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Daniel Cartin

Publications and source records attributed to Daniel Cartin.

18 recordsLinked to original sources
arXiv2024

Knotted 4-regular graphs II: Consistent application of the Pachner moves

A common choice for the evolution of the knotted graphs in loop quantum gravity is to use the Pachner moves, adapted to graphs from their dual triangulations. Here, we show that the natural way to consistently use these moves is on framed graphs with edge twists, where the Pachner moves can only be performed when the twists, and the vertices the edges are incident on, meet certain criteria. For other twists, one can introduce an algebraic object, which allow any knotted graph with framed edges to be written in terms of a generalized braid group.

gr-qc↗
arXiv2022

Knotted 4-regular graphs: polynomial invariants and the Pachner moves

In loop quantum gravity, states of quantum geometry are represented by classes of knotted graphs, equivalent under diffeomorphisms. Thus, it is worthwhile to enumerate and distinguish these classes. This paper looks at the case of 4-regular graphs, which have an interpretation as objects dual to triangulations of three-dimensional manifolds. Two different polynomial invariants are developed to characterize these graphs -- one inspired by the Kauffman bracket relations, and the other based on quandles. How the latter invariant changes under the Pachner moves acting on the graphs is then studied.

gr-qc↗
arXiv2015

Braids as a representation space of SU(5)

The Standard Model of particle physics provides very accurate predictions of phenomena occurring at the sub-atomic level, but the reason for the choice of symmetry group and the large number of particles considered elementary, is still unknown. Along the lines of previous preon models positing a substructure to explain these aspects, Bilson-Thompson showed how the first family of elementary particles is realized as the crossings of braids made of three strands, with charges resulting from twists of those strands with certain conditions; in this topological model, there are only two distinct neutrino states. Modeling the particles as braids implies these braids must be the representation space of a Lie algebra, giving the symmetries of the Standard Model. In this paper, this representation is made explicit, obtaining the raising operators associated with the Lie algebra of $SU(5)$, one of the earliest grand unified theories. Because the braids form a group, the action of these operators are braids themselves, leading to their identification as gauge bosons. Possible choices for the other two families are also given. Although this realization of particles as braids is lacking a dynamical framework, it is very suggestive, especially when considered as a natural method of adding matter to loop quantum gravity.

gr-qc↗
arXiv2015

Upper limits on the probability of an interstellar civilization arising in the local Solar neighborhood

At this point in time, there is very little empirical evidence on the likelihood of a space-faring species originating in the biosphere of a habitable world. However, there is a tension between the expectation that such a probability is relatively high (given our own origins on Earth), and the lack of any basis for believing the Solar System has ever been visited by an extraterrestrial colonization effort. This paper seeks to place upper limits on the probability of an interstellar civilization arising on a habitable planet in its stellar system, using a percolation model to simulate the progress of such a hypothetical civilization's colonization efforts in the local Solar neighborhood. To be as realistic as possible, the actual physical positions and characteristics of all stars within 40 parsecs of the Solar System are used as possible colony sites in the percolation process. If an interstellar civilization is very likely to have such colonization programs, and they can travel over large distances, then the upper bound on the likelihood of such a species arising per habitable world is on the order of $10^{-3}$; on the other hand, if civilizations are not prone to colonize their neighbors, or do not travel very far, then the upper limiting probability is much larger, even of order one.

