828 research outputs found
Quantising on a category
We review the problem of finding a general framework within which one can
construct quantum theories of non-standard models for space, or space-time. The
starting point is the observation that entities of this type can typically be
regarded as objects in a category whose arrows are structure-preserving maps.
This motivates investigating the general problem of quantising a system whose
`configuration space' (or history-theory analogue) is the set of objects
\Ob\Q in a category \Q.
We develop a scheme based on constructing an analogue of the group that is
used in the canonical quantisation of a system whose configuration space is a
manifold , where and are Lie groups. In particular, we
choose as the analogue of the monoid of `arrow fields' on \Q. Physically,
this means that an arrow between two objects in the category is viewed as an
analogue of momentum. After finding the `category quantisation monoid', we show
how suitable representations can be constructed using a bundle (or, more
precisely, presheaf) of Hilbert spaces over \Ob\Q. For the example of a
category of finite sets, we construct an explicit representation structure of
this type.Comment: To appear in a volume dedicated to the memory of James Cushin
The Canonical Approach to Quantum Gravity: General Ideas and Geometrodynamics
We give an introduction to the canonical formalism of Einstein's theory of
general relativity. This then serves as the starting point for one approach to
quantum gravity called quantum geometrodynamics. The main features and
applications of this approach are briefly summarized.Comment: 21 pages, 6 figures. Contribution to E. Seiler and I.-O. Stamatescu
(editors): `Approaches To Fundamental Physics -- An Assessment Of Current
Theoretical Ideas' (Springer Verlag, to appear
Video Pandemics: Worldwide Viral Spreading of Psy's Gangnam Style Video
Viral videos can reach global penetration traveling through international
channels of communication similarly to real diseases starting from a
well-localized source. In past centuries, disease fronts propagated in a
concentric spatial fashion from the the source of the outbreak via the short
range human contact network. The emergence of long-distance air-travel changed
these ancient patterns. However, recently, Brockmann and Helbing have shown
that concentric propagation waves can be reinstated if propagation time and
distance is measured in the flight-time and travel volume weighted underlying
air-travel network. Here, we adopt this method for the analysis of viral meme
propagation in Twitter messages, and define a similar weighted network distance
in the communication network connecting countries and states of the World. We
recover a wave-like behavior on average and assess the randomizing effect of
non-locality of spreading. We show that similar result can be recovered from
Google Trends data as well.Comment: 10 page
Factor ordering in standard quantum cosmology
The Wheeler-DeWitt equation of Friedmann models with a massless quantum field
is formulated with arbitrary factor ordering of the Hamiltonian constraint
operator. A scalar product of wave functions is constructed, giving rise to a
probability interpretation and making comparison with the classical solution
possible. In general the bahaviour of the wave function of the model depends on
a critical energy of the matter field, which, in turn, depends on the chosen
factor ordering. By certain choices of the ordering the critical energy can be
pushed down to zero.Comment: 15 pages, 3 figure
On the "renormalization" transformations induced by cycles of expansion and contraction in causal set cosmology
We study the ``renormalization group action'' induced by cycles of cosmic
expansion and contraction, within the context of a family of stochastic
dynamical laws for causal sets derived earlier. We find a line of fixed points
corresponding to the dynamics of transitive percolation, and we prove that
there exist no other fixed points and no cycles of length two or more. We also
identify an extensive ``basin of attraction'' of the fixed points but find that
it does not exhaust the full parameter space. Nevertheless, we conjecture that
every trajectory is drawn toward the fixed point set in a suitably weakened
sense.Comment: 22 pages, 1 firgure, submitted to Phys. Rev.
Comparing Formulations of Generalized Quantum Mechanics for Reparametrization-Invariant Systems
A class of decoherence schemes is described for implementing the principles
of generalized quantum theory in reparametrization-invariant `hyperbolic'
models such as minisuperspace quantum cosmology. The connection with
sum-over-histories constructions is exhibited and the physical equivalence or
inequivalence of different such schemes is analyzed. The discussion focuses on
comparing constructions based on the Klein-Gordon product with those based on
the induced (a.k.a. Rieffel, Refined Algebraic, Group Averaging, or Spectral
Analysis) inner product. It is shown that the Klein-Gordon and induced products
can be simply related for the models of interest. This fact is then used to
establish isomorphisms between certain decoherence schemes based on these
products.Comment: 21 pages ReVTe
Dirac Quantization of Parametrized Field Theory
Parametrized field theory (PFT) is free field theory on flat spacetime in a
diffeomorphism invariant disguise. It describes field evolution on arbitrary
foliations of the flat spacetime instead of only the usual flat ones, by
treating the `embedding variables' which describe the foliation as dynamical
variables to be varied in the action in addition to the scalar field. A formal
Dirac quantization turns the constraints of PFT into functional Schrodinger
equations which describe evolution of quantum states from an arbitrary Cauchy
slice to an infinitesimally nearby one.This formal Schrodinger picture- based
quantization is unitarily equivalent to the standard Heisenberg picture based
Fock quantization of the free scalar field if scalar field evolution along
arbitrary foliations is unitarily implemented on the Fock space. Torre and
Varadarajan (TV) showed that for generic foliations emanating from a flat
initial slice in spacetimes of dimension greater than 2, evolution is not
unitarily implemented, thus implying an obstruction to Dirac quantization.
We construct a Dirac quantization of PFT,unitarily equivalent to the standard
Fock quantization, using techniques from Loop Quantum Gravity (LQG) which are
powerful enough to super-cede the no- go implications of the TV results. The
key features of our quantization include an LQG type representation for the
embedding variables, embedding dependent Fock spaces for the scalar field, an
anomaly free representation of (a generalization of) the finite transformations
generated by the constraints and group averaging techniques. The difference
between 2 and higher dimensions is that in the latter, only finite gauge
transformations are defined in the quantum theory, not the infinitesimal ones.Comment: 33 page
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
Loop Variable Inequalities in Gravity and Gauge Theory
We point out an incompleteness of formulations of gravitational and gauge
theories that use traces of holonomies around closed curves as their basic
variables. It is shown that in general such loop variables have to satisfy
certain inequalities if they are to give a description equivalent to the usual
one in terms of local gauge potentials.Comment: 10pp., TeX, Syracuse SU-GP-93/3-
de Sitter symmetry of Neveu-Schwarz spinors
We study the relations between Dirac fields living on the 2-dimensional
Lorentzian cylinder and the ones living on the double-covering of the
2-dimensional de Sitter manifold, here identified as a certain coset space of
the group . We show that there is an extended notion of de Sitter
covariance only for Dirac fields having the Neveu-Schwarz anti-periodicity and
construct the relevant cocycle. Finally, we show that the de Sitter symmetry is
naturally inherited by the Neveu-Schwarz massless Dirac field on the cylinder.Comment: 24 page
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