13,340 research outputs found
Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity
We propose a description of open universes in the Chern-Simons formulation of
(2+1)-dimensional gravity where spatial infinity is implemented as a puncture.
At this puncture, additional variables are introduced which lie in the
cotangent bundle of the Poincar\'e group, and coupled minimally to the
Chern-Simons gauge field. We apply this description of spatial infinity to open
universes of general genus and with an arbitrary number of massive spinning
particles. Using results of [9] we give a finite dimensional description of the
phase space and determine its symplectic structure. In the special case of a
genus zero universe with spinless particles, we compare our result to the
symplectic structure computed by Matschull in the metric formulation of
(2+1)-dimensional gravity. We comment on the quantisation of the phase space
and derive a quantisation condition for the total mass and spin of an open
universe.Comment: 44 pages, 3 eps figure
Two-Level Type Theory and Applications
We define and develop two-level type theory (2LTT), a version of Martin-L\"of
type theory which combines two different type theories. We refer to them as the
inner and the outer type theory. In our case of interest, the inner theory is
homotopy type theory (HoTT) which may include univalent universes and higher
inductive types. The outer theory is a traditional form of type theory
validating uniqueness of identity proofs (UIP). One point of view on it is as
internalised meta-theory of the inner type theory.
There are two motivations for 2LTT. Firstly, there are certain results about
HoTT which are of meta-theoretic nature, such as the statement that
semisimplicial types up to level can be constructed in HoTT for any
externally fixed natural number . Such results cannot be expressed in HoTT
itself, but they can be formalised and proved in 2LTT, where will be a
variable in the outer theory. This point of view is inspired by observations
about conservativity of presheaf models.
Secondly, 2LTT is a framework which is suitable for formulating additional
axioms that one might want to add to HoTT. This idea is heavily inspired by
Voevodsky's Homotopy Type System (HTS), which constitutes one specific instance
of a 2LTT. HTS has an axiom ensuring that the type of natural numbers behaves
like the external natural numbers, which allows the construction of a universe
of semisimplicial types. In 2LTT, this axiom can be stated simply be asking the
inner and outer natural numbers to be isomorphic.
After defining 2LTT, we set up a collection of tools with the goal of making
2LTT a convenient language for future developments. As a first such
application, we develop the theory of Reedy fibrant diagrams in the style of
Shulman. Continuing this line of thought, we suggest a definition of
(infinity,1)-category and give some examples.Comment: 53 page
Consistent probabilities in loop quantum cosmology
A fundamental issue for any quantum cosmological theory is to specify how
probabilities can be assigned to various quantum events or sequences of events
such as the occurrence of singularities or bounces. In previous work, we have
demonstrated how this issue can be successfully addressed within the consistent
histories approach to quantum theory for Wheeler-DeWitt-quantized cosmological
models. In this work, we generalize that analysis to the exactly solvable loop
quantization of a spatially flat, homogeneous and isotropic cosmology sourced
with a massless, minimally coupled scalar field known as sLQC. We provide an
explicit, rigorous and complete decoherent histories formulation for this model
and compute the probabilities for the occurrence of a quantum bounce vs. a
singularity. Using the scalar field as an emergent internal time, we show for
generic states that the probability for a singularity to occur in this model is
zero, and that of a bounce is unity, complementing earlier studies of the
expectation values of the volume and matter density in this theory. We also
show from the consistent histories point of view that all states in this model,
whether quantum or classical, achieve arbitrarily large volume in the limit of
infinite `past' or `future' scalar `time', in the sense that the wave function
evaluated at any arbitrary fixed value of the volume vanishes in that limit.
Finally, we briefly discuss certain misconceptions concerning the utility of
the consistent histories approach in these models.Comment: 22 pages, 3 figures. Matches published versio
The Replication Argument for Incompatibilism
In this paper, I articulate an argument for incompatibilism about moral responsibility and determinism. My argument comes in the form of an extended story, modeled loosely on Peter van Inwagenâs ârollback argumentâ scenario. I thus call it âthe replication argument.â As I aim to bring out, though the argument is inspired by so-called âmanipulationâ and âoriginal designâ arguments, the argument is not a version of either such argumentâand plausibly has advantages over both. The result, I believe, is a more convincing incompatibilist argument than those we have considered previously
Vacuum Energy Sequestering: The Framework and Its Cosmological Consequences
Recently we suggested a reformulation of General Relativity which completely
sequesters from gravity {\it all} of the vacuum energy from a protected matter
sector, assumed to contain the Standard Model. Here we elaborate further on the
mechanism, presenting additional details of how it cancels all loop corrections
and renders all contributions from phase transitions automatically small. We
also consider cosmological consequences in more detail and show that the
mechanism is consistent with a variety of inflationary models that make a
universe big and old. We discuss in detail the underlying assumptions behind
the dynamics of our proposal, and elaborate on the relationship of the physical
interpretation of divergent operators in quantum field theory and the apparent
`acausality' which our mechanism seems to entail, which we argue is completely
harmless. It is merely a reflection of the fact that any UV sensitive quantity
in quantum field theory cannot be calculated from first principles, but is an
input whose numerical value must be measured. We also note that since the
universe should be compact in spacetime, and so will collapse in the future,
the current phase of acceleration with is just a transient.
This could be tested by future cosmological observations.Comment: 39 pages LaTeX, 1 .pdf figure v3: a small correction, version
published in PR
Fine tuning of parameters of the universe
The mechanism of production of a large number of universes is considered. It
is shown that universes with parameters suitable for creation of life are
necessarily produced as a result of quantum fluctuations. Fractal structures
are formed provided fluctuations take place near a maximum of the potential.
Several ways of formation of similar fractal structures within our universe are
discussed. Theoretical predictions are compared with observational data.Comment: 9 pages, 1 figur
Unifying Cubical Models of Univalent Type Theory
We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure
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