physics.pop-ph↗
arXiv2014

Quantifying the Fermi paradox in the local Solar neighborhood

The Fermi paradox highlights the dichotomy between the lack of physical contact with other civilizations and the expectation that technological civilizations are assumed likely to evolve in many locations in the Milky Way galaxy, given the large number of planetary systems within this galaxy. Work by Landis and others has modeled this question in terms of percolation theory and cellular automata, using this method to parametrize our ignorance about possible other civilizations as a function of the probability of one system to colonize another, and the maximum number of systems reachable from each starting location (i.e. the degree in the network used for percolation). These models used a fixed lattice of sites to represent a stellar region, so the degree of all sites were identical. In this paper, the question is examined again, but instead of using a pre-determined lattice, the actual physical positions of all known star systems within 40 parsecs of the Solar System are used as percolation sites; in addition, the number of sites accessible for further colonization from a given system is determined by a choice of maximum distance such efforts can travel across. The resulting simulations show that extraterrestrial colonization efforts may reach the Solar System, but only for certain values of the maximum travel distance and probability of an occupied system further colonizing other systems. Indeed, large numbers of systems may be colonized with either vessels that lack insufficient travel distance to reach the Solar System or else have a colonization probability where they are statistically unlikely to reach us.

physics.pop-ph↗
arXiv2013

Exploration of the local solar neighborhood I: Fixed number of probes

Previous work in studying interstellar exploration by one or several probes has focused primarily either on engineering models for a spacecraft targeting a single star system, or large-scale simulations to ascertain the time required for a civilization to completely explore the Milky Way galaxy. In this paper, a simulated annealing algorithm is used to numerically model the exploration of the local interstellar neighborhood (i.e. on the order of ten parsecs of the Solar System) by a fixed number of probes launched from the Solar System; these simulations use the observed masses, positions and spectral classes of targeted stars. Each probe visits a pre-determined list of target systems, maintains a constant cruise speed, and only changes direction from gravitational deflection at each target. From these simulations, it is examined how varying design choices -- differing the maximum cruise speed, number of probes launched, number of target stars to be explored, and probability of avoiding catastrophic system failure per parsec -- change the completion time of the exploration program and the expected number of stars successfully visited. In addition, it is shown that improving this success probability per parsec has diminishing returns beyond a certain point. Future improvements to the model and possible implications are discussed.

physics.pop-ph↗
arXiv2012

Conserved quantities in isotropic loop quantum cosmology

We develop an action principle for those models arising from isotropic loop quantum cosmology, and show that there is a natural conserved quantity $Q$ for the discrete difference equation arising from the Hamiltonian constraint. This quantity $Q$ relates the semi-classical limit of the wavefunction at large values of the spatial volume, but opposite triad orientations. Moreover, there is a similar quantity for generic difference equations of one parameter arising from a self-adjoint operator.

gr-qc↗
arXiv2012

Numerical Techniques in Loop Quantum Cosmology

In this article, we review the use of numerical techniques to obtain solutions for the quantum Hamiltonian constraint in loop quantum cosmology (LQC). First, we summarize the basic features of LQC, and describe features of the constraint equations to solve - generically, these are difference (rather than differential) equations. Important issues such as differing quantization methods, stability of the solutions, the semi-classical limit, and the relevance of lattice refinement in the difference equations are discussed. Finally, the cosmological models already considered in the literature are listed, along with typical features in these models and open issues.

gr-qc↗
arXiv2011

On the maximum sufficient range of interstellar vessels

This paper considers the likely maximum range of space vessels providing the basis of a mature interstellar transportation network. Using the principle of sufficiency, it is argued that this range will be less than three parsecs for the average interstellar vessel. This maximum range provides access from the Solar System to a large majority of nearby stellar systems, with total travel distances within the network not excessively greater than actual physical distance.

physics.pop-ph↗
arXiv2007

Lattice refining loop quantum cosmology, anisotropic models and stability

A general class of loop quantizations for anisotropic models is introduced and discussed, which enhances loop quantum cosmology by relevant features seen in inhomogeneous situations. The main new effect is an underlying lattice which is being refined during dynamical changes of the volume. In general, this leads to a new feature of dynamical difference equations which may not have constant step-size, posing new mathematical problems. It is discussed how such models can be evaluated and what lattice refinements imply for semiclassical behavior. Two detailed examples illustrate that stability conditions can put strong constraints on suitable refinement models, even in the absence of a fundamental Hamiltonian which defines changes of the underlying lattice. Thus, a large class of consistency tests of loop quantum gravity becomes available. In this context, it will also be seen that quantum corrections due to inverse powers of metric components in a constraint are much larger than they appeared recently in more special treatments of isotropic, free scalar models where they were artificially suppressed.

