arXiv subjects

Pietro Dona

Publications and source records attributed to Pietro Dona.

18 recordsLinked to original sources

arXiv2024

Tunneling of quantum geometries in spinfoams

Quantum gravitational tunneling effects are expected to give rise to a number of interesting observable phenomena, including, in particular, the evolution of black holes at the end of their existence or the emergence of the early universe from a quantum phase. Covariant Loop Quantum Gravity provides a framework to study these phenomena, yet a precise identification of tunneling processes is still not known. Motivated by this question, we consider a related, simpler case, that of Ponzano-Regge amplitudes: we find a surprising and detailed analogy of a class of simple transition amplitudes with tunneling processes in non-relativistic quantum mechanics.

gr-qc↗

arXiv2024

Non-abelian symmetry-resolved entanglement entropy

We introduce a mathematical framework for symmetry-resolved entanglement entropy with a non-abelian symmetry group. To obtain a reduced density matrix that is block-diagonal in the non-abelian charges, we define subsystems operationally in terms of subalgebras of invariant observables. We derive exact formulas for the average and the variance of the typical entanglement entropy for the ensemble of random pure states with fixed non-abelian charges. We focus on compact, semisimple Lie groups. We show that, compared to the abelian case, new phenomena arise from the interplay of locality and non-abelian symmetry, such as the asymmetry of the entanglement entropy under subsystem exchange, which we show in detail by computing the Page curve of a many-body system with SU(2) symmetry.

quant-ph↗

arXiv2023

Summing bulk quantum numbers with Monte Carlo in spin foam theories

We introduce a strategy to compute EPRL spin foam amplitudes with many internal faces numerically. We work with sl2cfoam-next, the state-of-the-art framework to numerically evaluate spin foam transition amplitudes. We find that uniform sampling Monte Carlo is exceptionally effective in approximating the sum over internal quantum numbers of a spin foam amplitude, considerably reducing the computational resources necessary. We apply it to compute large volume divergences of the theory and find surprising numerical evidence that the EPRL vertex renormalization amplitude is instead finite.

gr-qc↗

arXiv2022

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 $\texttt{sl2cfoam-next}$, 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.

gr-qc↗

arXiv2022

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.

gr-qc↗

arXiv2022

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.

gr-qc↗

arXiv2021

Asymptotics of $\mathrm{SL}(2,\mathbb{C})$ coherent invariant tensors

We study the semiclassical limit of a class of invariant tensors for infinite-dimensional unitary representations of $\mathrm{SL}(2,\mathbb{C})$ of the principal series, corresponding to generalized Clebsch-Gordan coefficients with $n\geq3$ legs. We find critical configurations of the quantum labels with a power-law decay of the invariants. They describe 3d polygons that can be deformed into one another via a Lorentz transformation. This is defined viewing the edge vectors of the polygons are the electric part of bivectors satisfying a (frame-dependent) relation between their electric and magnetic parts known as $γ$-simplicity in the loop quantum gravity literature. The frame depends on the SU(2) spin labelling the basis elements of the invariants. We compute a saddle point approximation using the critical points and provide a leading-order approximation of the invariants. The power-law is universal if the SU(2) spins have their lowest value, and $n$-dependent otherwise. As a side result, we provide a compact formula for $γ$-simplicity in arbitrary frames. The results have applications to the current EPRL model, but also to future research aiming at going beyond the use of fixed time gauge in spin foam models.

gr-qc↗

arXiv2021

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 $Spin(4)$ and $SL(2,\mathbb{C})$, applied to the decomposition of the two models in terms of $SU(2)$ invariants and booster functions.

gr-qc↗

arXiv2021

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.

gr-qc↗

arXiv2021

Numerical analysis of spin foam dynamics and the flatness problem

In this paper, we apply a recently proposed numerical algorithm for finding stationary phase points in spin foam amplitudes. We study a spin foam amplitude with three vertices and a bulk face in 4d BF theory. We fix the boundary coherent states to three possible triangulations, one with zero deficit angle on the bulk face and two with non-zero deficit angle. We compute the amplitude numerically and we find a stationary phase point already at low spins in all the three cases. We comment on how this result contrasts with the claims of flatness problem in spin foam theories. We point out where these arguments may be misleading and we propose further computations to definitively answer the question.

gr-qc↗

arXiv2020

Numerical study of the Lorentzian Engle-Pereira-Rovelli-Livine spin foam amplitude

The Lorentzian EPRL spin foam amplitude for loop quantum gravity is a multi-dimensional non-compact integral of highly oscillating functions. Using a method based on the decomposition of Clebsch-Gordan coefficients for the unitary infinite-dimensional representations of SL(2,C) in terms of those of SU(2), we are able to provide for the first time numerical evaluations of the vertex amplitude. The values obtained support the asymptotic formula obtained by Barrett and collaborators with a saddle point approximation, showing, in particular, a power-law decay and oscillations related to the Regge action. The comparison offers a test of the efficiency of the method. Truncating the decomposition to the first few terms provides a qualitative matching of the power-law decay and oscillations. For vector and Euclidean Regge boundary data, a qualitative matching is obtained with just the first term, which corresponds to the simplified EPRL model. We comment on future developments for the numerics and extension to higher vertices. We complete our work with some analytic results: We provide an algorithm and explicit configurations for the different geometries that can arise as boundary data, and explain the geometric consequences of the decomposition used.

