arXiv subjects

Pedro Naranjo

Publications and source records attributed to Pedro Naranjo.

14 recordsLinked to original sources

arXiv2025

Pure Shape Dynamics: Relational General Relativity

We present a Pure Shape Dynamics (PSD) formulation of General Relativity (GR), which implements full relationalism by eliminating absolute scale and external time references from the fundamental description of gravity. Starting from the Arnowitt-Deser-Misner (ADM) formulation, we derive a decoupled dynamical system that governs the evolution of the spatial conformal geometry and relational matter degrees of freedom, while eliminating the total volume and York time as independent dynamical variables. This results in an autonomous subsystem describing an unparametrized trajectory in the conformal superspace of metric and matter configurations, with its evolution encoded in an equation of state that characterises the intrinsic geometric properties of the curve in shape space. We show that this equation of state is structurally analogous to the corresponding PSD description of the Newtonian $N$-body problem, reinforcing the fundamental similarity between gravity and relational particle dynamics. Our framework is applied to the homogeneous Bianchi IX cosmological model, demonstrating that the Janus point evolution through the Big Bang, as previously found in a symmetry-reduced setting, is a generic feature of the full inhomogeneous PSD description. This work establishes PSD as a fully scale- and reparametrization-invariant formulation of classical gravity and lays the foundation for addressing key open questions that are discussed at the end of the paper.

gr-qc↗

arXiv2024

A proposal for a metaphysics of self-subsisting structures. II. Quantum physics

The paper presents an extension of the metaphysics of self-subsisting structures set out in a companion paper to the realm of non-relativistic quantum physics. The discussion is centered around a Pure Shape Dynamics model representing a relational implementation of a de Broglie-Bohm $N$-body system. An interpretation of this model in terms of self-subsisting structures is proposed and assessed against the background of the debate on the metaphysics of quantum physics, with a particular emphasis on the nature of the wave function. The analysis shows that elaborating an appropriate Leibnizian/Machian metaphysics of the quantum world requires a substantial revision of the notion of world-building relation.

quant-ph↗

arXiv2024

Pure shape dynamics, self-subsisting structures, and the nature of time

The paper discusses the possible implications of the relational framework of Pure Shape Dynamics for the metaphysics of time. The starting point of the analysis is an interpretation of shapes in ontic structural realist terms, which gives rise to the notion of self-subsisting structure. The relational version of a Newtonian-particle toy model is introduced and discussed as a concrete example.

physics.hist-ph↗

arXiv2023

On the Prospects of a de Broglie-Bohm-Barbour-Bertotti Theory

Pure shape dynamics (PSD) is a novel implementation of the relational framework originally proposed by Julian Barbour and Bruno Bertotti. PSD represents a Leibnizian/Machian approach to physics in that it completely describes the dynamical evolution of a physical system without resorting to any structure external to the system itself. The chapter discusses how PSD effectively describes a de Broglie-Bohm N-body system and the conceptual benefits of such a relational description. The analysis will highlight the new directions in the quest for an understanding of the nature of the wave function that are opened up by a modern relationalist elaboration on de Broglie's and Bohm's original insights.

quant-ph↗

arXiv2023

A de Broglie-Bohm Model of Pure Shape Dynamics: $N$-body system

We provide the construction of a de Broglie-Bohm model of the $N$-body system within the framework of Pure Shape Dynamics. The equation of state of the curve in shape space is worked out, with the instantaneous shape being guided by a wave function. In order to get a better understanding of the dynamical system, we also give some numerical analysis of the 3-body case. Remarkably enough, our simulations typically show the attractor-driven behaviour of complexity, well known in the classical case, thereby providing further evidence for the claim that the arrow of complexity is the ultimate cause of the experienced arrow of time.

gr-qc↗

arXiv2022

Pure shape dynamics: General framework

We put forward a general framework for describing relational physical theories, which we call Pure Shape Dynamics (PSD). Elaborating on the original insights brought about by the Shape Dynamics program, PSD's novel take on relationalism is its insistence on describing any dynamical system by means of the intrinsic geometry of its associated curve in the suitable relational configuration space of the theory, namely shape space, whereby the corresponding equation of state of the curve expresses the ratio of change of one of its geometric degrees of freedom with respect to another one. The mathematical structure underlying the equation of state is a local section over a natural generalization of the unit tangent bundle, which we call shape phase space.

