Quantum Mechanics of a Point Particle in 2+1 Dimensional Gravity

We study the phase space structure and the quantization of a pointlike particle in 2+1 dimensional gravity. By adding boundary terms to the first order Einstein Hilbert action, and removing all redundant gauge degrees of freedom, we arrive at a reduced action for a gravitating particle in 2+1 dimensions, which is invariant under Lorentz transformations and a group of generalized translations. The momentum space of the particle turns out to be the group manifold SL(2). Its position coordinates have non-vanishing Poisson brackets, resulting in a non-commutative quantum spacetime. We use the representation theory of SL(2) to investigate its structure. We find a discretization of time, and some semi-discrete structure of space. An uncertainty relation forbids a fully localized particle. The quantum dynamics is described by a discretized Klein Gordon equation.Comment: 58 pages, 3 eps figures, presentation of the classical theory improve

Coherent States for 3d Deformed Special Relativity: semi-classical points in a quantum flat spacetime

We analyse the quantum geometry of 3-dimensional deformed special relativity (DSR) and the notion of spacetime points in such a context, identified with coherent states that minimize the uncertainty relations among spacetime coordinates operators. We construct this system of coherent states in both the Riemannian and Lorentzian case, and study their properties and their geometric interpretation.Comment: RevTeX4, 20 page

Two particle Quantummechanics in 2+1 Gravity using Non Commuting Coordinates

We find that the momentum conjugate to the relative distance between two gravitating particles in their center of mass frame is a hyperbolic angle. This fact strongly suggests that momentum space should be taken to be a hyperboloid. We investigate the effect of quantization on this curved momentum space. The coordinates are represented by non commuting, Hermitian operators on this hyperboloid. We also find that there is a smallest distance between the two particles of one half times the Planck length.Comment: 18 pages Latex, 2 eps figure

Tall tales from de Sitter space II: Field theory dualities

We consider the evolution of massive scalar fields in (asymptotically) de Sitter spacetimes of arbitrary dimension. Through the proposed dS/CFT correspondence, our analysis points to the existence of new nonlocal dualities for the Euclidean conformal field theory. A massless conformally coupled scalar field provides an example where the analysis is easily explicitly extended to 'tall' background spacetimes.Comment: 31 pages, 2 figure

About Lorentz invariance in a discrete quantum setting

A common misconception is that Lorentz invariance is inconsistent with a discrete spacetime structure and a minimal length: under Lorentz contraction, a Planck length ruler would be seen as smaller by a boosted observer. We argue that in the context of quantum gravity, the distance between two points becomes an operator and show through a toy model, inspired by Loop Quantum Gravity, that the notion of a quantum of geometry and of discrete spectra of geometric operators, is not inconsistent with Lorentz invariance. The main feature of the model is that a state of definite length for a given observer turns into a superposition of eigenstates of the length operator when seen by a boosted observer. More generally, we discuss the issue of actually measuring distances taking into account the limitations imposed by quantum gravity considerations and we analyze the notion of distance and the phenomenon of Lorentz contraction in the framework of ``deformed (or doubly) special relativity'' (DSR), which tentatively provides an effective description of quantum gravity around a flat background. In order to do this we study the Hilbert space structure of DSR, and study various quantum geometric operators acting on it and analyze their spectral properties. We also discuss the notion of spacetime point in DSR in terms of coherent states. We show how the way Lorentz invariance is preserved in this context is analogous to that in the toy model.Comment: 25 pages, RevTe