gr-qc↗

arXiv2020

Non Bunch Davies group coherent states, and their quantum signatures in CMB observables

We approach the problem of finding generalized states for matter fields in a de Sitter universe, moving from a group theoretical point of view. This has profound consequences for cosmological perturbations during inflation, and for other CMB observables. In a systematic derivation, we find all the allowed generalities that such a state may have, within the requirements of coherent states. Furthermore, we show their relation to the more familiar excited initial states, often used in inflation.

gr-qc↗

arXiv2020

Searching for classical geometries in spin foam amplitudes: a numerical method

We develop a numerical method to investigate the semiclassical limit of spin foam amplitudes with many vertices. We test it using the Ponzano-Regge model, a spin foam model for three-dimensional euclidean gravity, and a transition amplitude with three vertices. We study the summation over bulk spins, and we identify the stationary phase points that dominate it and that correspond to classical geometries. We complement with the numerical analysis of a four vertex transition amplitude and with a modification of the model that includes local curvature. We discuss the generalization of our results to the four-dimensional EPRL spin foam model, and we provide suggestions for new computations.

gr-qc↗

arXiv2019

Typical entanglement entropy in the presence of a center: Page curve and its variance

In a quantum system in a pure state, a subsystem generally has a nonzero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? We present a method to compute the exact formula of the momenta of the probability $P(S_A) \mathrm{d}S_A$ that a subsystem has entanglement entropy $S_A$. The method applies to subsystems defined by a subalgebra of observables with a center. In the case of a trivial center, we reobtain the well-known result for the average entropy and the formula for the variance. In the presence of a nontrivial center, the Hilbert space does not have a tensor product structure and the well-known formula does not apply. We present the exact formula for the average entanglement entropy and its variance in the presence of a center. We show that for large systems the variance is small, $ΔS_A/\langle{S_{A}}\rangle\ll 1$, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems where a center arises. In particular, for a system of noninteracting spins in a magnetic field and for a free quantum field, we show how the thermal entropy arises as the typical entanglement entropy of energy eigenstates.

hep-th↗

arXiv2019

Entanglement entropy of Bell-network states in LQG: Analytical and numerical results

Bell-network states are loop-quantum-gravity states that glue quantum polyhedra with entanglement. We present an algorithm and a code that evaluates the reduced density matrix of a Bell-network state and computes its entanglement entropy. In particular, we use our code for simple graphs to study properties of Bell-network states and to show that they are non-typical in the Hilbert space. Moreover, we investigate analytically Bell-network states on arbitrary finite graphs. We develop methods to compute the Rényi entropy of order p for a restriction of the state to an arbitrary region. In the uniform large-spin regime, we determine bounds on the entanglement entropy and show that it obeys an area law. Finally, we discuss the implications of our results for correlations of geometric observables.

gr-qc↗

arXiv2018

Numerical methods for EPRL spin foam transition amplitudes and Lorentzian recoupling theory

The intricated combinatorial structure and the non-compactness of the Lorentz group have always made the computation of $SL(2,\mathbb{C})$ EPRL spin foam transition amplitudes a very hard and resource demanding task. With \texttt{sl2cfoam} we provide a C-coded library for the evaluation of the Lorentzian EPRL vertex amplitude. We provide a tool to compute the Lorentzian EPRL 4-simplex vertex amplitude in the intertwiner basis and some utilities to evaluate SU(2) invariants, booster functions and $SL(2,\mathbb{C})$ Clebsch-Gordan coefficients. We discuss the data storage, parallelizations, time, and memory performances and possible future developments.

gr-qc↗

arXiv2018

Infrared divergences in the EPRL-FK Spin Foam model

We provide an algorithm to estimate the divergence degree of the Lorentzian EPRL-FK spin foam amplitudes for arbitrary 2-complexes. We focus on the "self-energy" and "vertex renormalization" diagrams and find an upper bound estimate. We argue that our upper bound must be close to the actual value, and explain what numerical improvements are needed to verify this numerically. For the self-energy, this turns out to be significantly more divergent than the lower bound estimate present in the literature. We support the validity of our algorithm using 3-stranded versions of the amplitudes (corresponding to a toy 3d model) for which our estimates are confirmed numerically. We also apply our methods to the simplified EPRLs model, finding a completely convergent behavior, and to BF theory, independently recovering the divergent estimates present in the literature.

gr-qc↗

arXiv2016

A note on the semiclassicality of cosmological perturbations

Moving from the consideration that matter fields must be treated in terms of their fundamental quantum counterparts, we show straightforward arguments, within the framework of ordinary quantum mechanics and quantum field theory, in order to convince readers that cosmological perturbations must be addressed in term of the semiclassical limit of the expectation value of quantum fields. We first take into account cosmological perturbations originated by a quantum scalar field, and then extend our treatment in order to account for the expectation values of bilinears of Dirac fermion fields. The latter can indeed transform as scalar quantities under diffeomorphisms, as well as all the other bilinear of the Dirac fields that belong to the Clifford algebra. This is the first of a series of works that is intended to prove that cosmological quantum perturbations can actually be accounted for in terms of Dirac fermion fields, which must be treated as fundamental quantum objects, and their dynamics.

gr-qc↗