Relational particle models are useful toy models for quantum cosmology and the problem of time in quantum general relativity. This paper shows how to extend existing work on concrete examples of relational particle models in 1-d to include a notion of scale. This is useful as regards forming a tight analogy with quantum cosmology and the emergent semiclassical time and hidden time approaches to the problem of time. This paper shows furthermore that the correspondence between relational particle models and classical and quantum cosmology can be strengthened using judicious choices of the mechanical potential. This gives relational particle mechanics models with analogues of spatial curvature, cosmological constant, dust and radiation terms. A number of these models are then tractable at the quantum level. These models can be used to study important issues 1) in canonical quantum gravity: the problem of time, the semiclassical approach to it and timeless approaches to it (such as the naive Schrodinger interpretation and records theory). 2) In quantum cosmology, such as in the investigation of uniform states, robustness, and the qualitative understanding of the origin of structure formation.
We discuss the relational strategy to solve the problem of time in quantum gravity and different ways in which it could be implemented, pointing out in particular the fundamentally new dimension that the problem takes in a quantum gravity context in which spacetime and geometry are understood as emergent. We realize concretely the relational strategy we have advocated in the context of the tensorial group field theory formalism for quantum gravity, leading to the extraction of an effective relational cosmological dynamics from quantum geometric models. We analyze in detail the emergent cosmolo…
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 mathem…
Suppose one seeks to free oneself from a symmetric absolute space by quotienting out its symmetry group. This in general however fails to erase all memory of this absolute space's symmetry properties. Stratification is one major reason for this, which is present in both a) Kendall-type Shape Theory and subsequent Relational Mechanics, and b) General Relativity configuration spaces. We consider the alternative starting point with a generic absolute space, meaning with no nontrivial generalized Killing vectors whatsoever. In this approach, generically Shape-and-Scale Theory is but trivially real…
Kendall's Similarity Shape Theory for constellations of N points in the carrier space $\mathbb{R}^d$ as quotiented by the similarity group was developed for use in Probability and Statistics. It was subsequently shown to reside within Mechanics' Shape-and-Scale Theory, in which points are interpreted as particles, carrier space plays the role of absolute space, and the Euclidean group is quotiented out. Let us jointly refer to Shape(-and-Scale) Theory as Relational Theory, and to its reduced configuration spaces as relational spaces. We now consider a less structured version: the Topological R…
On the possibility of emergent light cones from relational shape dynamics
Instituto de Astrofísica e Ciências do Espaço（空间天文与天体物理研究所）；Faculdade de Ciências da Universidade de Lisboa（里斯本大学理学院）；Departamento de Física, Faculdade de Ciências da Universidade de Lisboa（里斯本大学理学院物理系）
Shape Theory, together with Shape-and-Scale Theory, comprise Relational Theory. This consists of $N$-point models on a manifold $M$, for which some geometrical automorphism group $G$ is regarded as meaningless and is thus quotiented out from the $N$-point model's product space $\times_{I = 1}^N M$. Each such model has an associated function space of preserved quantities, solving the PDE system for zero brackets with the sums over $N$ of each of $G$'s generators. These are smooth functions of the $N$-point geometrical invariants. Each $(M, G)$ pair has moreover a `minimal nontrivially relationa…
We analyze the size and evolution of quantum fluctuations of cosmologically relevant geometric observables, in the context of the effective relational cosmological dynamics of GFT models of quantum gravity. We consider the fluctuations of the matter clock observables, to test the validity of the relational evolution picture itself. Next, we compute quantum fluctuations of the universe volume and of other operators characterizing its evolution (number operator for the fundamental GFT quanta, effective Hamiltonian and scalar field momentum). In particular, we focus on the late (clock) time regim…
Using a massless scalar field as a clock variable, the Legendre transform of the group field theory Lagrangian gives a relational Hamiltonian. In the classical theory, it is natural to define 'equal relational time' Poisson brackets, where 'equal time' corresponds to equal values of the scalar field clock. The quantum theory can then be defined by imposing 'equal relational time' commutation relations for the fundamental operators of the theory, with the states being elements of a Fock space with their evolution determined by the relational Hamiltonian operator. A particularly interesting fami…