Kemijski profil sedimenata Plominskog zaljeva

Granulometric, chemical, and leaching properties of sediments dredged in the Plomin Bay (Northern Adriatic Sea, Croatia) were investigated in order to asses the risk of remobilisation of heavy metals into the water column. In total 65 samples from 65 sampling sites were taken from different sediment depths within the bay. Analysis of variance confirmed the homogeneity of granulometric and elemental composition of the investigated sediment throughout its volume. Granulometric analysis showed that all samples corresponded to a pelitic fraction (<0.063 mm). Bulk elemental mass fractions in the sediments were similar to literature data on relatively unpolluted areas of the Adriatic Sea. High sedimentation rate caused by constant infl ow of material from the Boljunčica River drainage may be responsible for low levels of heavy metals and negligible infl uence of fl y and bottom ash from a nearby disposal site on the chemical composition of the sediments. In contact with sea water only 0.29 mg kg-1 of V, 0.04 mg kg-1 of Cr, 0.07 mg kg-1 of Ni, 0.33 mg kg-1 of Cu, 0.67 mg kg-1 of Zn and 0.06 mg kg-1 of Pb could be remobilised from sediment material into the water column. However, these values increased three to ten times in case of leaching with organic acids.Granulometrijska i kemijska svojstva te mogućnost otpuštanja teških metala ispitivani su u sedimentima Plominskog zaljeva (Sjeverni Jadran, Hrvatska) u svrhu utvrđivanja rizika od remobilizacije teških metala iz sedimenta u stupac vode. Uzeto je 65 uzoraka s različitih točaka i dubina unutar zaljeva. Analizom varijance potvrđena je granulometrijska i kemijska homogenost cijelog volumena sedimenta, što upućuje na jedan prevladavajući izvor tijekom cijeloga sedimentacijskog razdoblja. Granulometrijskom analizom je utvrđeno da u svim uzorcima prevladava sitnozrnata frakcija (<0,063 mm). Koncentracije elemenata u ukupnim uzorcima sedimenata slične su literaturnim vrijednostima objavljenim za relativno onečišćena područja Jadranskog mora. Velika brzina sedimentacije uzrokovana konstantnim donosom materijala iz slijevnog područja Boljunčice vjerojatan je uzrok niskih koncentracija teških metala i slabo vidljivog utjecaja odlagališta šljake i pepela na sastav sedimenata. U kontaktu s morskom vodom moguća je remobilizacija samo 0,29 mg kg-1 V, 0,04 mg kg-1 Cr, 0,07 mg kg-1 Ni, 0,33 mg kg-1 Cu, 0,67 mg kg-1 Zn i 0,06 mg kg-1 Pb iz sedimenta u stupac morske vode. Ipak ove vrijednosti su tri do deset puta povećane u slučaju izluživanja s pomoću organskih kiselina

A small-time coupling between Λ-coalescents and branching processes

We describe a new general connection between Λ-coalescents and genealogies of continuous-state branching processes. This connection is based on the construction of an explicit coupling using a particle representation inspired by the lookdown process of Donnelly and Kurtz. This coupling has the property that the coalescent comes down from infinity if and only if the branching process becomes extinct, thereby answering a question of Bertoin and Le Gall. The coupling also offers new perspective on the speed of coming down from infinity and allows us to relate power-law behavior for NΛ(t) to the classical upper and lower indices arising in the study of pathwise properties of Lévy processes. © Institute of Mathematical Statistics, 2014

Asymptotic sampling formulae for Λ-coalescents

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The Λ-coalescent speed of coming down from infinity

Consider a Λ-coalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number Nt of blocks at any positive time t&gt;0). We exhibit a deterministic function υ: (0, ∞) → (0, ∞) such that Nt/υ(t) → 1, almost surely, and in Lp for any p ≥ 1, as t → 0. Our approach relies on a novel martingale technique. © Institute of Mathematical Statistics, 2010

Asymptotic sampling formulae for Λ-coalescents

We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a Λ-coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to ∞. Some of our results hold in the case of a general Λ-coalescent that comes down from infinity, but we obtain more precise information under a regular variation assumption. In this case, we obtain results of independent interest for the time at which a mutation uniformly chosen at random was generated. This exhibits a phase transition at α = 3/2, where α ∈ (1, 2) is the exponent of regular variation. © Association des Publications de l'Institut Henri Poincaré, 2014