gr-qc↗
arXiv2006

Wave functions for the Schwarschild black hole interior

Using the Hamiltonian constraint derived by Ashtekar and Bojowald, we look for pre-classical wave functions in the Schwarzschild interior. In particular, when solving this difference equation by separation of variables, an inequality is obtained relating the Immirzi parameter $γ$ to the quantum ambiguity $δ$ appearing in the model. This bound is violated when we use a natural value for $δ$ based on loop quantum gravity together with a recent proposal for $γ$. We also present numerical solutions of the constraint.

gr-qc↗
arXiv2005

Separable sequences in Bianchi I loop quantum cosmology

In this paper, we discuss the properties of one-parameter sequences that arise when solving the Hamiltonian constraint in Bianchi I loop quantum cosmology using a separation of variables method. In particular, we focus on finding an expression for the sequence for all real values of the parameter, and discuss the pre-classicality of this function. We find that the behavior of these preclassical sequences imply time asymmetry on either side of the classical singularity in Bianchi I cosmology.

gr-qc↗
arXiv2005

Absence of pre-classical solutions in Bianchi I loop quantum cosmology

Loop quantum cosmology, the symmetry reduction of quantum geometry for the study of various cosmological situations, leads to a difference equation for its quantum evolution equation. To ensure that solutions of this equation act in the expected classical manner far from singularities, additional restrictions are imposed on the solution. In this paper, we consider the Bianchi I model, both the vacuum case and the addition of a cosmological constant, and show using generating function techniques that only the zero solution satisfies these constraints. This implies either that there are technical difficulties with the current method of quantizing the evolution equation, or else loop quantum gravity imposes strong restrictions on the physically allowed solutions.

gr-qc↗
arXiv2004

Generating function techniques for loop quantum cosmology

Loop quantum cosmology leads to a difference equation for the wave function of a universe, which in general has solutions changing rapidly even when the volume changes only slightly. For a semiclassical regime such small-scale oscillations must be suppressed, by choosing the parameters of the solution appropriately. For anisotropic models this is not possible to do numerically by trial and error; instead, it is shown here for the Bianchi I LRS model how this can be done analytically, using generating function techniques. Those techniques can also be applied to more complicated models, and the results gained allow conclusions about initial value problems for other systems.

gr-qc↗
arXiv2003

The Lanczos potential as a spin-2 field

The Lanczos potential $L_{abc}$ acts as a tensor potential for the spin-2 field strength $W_{abcd}$ in an role similar to that of the vector potential $A_a$ for the Maxwell tensor $F_{ab}$. After some general considerations inspired by the example of electromagnetism, we consider the linear spin-2 theory and a Born-Infeld type action in terms of $L_{abc}$.

hep-th↗
arXiv1999

Linearized general relativity and the Lanczos potential

Recently, there has been a revival of interest in the Lanczos potential of the Weyl conformal tensor. Previous work by Novello and Neto has been done with the linearized Lanczos potential as a model of a spin-2 field, which depends on a massless limit of the field. In this paper, we look at an action based on a massless potential, and show that it is classically equivalent to the linearized regime of general relativity, without reference to a massless limit.

gr-qc↗
arXiv1997

Generalized Symplectic Manifolds

This paper uses a generalization of symplectic geometry, known as $n$-symplectic geometry and developed by Norris, to find observables on three-dimensional manifolds. It will be seen that for the cases considered, the $n$-symplectic observables are derivable from the symplectic observables of $C^2$. The quantization of these observables, as well as those on the frame bundle of $R^n$, is also examined.

dg-ga↗