gr-qc↗

arXiv2022

A Proposal for a Metaphysics of Self-Subsisting Structures. I. Classical Physics

We present a new metaphysical framework for physics that is conceptually clear, ontologically parsimonious, and empirically adequate. This framework relies on the notion of self-subsisting structure, that is, a set of fundamental physical elements whose individuation and behavior are described in purely relational terms, without any need for a background spacetime. Although the specification of the fundamental elements of the ontology depends on the particular physical domain considered -- and is thus susceptible to scientific progress -- , the empirically successful structural features of the framework are preserved through theory change. The kinematics and dynamics of these self-subsisting structures are technically implemented using the theoretical framework of Pure Shape Dynamics, which provides a completely relational physical description of a system in terms of the intrinsic geometry of a suitably defined space called shape space.

physics.hist-ph↗

arXiv2022

The Physics and Metaphysics of Pure Shape Dynamics

The goal of this essay is twofold. First, it provides a quick look at the foundations of modern relational mechanics by tracing its development from Julian Barbour and Bruno Bertotti's original ideas until present-day's pure shape dynamics. Secondly, it discusses the most appropriate metaphysics for pure shape dynamics, showing that relationalism is more of a nuanced thesis rather than an elusive one. The chapter ends with a brief assessment of the prospects of pure shape dynamics in light of quantum physics.

physics.hist-ph↗

arXiv2017

AdS Poisson homogeneous spaces and Drinfel'd doubles

The correspondence between Poisson homogeneous spaces over a Poisson-Lie group $G$ and Lagrangian Lie subalgebras of the classical double $D({\mathfrak g})$ is revisited and explored in detail for the case in which ${\mathfrak g}=D(\mathfrak a)$ is a classical double itself. We apply these results to give an explicit description of some coisotropic 2d Poisson homogeneous spaces over the group $\mathrm{SL}(2,R)\cong\mathrm{SO}(2,1)$, namely 2d anti de Sitter space, 2d hyperbolic space and the lightcone in 3d Minkowski space. We show how each of these spaces is obtained as a quotient with respect to a Poisson-subgroup for one of the three inequivalent Lie bialgebra structures on ${sl}(2,R)$ and as a coisotropic one for the others. We then construct families of coisotropic Poisson homogeneous structures for 3d anti de Sitter space $\mathrm{AdS}_3$ and show that the ones that are quotients by a Poisson subgroup are determined by a three-parameter family of classical $r$-matrices for ${so}(2,2)$, while the non Poisson-subgroup cases are much more numerous. In particular, we present the two Poisson homogeneous structures on $\mathrm{AdS}_3$ that arise from two Drinfel'd double structures on $\mathrm{SO}(2,2)$. The first one realises $\mathrm{AdS}_3$ as a quotient of $\mathrm{SO}(2,2)$ by the Poisson-subgroup $\mathrm{SL}(2,R)$, while the second one, the non-commutative spacetime of the twisted $κ$-AdS deformation, realises $\mathrm{AdS}_3$ as a coisotropic Poisson homogeneous space.

math-ph↗

arXiv2017

On Hamiltonians with position-dependent mass from Kaluza-Klein compactifications

In a recent paper (J.R. Morris, Quant. Stud. Math. Found. 2 (2015) 359), an inhomogeneous compactification of the extra dimension of a five-dimensional Kaluza-Klein metric has been shown to generate a position-dependent mass (PDM) in the corresponding four-dimensional system. As an application of this dimensional reduction mechanism, a specific static dilatonic scalar field has been connected with a PDM Lagrangian describing a well-known nonlinear PDM oscillator. Here we present more instances of this construction that lead to PDM systems with radial symmetry, and the properties of their corresponding inhomogeneous extra dimensions are compared with the ones in the nonlinear oscillator model. Moreover, it is also shown how the compactification introduced in this type of models can alternatively be interpreted as a novel mechanism for the dynamical generation of curvature.

