dimensional but
separable (particle on a circle) and infinite dimensional but non-separable
(Polymer quantization). To construct a corresponding classical dynamics, one
needs to choose a suitable section and identify an effective Hamiltonian. The
effective dynamics mirrors the quantum dynamics provided the section satisfies
conditions of semiclassicality and tangentiality.Comment: revtex4, 24 pages, no figures. In the version 2 certain technical
errors in section I-B are corrected, the part on WKB (and section II-B) is
removed, discussion of dynamics and semiclassicality is extended and
references are added. Accepted for publication on Classical and Quantum
Gravit
Reality conditions for Ashtekar variables as Dirac constraints
We show that the reality conditions to be imposed on Ashtekar variables to
recover real gravity can be implemented as second class constraints a la Dirac.
Thus, counting gravitational degrees of freedom follows accordingly. Some
constraints of the real theory turn out to be non-polynomial, regardless of the
form, polynomial or non-polynomial, taken for the reality conditions. We
comment upon the compatibility of our approach with the recently proposed Wick
transform point of view, as well as on some alternatives for dealing with such
second class constraints.Comment: 16 pages, plain LaTeX, submitted to Class. Quant. Grav. E-mail:
[email protected]
Numerical loop quantum cosmology: an overview
A brief review of various numerical techniques used in loop quantum cosmology
and results is presented. These include the way extensive numerical simulations
shed insights on the resolution of classical singularities, resulting in the
key prediction of the bounce at the Planck scale in different models, and the
numerical methods used to analyze the properties of the quantum difference
operator and the von Neumann stability issues. Using the quantization of a
massless scalar field in an isotropic spacetime as a template, an attempt is
made to highlight the complementarity of different methods to gain
understanding of the new physics emerging from the quantum theory. Open
directions which need to be explored with more refined numerical methods are
discussed.Comment: 33 Pages, 4 figures. Invited contribution to appear in Classical and
Quantum Gravity special issue on Non-Astrophysical Numerical Relativit
Loop Quantum Cosmology: A Status Report
The goal of this article is to provide an overview of the current state of
the art in loop quantum cosmology for three sets of audiences: young
researchers interested in entering this area; the quantum gravity community in
general; and, cosmologists who wish to apply loop quantum cosmology to probe
modifications in the standard paradigm of the early universe. An effort has
been made to streamline the material so that, as described at the end of
section I, each of these communities can read only the sections they are most
interested in, without a loss of continuity.Comment: 138 pages, 15 figures. Invited Topical Review, To appear in Classical
and Quantum Gravity. Typos corrected, clarifications and references adde
Symplectic approach to quantum constraints
A general prescription for the treatment of constrained quantum motion is
outlined. We consider in particular constraints defined by algebraic
submanifolds of the quantum state space. The resulting formalism is applied to
obtain solutions to the constrained dynamics of systems of multiple spin-1/2
particles. When the motion is constrained to a certain product space containing
all of the energy eigenstates, the dynamics thus obtained are quasi-unitary in
the sense that the equations of motion take a form identical to that of unitary
motion, but with different boundary conditions. When the constrained subspace
is a product space of disentangled states, the associated motion is more
intricate. Nevertheless, the equations of motion satisfied by the dynamical
variables are obtained in closed form.Comment: 11 page
Spin and Rotations in Galois Field Quantum Mechanics
We discuss the properties of Galois Field Quantum Mechanics constructed on a
vector space over the finite Galois field GF(q). In particular, we look at
2-level systems analogous to spin, and discuss how SO(3) rotations could be
embodied in such a system. We also consider two-particle `spin' correlations
and show that the Clauser-Horne-Shimony-Holt (CHSH) inequality is nonetheless
not violated in this model.Comment: 21 pages, 11 pdf figures, LaTeX. Uses iopart.cls. Revised
introduction. Additional reference
Is quantum mechanics based on an invariance principle?
Non-relativistic quantum mechanics for a free particle is shown to emerge
from classical mechanics through an invariance principle under transformations
that preserve the Heisenberg position-momentum inequality. These
transformations are induced by isotropic space dilations. This invariance
imposes a change in the laws of classical mechanics that exactly corresponds to
the transition to quantum mechanics. The Schroedinger equation appears jointly
with a second nonlinear equation describing non-unitary processes. Unitary and
non-unitary evolutions are exclusive and appear sequentially in time. The
non-unitary equation admits solutions that seem to correspond to the collapse
of the wave function.Comment: 15 page
Practical recipes for the model order reduction, dynamical simulation, and compressive sampling of large-scale open quantum systems
This article presents numerical recipes for simulating high-temperature and
non-equilibrium quantum spin systems that are continuously measured and
controlled. The notion of a spin system is broadly conceived, in order to
encompass macroscopic test masses as the limiting case of large-j spins. The
simulation technique has three stages: first the deliberate introduction of
noise into the simulation, then the conversion of that noise into an equivalent
continuous measurement and control process, and finally, projection of the
trajectory onto a state-space manifold having reduced dimensionality and
possessing a Kahler potential of multi-linear form. The resulting simulation
formalism is used to construct a positive P-representation for the thermal
density matrix. Single-spin detection by magnetic resonance force microscopy
(MRFM) is simulated, and the data statistics are shown to be those of a random
telegraph signal with additive white noise. Larger-scale spin-dust models are
simulated, having no spatial symmetry and no spatial ordering; the
high-fidelity projection of numerically computed quantum trajectories onto
low-dimensionality Kahler state-space manifolds is demonstrated. The
reconstruction of quantum trajectories from sparse random projections is
demonstrated, the onset of Donoho-Stodden breakdown at the Candes-Tao sparsity
limit is observed, a deterministic construction for sampling matrices is given,
and methods for quantum state optimization by Dantzig selection are given.Comment: 104 pages, 13 figures, 2 table