874 research outputs found
A Topos Perspective on State-Vector Reduction
A preliminary investigation is made of possible applications in quantum
theory of the topos formed by the collection of all -sets, where is a
monoid. Earlier results on topos aspects of quantum theory can be rederived in
this way. However, the formalism also suggests a new way of constructing a
`neo-realist' interpretation of quantum theory in which the truth values of
propositions are determined by the actions of the monoid of strings of finite
projection operators. By these means, a novel topos perspective is gained on
the concept of state-vector reduction
Topos theory and `neo-realist' quantum theory
Topos theory, a branch of category theory, has been proposed as mathematical
basis for the formulation of physical theories. In this article, we give a
brief introduction to this approach, emphasising the logical aspects. Each
topos serves as a `mathematical universe' with an internal logic, which is used
to assign truth-values to all propositions about a physical system. We show in
detail how this works for (algebraic) quantum theory.Comment: 22 pages, no figures; contribution for Proceedings of workshop
"Recent Developments in Quantum Field Theory", MPI MIS Leipzig, July 200
Entropy of Classical Histories
We consider a number of proposals for the entropy of sets of classical
coarse-grained histories based on the procedures of Jaynes, and prove a series
of inequalities relating these measures. We then examine these as a function of
the coarse-graining for various classical systems, and show explicitly that the
entropy is minimized by the finest-grained description of a set of histories.
We propose an extension of the second law of thermodynamics to the entropy of
histories. We briefly discuss the implications for decoherent or consistent
history formulations of quantum mechanics.Comment: 35 pages RevTeX 3.0 + 5 figures (postscript). Minor corrections and
typos. To appear in Physical Review
Continuous Time and Consistent Histories
We discuss the use of histories labelled by a continuous time in the approach
to consistent-histories quantum theory in which propositions about the history
of the system are represented by projection operators on a Hilbert space. This
extends earlier work by two of us \cite{IL95} where we showed how a continuous
time parameter leads to a history algebra that is isomorphic to the canonical
algebra of a quantum field theory. We describe how the appropriate
representation of the history algebra may be chosen by requiring the existence
of projection operators that represent propositions about time average of the
energy. We also show that the history description of quantum mechanics contains
an operator corresponding to velocity that is quite distinct from the momentum
operator. Finally, the discussion is extended to give a preliminary account of
quantum field theory in this approach to the consistent histories formalism.Comment: Typeset in RevTe
Topos-Theoretic Extension of a Modal Interpretation of Quantum Mechanics
This paper deals with topos-theoretic truth-value valuations of quantum
propositions. Concretely, a mathematical framework of a specific type of modal
approach is extended to the topos theory, and further, structures of the
obtained truth-value valuations are investigated. What is taken up is the modal
approach based on a determinate lattice \Dcal(e,R), which is a sublattice of
the lattice \Lcal of all quantum propositions and is determined by a quantum
state and a preferred determinate observable . Topos-theoretic extension
is made in the functor category \Sets^{\CcalR} of which base category
\CcalR is determined by . Each true atom, which determines truth values,
true or false, of all propositions in \Dcal(e,R), generates also a
multi-valued valuation function of which domain and range are \Lcal and a
Heyting algebra given by the subobject classifier in \Sets^{\CcalR},
respectively. All true propositions in \Dcal(e,R) are assigned the top
element of the Heyting algebra by the valuation function. False propositions
including the null proposition are, however, assigned values larger than the
bottom element. This defect can be removed by use of a subobject
semi-classifier. Furthermore, in order to treat all possible determinate
observables in a unified framework, another valuations are constructed in the
functor category \Sets^{\Ccal}. Here, the base category \Ccal includes all
\CcalR's as subcategories. Although \Sets^{\Ccal} has a structure
apparently different from \Sets^{\CcalR}, a subobject semi-classifier of
\Sets^{\Ccal} gives valuations completely equivalent to those in
\Sets^{\CcalR}'s.Comment: LaTeX2
Bimetric Gravity Theory, Varying Speed of Light and the Dimming of Supernovae
In the bimetric scalar-tensor gravitational theory there are two frames
associated with the two metrics {\hat g}_{\mu\nu} and g_{\mu\nu}, which are
linked by the gradients of a scalar field \phi. The choice of a comoving frame
for the metric {\hat g}_{\mu\nu} or g_{\mu\nu} has fundamental consequences for
local observers in either metric spacetimes, while maintaining diffeomorphism
invariance. When the metric g_{\mu\nu} is chosen to be associated with comoving
coordinates, then the speed of light varies in the frame with the metric {\hat
g}_{\mu\nu}. Observers in this frame see the dimming of supernovae because of
the increase of the luminosity distance versus red shift, due to an increasing
speed of light in the early universe. Moreover, in this frame the scalar field
\phi describes a dark energy component in the Friedmann equation for the cosmic
scale without acceleration. If we choose {\hat g}_{\mu\nu} to be associated
with comoving coordinates, then an observer in the g_{\mu\nu} metric frame will
observe the universe to be accelerating and the supernovae will appear to be
farther away. The theory predicts that the gravitational constant G can vary in
spacetime, while the fine-structure constant \alpha=e^2/\hbar c does not vary.