hep-th↗

arXiv2016

The kappa-(A)dS quantum algebra in (3+1) dimensions

The quantum duality principle is used to obtain explicitly the Poisson analogue of the kappa-(A)dS quantum algebra in (3+1) dimensions as the corresponding Poisson-Lie structure on the dual solvable Lie group. The construction is fully performed in a kinematical basis and deformed Casimir functions are also explicitly obtained. The cosmological constant $Λ$ is included as a Poisson-Lie group contraction parameter, and the limit $Λ\to 0$ leads to the well-known kappa-Poincaré algebra in the bicrossproduct basis. A twisted version with Drinfel'd double structure of this kappa-(A)dS deformation is sketched.

hep-th↗

arXiv2015

Towards (3+1) gravity through Drinfel'd doubles with cosmological constant

We present the generalisation to (3+1) dimensions of a quantum deformation of the (2+1) (Anti)-de Sitter and Poincaré Lie algebras that is compatible with the conditions imposed by the Chern-Simons formulation of (2+1) gravity. Since such compatibility is automatically fulfilled by deformations coming from Drinfel'd double structures, we believe said structures are worth being analysed also in the (3+1) scenario as a possible guiding principle towards the description of (3+1) gravity. To this aim, a canonical classical $r$-matrix arising from a Drinfel'd double structure for the three (3+1) Lorentzian algebras is obtained. This $r$-matrix turns out to be a twisted version of the one corresponding to the (3+1) $κ$-deformation, and the main properties of its associated noncommutative spacetime are analysed. In particular, it is shown that this new quantum spacetime is not isomorphic to the $κ$-Minkowski one, and that the isotropy of the quantum space coordinates can be preserved through a suitable change of basis of the quantum algebra generators. Throughout the paper the cosmological constant appears as an explicit parameter, thus allowing the (flat) Poincaré limit to be straightforwardly obtained.

gr-qc↗

arXiv2014

From Lorentzian to Galilean (2+1) gravity: Drinfel'd doubles, quantisation and noncommutative spacetimes

It is shown that the canonical classical $r$-matrix arising from the Drinfel'd double structure underlying the two-fold centrally extended (2+1) Galilean and Newton-Hooke Lie algebras (with either zero or non-zero cosmological constant $Λ$, respectively) originates as a well-defined non-relativistic contraction of a specific class of canonical $r$-matrices associated with the Drinfel'd double structure of the (2+1) (anti)-de Sitter Lie algebra. The full quantum group structure associated with such (2+1) Galilean and Newton-Hooke Drinfel'd doubles is presented, and the corresponding noncommutative spacetimes are shown to contain a commuting 'absolute time' coordinate ${\hat x}_0$ together with two noncommutative space coordinates $({\hat x}_1,{\hat x}_2)$, whose commutator is a function of the cosmological constant $Λ$ and of the (central) 'quantum time' coordinate ${\hat x}_0$. Thus, the Chern-Simons approach to Galilean (2+1) gravity can be consistently understood as the appropriate non-relativistic limit of the Lorentzian theory, and their associated quantum group symmetries (which do not fall into the family of so-called kappa-deformations) can also be derived from the (anti)-de Sitter quantum doubles through a well-defined quantum group contraction procedure.

gr-qc↗

arXiv2014

Twisted (2+1) $κ$-AdS Algebra, Drinfel'd Doubles and Non-Commutative Spacetimes

We construct the full quantum algebra, the corresponding Poisson-Lie structure and the associated quantum spacetime for a family of quantum deformations of the isometry algebras of the (2+1)-dimensional anti-de Sitter (AdS), de Sitter (dS) and Minkowski spaces. These deformations correspond to a Drinfel'd double structure on the isometry algebras that are motivated by their role in (2+1)-gravity. The construction includes the cosmological constant $Λ$ as a deformation parameter, which allows one to treat these cases in a common framework and to obtain a twisted version of both space- and time-like $κ$-AdS and dS quantum algebras; their flat limit $Λ\to 0$ leads to a twisted quantum Poincaré algebra. The resulting non-commutative spacetime is a nonlinear $Λ$-deformation of the $κ$-Minkowski one plus an additional contribution generated by the twist. For the AdS case, we relate this quantum deformation to two copies of the standard (Drinfel'd-Jimbo) quantum deformation of the Lorentz group in three dimensions, which allows one to determine the impact of the twist.

math-ph↗