684 research outputs found
Continuous-time histories: observables, probabilities, phase space structure and the classical limit
In this paper we elaborate on the structure of the continuous-time histories
description of quantum theory, which stems from the consistent histories
scheme. In particular, we examine the construction of history Hilbert space,
the identification of history observables and the form of the decoherence
functional (the object that contains the probability information). It is shown
how the latter is equivalent to the closed-time-path (CTP) generating
functional. We also study the phase space structure of the theory first through
the construction of general representations of the history group (the analogue
of the Weyl group) and the implementation of a histories Wigner-Weyl transform.
The latter enables us to write quantum theory solely in terms of phase space
quantities. These results enable the implementation of an algorithm for
identifying the classical (stochastic) limit of a general quantum system.Comment: 46 pages, latex; in this new version typographical errors have been
removed and the presentation has been made cleare
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
Discovery of Information in Possession of Government Agencies and Bureaus Over a Claim of Privilege by the Government
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
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
A Topos Foundation for Theories of Physics: IV. Categories of Systems
This paper is the fourth in a series whose goal is to develop a fundamentally
new way of building theories of physics. The motivation comes from a desire to
address certain deep issues that arise in the quantum theory of gravity. Our
basic contention is that constructing a theory of physics is equivalent to
finding a representation in a topos of a certain formal language that is
attached to the system. Classical physics arises when the topos is the category
of sets. Other types of theory employ a different topos. The previous papers in
this series are concerned with implementing this programme for a single system.
In the present paper, we turn to considering a collection of systems: in
particular, we are interested in the relation between the topos representation
for a composite system, and the representations for its constituents. We also
study this problem for the disjoint sum of two systems. Our approach to these
matters is to construct a category of systems and to find a topos
representation of the entire category.Comment: 38 pages, no figure
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
Lightcone fluctuations in flat spacetimes with nontrivial topology
The quantum lightcone fluctuations in flat spacetimes with compactified
spatial dimensions or with boundaries are examined. The discussion is based
upon a model in which the source of the underlying metric fluctuations is taken
to be quantized linear perturbations of the gravitational field. General
expressions are derived, in the transverse trace-free gauge, for the summation
of graviton polarization tensors, and for vacuum graviton two-point functions.
Because of the fluctuating light cone, the flight time of photons between a
source and a detector may be either longer or shorter than the light
propagation time in the background classical spacetime. We calculate the mean
deviations from the classical propagation time of photons due to the changes in
the topology of the flat spacetime. These deviations are in general larger in
the directions in which topology changes occur and are typically of the order
of the Planck time, but they can get larger as the travel distance increases.Comment: 25 pages, 5 figures, some discussions added and a few typos
corrected, final version to appear in Phys. Rev.
(Quantum) Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces
We develop a kind of pregeometry consisting of a web of overlapping fuzzy
lumps which interact with each other. The individual lumps are understood as
certain closely entangled subgraphs (cliques) in a dynamically evolving network
which, in a certain approximation, can be visualized as a time-dependent random
graph. This strand of ideas is merged with another one, deriving from ideas,
developed some time ago by Menger et al, that is, the concept of probabilistic-
or random metric spaces, representing a natural extension of the metrical
continuum into a more microscopic regime. It is our general goal to find a
better adapted geometric environment for the description of microphysics. In
this sense one may it also view as a dynamical randomisation of the causal-set
framework developed by e.g. Sorkin et al. In doing this we incorporate, as a
perhaps new aspect, various concepts from fuzzy set theory.Comment: 25 pages, Latex, no figures, some references added, some minor
changes added relating to previous wor
A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
This paper is the second in a series whose goal is to develop a fundamentally
new way of constructing theories of physics. The motivation comes from a desire
to address certain deep issues that arise when contemplating quantum theories
of space and time. Our basic contention is that constructing a theory of
physics is equivalent to finding a representation in a topos of a certain
formal language that is attached to the system. Classical physics arises when
the topos is the category of sets. Other types of theory employ a different
topos. In this paper, we study in depth the topos representation of the
propositional language, PL(S), for the case of quantum theory. In doing so, we
make a direct link with, and clarify, the earlier work on applying topos theory
to quantum physics. The key step is a process we term `daseinisation' by which
a projection operator is mapped to a sub-object of the spectral presheaf--the
topos quantum analogue of a classical state space. In the second part of the
paper we change gear with the introduction of the more sophisticated local
language L(S). From this point forward, throughout the rest of the series of
papers, our attention will be devoted almost entirely to this language. In the
present paper, we use L(S) to study `truth objects' in the topos. These are
objects in the topos that play the role of states: a necessary development as
the spectral presheaf has no global elements, and hence there are no
microstates in the sense of classical physics. Truth objects therefore play a
crucial role in our formalism.Comment: 34 pages, no figure
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