35 research outputs found
Polytopes and Loop Quantum Gravity
The main aim of this thesis is to give a geometrical interpretation of ``spacetime grains'' at Planck scales in the framework of Loop Quantum Gravity.
My work consisted in analyzing the details of the interpretation of the quanta of space in terms of polytopes. The main results I obtained are the following:
We clarified details on the relation between polytopes and interwiners, and concluded that an intertwiner can be seen unambiguously as the state of a \emph{quantum polytope}.
Next we analyzed the properties of these polytopes: studying how to reconstruct the solid figure from LQG variables, the possible shapes and the volume. We adapted existing algorithms to express the geometry of the polytopes in terms of the holonomy-fluxes variables of LQG, thus providing an explicit bridge between the original variables and the interpretation in terms of polytopes of the phase space.
Finally we present some direct application of this geometrical picture. We defined a volume operator such as in the large spin limit it reproduce the geometrical volume of a polytope, we computed numerically his spectrum for some elementary cases and we pointed out some asymptotic property of his spectrum. We discuss applications of the picture in terms of polytopes to the study of the semiclassical limit of LQG, in particular commenting a connection between the quantum dynamics and a generalization of Regge calculus on polytopes
Geometry from local flatness in Lorentzian spin foam theories
Local flatness is a property shared by all the spin foam models. It ensures
that the theory's fundamental building blocks are flat by requiring locally
trivial parallel transport. In the context of simplicial Lorentzian spin foam
theory, we show that local flatness is the main responsible for the emergence
of geometry independently of the details of the spin foam model. We discuss the
asymptotic analysis of the EPRL spin foam amplitudes in the large quantum
number regime, highlighting the interplay with local flatness.Comment: 20 pages, 4 figures. Comments are welcom
How-To compute EPRL spin foam amplitudes
Spin foam theory is a concrete framework for quantum gravity where numerical
calculations of transition amplitudes are possible. Recently, the field became
very active, but the entry barrier is steep, mainly because of its unusual
language and notions scattered around the literature. This paper is a
pedagogical guide to spin foam transition amplitude calculations. We show how
to write an EPRL-FK transition amplitude, from the definition of the 2-complex
to its numerical implementation using \texttt{sl2cfoam-next}. We guide the
reader using an explicit example balancing mathematical rigor with a practical
approach. We discuss the advantages and disadvantages of this approach and
provide a novel look at a recently proposed approximation scheme.Comment: 28 pages with many colored figures. Paper published in the special
issue "Probing the Quantum Space-Time" of Universe. v-2 Added introductory
section. Matching published versio
Asymptotics of lowest unitary SL(2,C) invariants on graphs
We describe a technique to study the asymptotics of SL(2,C) invariant tensors
associated to graphs, with unitary irreps and lowest SU(2) spins, and apply it
to the Lorentzian EPRL-KKL (Engle, Pereira, Rovelli, Livine; Kaminski,
Kieselowski, Lewandowski) model of quantum gravity. We reproduce the known
asymptotics of the 4-simplex graph with a different perspective on the
geometric variables and introduce an algorithm valid for any graph. On general
grounds, we find that critical configurations are not just Regge geometries,
but a larger set corresponding to conformal twisted geometries. These can be
either Euclidean or Lorentzian, and include curved and flat 4d polytopes as
subsets. For modular graphs, we show that multiple pairs of critical points
exist, and there exist critical configurations of mixed signature, Euclidean
and Lorentzian in different subgraphs, with no 4d embedding possible.Comment: 40 Pages + 5 Appendices. 11 Figures. v2: Refined presentation of the
general algorithm, additional minor amendments. v3: paragraph added in
section 5 about curved embedding
A Wick rotation for EPRL spin foam models
We show that the Euclidean and Lorentzian EPRL vertex amplitudes of covariant
Loop Quantum Gravity are related through a ``Wick rotation'' of the real
Immirzi parameter to purely imaginary values. Our result follows from the
simultaneous analytic continuation of the algebras, group elements and unitary
irreducible representations of the gauge groups and
, applied to the decomposition of the two models in terms of
invariants and booster functions.Comment: 25 pages, 1 figur
Spinfoams and high performance computing
Numerical methods are a powerful tool for doing calculations in spinfoam
theory. We review the major frameworks available, their definition, and various
applications. We start from , the state-of-the-art
library to efficiently compute EPRL spin foam amplitudes based on the booster
decomposition. We also review two alternative approaches based on the
integration representation of the spinfoam amplitude: Firstly, the numerical
computations of the complex critical points discover the curved geometries from
the spinfoam amplitude and provides important evidence of resolving the
flatness problem in the spinfoam theory. Lastly, we review the numerical
estimation of observable expectation values based on the Lefschetz thimble and
Markov-Chain Monte Carlo method, with the EPRL spinfoam propagator as an
example.Comment: 33 pages, 11 figures. Invited chapter for the book "Handbook of
Quantum Gravity" (Eds. C. Bambi, L. Modesto and I.L. Shapiro, Springer
Singapore, expected in 2023