Effective Quantum Extended Spacetime of Polymer Schwarzschild Black Hole

The physical interpretation and eventual fate of gravitational singularities in a theory surpassing classical general relativity are puzzling questions that have generated a great deal of interest among various quantum gravity approaches. In the context of loop quantum gravity (LQG), one of the major candidates for a non-perturbative background-independent quantisation of general relativity, considerable effort has been devoted to construct effective models in which these questions can be studied. In these models, classical singularities are replaced by a "bounce" induced by quantum geometry corrections. Undesirable features may arise however depending on the details of the model. In this paper, we focus on Schwarzschild black holes and propose a new effective quantum theory based on polymerisation of new canonical phase space variables inspired by those successful in loop quantum cosmology. The quantum corrected spacetime resulting from the solutions of the effective dynamics is characterised by infinitely many pairs of trapped and anti-trapped regions connected via a space-like transition surface replacing the central singularity. Quantum effects become relevant at a unique mass independent curvature scale, while they become negligible in the low curvature region near the horizon. The effective quantum metric describes also the exterior regions and asymptotically classical Schwarzschild geometry is recovered. We however find that physically acceptable solutions require us to select a certain subset of initial conditions, corresponding to a specific mass (de-)amplification after the bounce. We also sketch the corresponding quantum theory and explicitly compute the kernel of the Hamiltonian constraint operator.Comment: 50 pages, 10 figures; v2: journal version, minor comment and references added; v3: minor corrections in section 5.3 to match journal versio

A note on the Hamiltonian as a polymerisation parameter

In effective models of loop quantum gravity, the onset of quantum effects is controlled by a so-called polymerisation scale. It is sometimes necessary to make this scale phase space dependent in order to obtain sensible physics. A particularly interesting choice recently used to study quantum corrected black hole spacetimes takes the generator of time translations itself to set the scale. We review this idea, point out errors in recent treatments, and show how to fix them in principle.Comment: 7 pages, 2 figures; v2: journal version, minor clarification

Fisher Metric, Geometric Entanglement and Spin Networks

Starting from recent results on the geometric formulation of quantum mechanics, we propose a new information geometric characterization of entanglement for spin network states in the context of quantum gravity. For the simple case of a single-link fixed graph (Wilson line), we detail the construction of a Riemannian Fisher metric tensor and a symplectic structure on the graph Hilbert space, showing how these encode the whole information about separability and entanglement. In particular, the Fisher metric defines an entanglement monotone which provides a notion of distance among states in the Hilbert space. In the maximally entangled gauge-invariant case, the entanglement monotone is proportional to a power of the area of the surface dual to the link thus supporting a connection between entanglement and the (simplicial) geometric properties of spin network states. We further extend such analysis to the study of non-local correlations between two non-adjacent regions of a generic spin network graph characterized by the bipartite unfolding of an Intertwiner state. Our analysis confirms the interpretation of spin network bonds as a result of entanglement and to regard the same spin network graph as an information graph, whose connectivity encodes, both at the local and non-local level, the quantum correlations among its parts. This gives a further connection between entanglement and geometry.Comment: 29 pages, 3 figures, revised version accepted for publicatio

On the Role of Fiducial Structures in Minisuperspace Reduction and Quantum Fluctuations in LQC

We study the homogeneous minisuperspace reduction within the canonical framework for a scalar field theory and gravity. Symmetry reduction is implemented via second class constraints for the field modes over a partitioning of the non-compact spatial slice $\Sigma$ into disjoint cells. The canonical structure of the resulting homogeneous theories is obtained via the associated Dirac bracket which can only be defined on a finite number of cells homogeneously patched together and agrees with the full theory Poisson bracket for the averaged fields. This identifies a finite region $V_o$, the fiducial cell, whose size $L$ sets the physical scale over which homogeneity is imposed, namely a wavelength cutoff. The reduced theory results from 1) selecting a subset of $V_o$-averaged observables of the full theory; 2) neglecting inhomogeneous $\vec k\neq\mathbf0$ modes with wavelengths $\lambda\geq L$ and $\lambda< L$; 3) neglecting boundary terms encoding interactions between neighbouring cells. The error made is of order $\mathcal O(1/kL)$. As a result, the off-shell structures of the reduced theory depend on the size of $V_o$ and different $V_o$ identify canonically inequivalent theories whose dynamics though is $V_o$-independent. Their quantisation leads then to a family of $V_o$-labeled quantum representations and the quantum version of an active rescaling of $V_o$ is implemented via a suitable dynamics-preserving isomorphism between the different theories. We discuss the consequences for statistical moments, fluctuations, and semiclassical states in both a standard and polymer quantisation. For a scalar field of mass $m$, we also sketch the quantum reduction and identify a subsector of the QFT where the results of the"first reduced, then quantised" theories can be reproduced with good approximation as long as $m\gg1/L$. Finally, a strategy to include inhomogeneities in cosmology is outlined.Comment: 71 + 13 pages, 4 figure

A Pedagogical Intrinsic Approach to Relative Entropies as Potential Functions of Quantum Metrics: the qq-zz Family