The problem of cosmological horizons as viewed in the two frames is discussed.Comment: 22 pages, Latex file. No figures. Corrected typos. Added reference.
Further references added. Further corrections. To be published in Int. J.
Mod. Phys. D, 200
Reply to comment by S. Nadarajah on "Space-time modeling of soil moisture: Stochastic rainfall forcing with heterogeneous vegetation"
Persistence of Tripartite Nonlocality for Non-inertial Observers
We consider the behaviour of bipartite and tripartite non-locality between
fermionic entangled states shared by observers, one of whom uniformly
accelerates. We find that while fermionic entanglement persists for arbitrarily
large acceleration, the Bell/CHSH inequalities cannot be violated for
sufficiently large but finite acceleration. However the Svetlichny inequality,
which is a measure of genuine tripartite non-locality, can be violated for any
finite value of the acceleration.Comment: 4 pages, pdflatex, 2 figure
Space-time modeling of soil moisture: Stochastic rainfall forcing with heterogeneous vegetation
The present paper complements that of Isham et al. (2005), who introduced a space-time soil moisture model driven by stochastic space-time rainfall forcing with homogeneous vegetation and in the absence of topographical landscape effects. However, the spatial variability of vegetation may significantly modify the soil moisture dynamics with important implications for hydrological modeling. In the present paper, vegetation heterogeneity is incorporated through a two dimensional Poisson process representing the coexistence of two functionally different types of plants (e.g., trees and grasses). The space-time statistical structure of relative soil moisture is characterized through its covariance function which depends on soil, vegetation, and rainfall patterns. The statistical properties of the soil moisture process averaged in space and time are also investigated. These properties are especially important for any modeling that aggregates soil moisture characteristics over a range of spatial and temporal scales. It is found that particularly at small scales, vegetation heterogeneity has a significant impact on the averaged process as compared with the uniform vegetation case. Also, averaging in space considerably smoothes the soil moisture process, but in contrast, averaging in time up to 1 week leads to little change in the variance of the averaged process
Topos Theory and Consistent Histories: The Internal Logic of the Set of all Consistent Sets
A major problem in the consistent-histories approach to quantum theory is
contending with the potentially large number of consistent sets of history
propositions. One possibility is to find a scheme in which a unique set is
selected in some way. However, in this paper we consider the alternative
approach in which all consistent sets are kept, leading to a type of `many
world-views' picture of the quantum theory. It is shown that a natural way of
handling this situation is to employ the theory of varying sets (presheafs) on
the space \B of all Boolean subalgebras of the orthoalgebra \UP of history
propositions. This approach automatically includes the feature whereby
probabilistic predictions are meaningful only in the context of a consistent
set of history propositions. More strikingly, it leads to a picture in which
the `truth values', or `semantic values' of such contextual predictions are not
just two-valued (\ie true and false) but instead lie in a larger logical
algebra---a Heyting algebra---whose structure is determined by the space \B
of Boolean subalgebras of \UP.Comment: 28 pages, LaTe
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