The so-called $q$-z-\textit{R\'enyi Relative Entropies} provide a huge two-parameter family of relative entropies which includes almost all well-known examples of quantum relative entropies for suitable values of the parameters. In this paper we consider a log-regularized version of this family and use it as a family of potential functions to generate covariant $(0,2)$ symmetric tensors on the space of invertible quantum states in finite dimensions. The geometric formalism developed here allows us to obtain the explicit expressions of such tensor fields in terms of a basis of globally defined differential forms on a suitable unfolding space without the need to introduce a specific set of coordinates. To make the reader acquainted with the intrinsic formalism introduced, we first perform the computation for the qubit case, and then, we extend the computation of the metric-like tensors to a generic $n$-level system. By suitably varying the parameters $q$ and $z$, we are able to recover well-known examples of quantum metric tensors that, in our treatment, appear written in terms of globally defined geometrical objects that do not depend on the coordinates system used. In particular, we obtain a coordinate-free expression for the von Neumann-Umegaki metric, for the Bures metric and for the Wigner-Yanase metric in the arbitrary $n$-level case.Comment: 50 pages, 1 figur

(b,v)-type variables for black to white hole transitions in effective loop quantum gravity

Quantum gravity effects in effective models of loop quantum gravity, such as loop quantum cosmology, are encoded in the choice of so-called polymerisation schemes. Physical viability of the models, such as an onset of quantum effects at curvature scales near the Planck curvature, severely restrict the possible choices. An alternative point of view on the choice of polymerisation scheme is to choose adapted variables so that the scheme is the simplest possible one, known as $\mu_0$-scheme in loop quantum cosmology. There, physically viable models with $\mu_0$-scheme polymerise the Hubble rate $b$ that is directly related to the Ricci scalar and the matter energy density on-shell. Consequently, the onset of quantum effects depends precisely on those parameters. In this letter, we construct similar variables for black to white hole transitions modelled using the description of the Schwarzschild interior as a Kantowski-Sachs cosmology. The resulting model uses the $\mu_0$-scheme and features sensible physics for a broad range of initial conditions (= choices of black and white hole masses) and favours symmetric transitions upon invoking additional qualitative arguments. The resulting Hamiltonian is very simple and at most quadratic in its arguments, allowing for a straight forward quantisation.Comment: 13 pages, 1 figure, published version, minor improvements, references added, conclusions expande

Is limiting curvature mimetic gravity an effective polymer quantum gravity?

A recently proposed version of mimetic gravity incorporates a limiting curvature into general relativity by means of a specific potential depending on the d'Alembertian of the scalar field. In the homogeneous and isotropic setting, the resulting theory agrees with the simplest incarnation of effective loop quantum cosmology (LQC) once the limiting curvature is identified with a multiple of the Planck scale. In this paper, we answer the question of whether such a relation can hold in the context of Bianchi I models. Our result is negative: it turns out to be impossible to view the Hamiltonian of limiting curvature mimetic gravity as an effective LQC Hamiltonian due to the appearance of terms that cannot be supported on the polymer Hilbert space underlying LQC. The present analysis complements a related result in the context of spherical symmetry.Comment: 13 pages; v2: references added; v3: journal version, minor clarification

Quantum Frame Relativity of Subsystems, Correlations and Thermodynamics

It was recently noted that different internal quantum reference frames (QRFs) partition a system in different ways into subsystems, much like different inertial observers in special relativity decompose spacetime in different ways into space and time. Here we expand on this QRF relativity of subsystems and elucidate that it is the source of all novel QRF dependent effects, just like the relativity of simultaneity is the origin of all characteristic special relativistic phenomena. We show that subsystem relativity, in fact, also arises in special relativity with internal frames and, by implying the relativity of simultaneity, constitutes a generalisation of it. Physical consequences of the QRF relativity of subsystems, which we explore here systematically, and the relativity of simultaneity may thus be seen in similar light. We focus on investigating when and how subsystem correlations and entropies, interactions and types of dynamics (open vs. closed), as well as quantum thermodynamical processes change under QRF transformations. We show that thermal equilibrium is generically QRF relative and find that, remarkably, $\textit{QRF transformations not only can change a subsystem temperature, but even map positive into negative temperature states}$. We further examine how non-equilibrium notions of heat and work exchange, as well as entropy production and flow depend on the QRF. Along the way, we develop the first study of how reduced subsystem states transform under QRF changes. Focusing on physical insights, we restrict to ideal QRFs associated with finite abelian groups. Besides being conducive to rigour, the ensuing finite-dimensional setting is where quantum information-theoretic quantities and quantum thermodynamics are best developed. We anticipate, however, that our results extend qualitatively to more general groups and frames, and even to subsystems in gauge theory and gravity.Comment: 49 pages + appendices, 12 figures. Comments welcom

The rate of sedimentation from turbulent suspension: an experimental model with application to pyroclastic density currents and discussion on the grain-size dependence of flow runout

Large‐scale experiments generating ground‐hugging multiphase flows were carried out with the aim of modelling the rate of sedimentation, of pyroclastic density currents. The current was initiated by the impact on the ground of a dense gas‐particle fountain issuing from a vertical conduit. On impact, a thick massive deposit was formed. The grain size of the massive deposit was almost identical to that of the mixture feeding the fountain, suggesting that similar layers formed at the impact of a natural volcanic fountain should be representative of the parent grain‐size distribution of the eruption. The flow evolved laterally into a turbulent suspension current that sedimented a thin, tractive layer. A good correlation was found between the ratio of transported/sedimented load and the normalized Rouse number of the turbulent current. A model of the sedimentation rate was developed, which shows a relationship between grain size and flow runout. A current fed with coarser particles has a higher sedimentation rate, a larger grain‐size selectivity and runs shorter than a current fed with finer particles. Application of the model to pyroclastic deposits of Vesuvius and Campi Flegrei of Southern Italy resulted in sedimentation rates falling inside the range of experiments and allowed definition of the duration of pyroclastic density currents which add important information on the hazard of such dangerous flows. The model could possibly be extended, in the future, to other geological density currents as, for example, turbidity currents