A suite of relational notions of shape are presented at the level of configuration space geometry, with corresponding new theories of shape mechanics and shape statistics. These further generalize two quite well known examples: --1) Kendall's (metric) shape space with his shape statistics and Barbour's mechanics thereupon. 0) Leibnizian relational space alias metric scale-and-shape space to which corresponds Barbour-Bertotti mechanics. This paper's new theories include, using the invariant and group namings, 1) $Angle$ alias $conformal$ $shape$ $mechanics$. 2) $Area ratio$ alias $affine$ $shap…
Relational Quadrilateralland. I. The Classical Theory
相对粒子力学模型支持了绝对运动与相对运动之争的关系方面，并且是广义相对论及其时间问题的动力学公式的玩具模型。它们涵盖了两个更普遍研究的小超空间广义相对论模型所没有的方面：1）具有非平凡的结构概念，从而具有宇宙结构形成和定域记录的概念。2)它们具有线性和二次约束，这对于建模许多时间问题方面至关重要。我以前解决了关系三角和经典，量子力学和关于时间问题的局部解决方案。这依赖于三角形的形状空间是S^2，等距群SO（3），允许使用广为人知的几何，方法和原子/分子物理学类比。我现在将这项工作扩展到关系四边形，这是更典型的一般N-a-gon，代表了一个“对角到非对角Bianchi IX minissuperspace”的复杂性，并编码进一步的时间问题的微妙之处。形状空间现在是CP ^2，等距群SU（3）/Z_3，我现在需要利用几何学，形状统计学和粒子物理学来解决这个模型;因此，这是一个跨学科的论文。本文在经典水平上处理四边形和，然后论文II提供了量子处理。
We discuss the construction of relational observables in time-reparametrization invariant quantum mechanics and we argue that their physical interpretation can be understood in terms of conditional probabilities, which are defined from the solutions of the quantum constraint equation in a generalization of the Page-Wootters formalism. In this regard, we show how conditional expectation values of worldline tensor fields are related to quantum averages of suitably defined relational observables. We also comment on how the dynamics of these observables can be related to a notion of quantum refere…
We use the separate universe framework to study cosmological perturbations within the group field theory formalism for quantum gravity, based on multi-condensate quantum states. Working with a group field theory action for gravity minimally coupled to four scalar fields that can act as a set of relational clock and rods, we argue that these multi-condensate states correspond to cosmological space-times with small long-wavelength scalar perturbations. Equations of motion for the cosmological perturbations are derived, which in the classical limit agree with the standard results of general relat…
We derive an effective dynamics for scalar cosmological perturbations from quantum gravity, in the framework of group field theory (GFT) condensate cosmology. The emergent spacetime picture is obtained from the mean field hydrodynamic regime of the fundamental theory, and physical observables are defined using a relational strategy applied at the same level of approximation, in terms of suitable collective states of the GFT field. The dynamical equations we obtain for volume and matter perturbations lead to the same solutions as those of classical general relativity in the long-wavelength, sup…
Emergent scalar field dynamics in a cosmological spacetime from GFT quantum gravity
Nordita, KTH Royal Institute of Technology and Stockholm University（北欧理论天体物理学研究所，瑞典皇家理工学院和斯德哥尔摩大学）；Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova（帕多瓦大学伽利略物理与天文系）；INFN, Sezione di Padova（意大利国家核物理研究所帕多瓦分部）；SISSA（国际高等研究学院）；INFN sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；IFPU, Institute for Fundamental Physics of the Universe（宇宙基础物理研究所）；Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense de Madrid（马德里康普顿斯大学物理科学学院理论物理系）
Primordial black holes may have formed in the radiative era of the early Universe from the collapse of large enough amplitude perturbations of the metric. These correspond to non linear energy density perturbations characterized by an amplitude larger than a certain threshold, measured when the perturbations reenter the cosmological horizon. The process of primordial black hole formation is studied here within spherical symmetry, using the gradient expansion approximation in the long wavelength limit, where the pressure gradients are small, and the initial perturbations are functions only of a…
We explore non-Gaussian features of a massless spin-two field in the Vasiliev theory of higher-spin gravity. The theory contains an infinite tower of interacting gauge fields with increasing spin, and admits four-dimensional asymptotically de Sitter configurations. Using a recent proposal for calculating late-time quantum correlations in Vasiliev theory, we provide an exact formula for the tensor non-Gaussianities of the massless spin-two graviton field. By general symmetry considerations, we relate our result to that produced by a tree-level calculation in a gravitational theory containing an…
The Shapere-Wilczek model \cite{wil}, or so called $ fgh$ model, enjoys the remarkable features of a Time Crystal (TC) that has a non-trivial time dependence in its lowest energy state (or the classical ground state). We construct a particular form of $ fgh$ model (with specified $f,g,h$ functions) that is derived from a Mini-superspace version of a quadratic $f(R,R_{μν})$ gravity theory. Main part of the investigation deals with thermodynamic properties of such systems from classical statistical mechanics perspective. Our analysis reveals the possibility of a {\it phase transition}. Because o…
We consider the total nonlocal energy associated with a particle at rest in the Hubble flow, i.e., the relational energy between this particle and all connected particles within the causal horizon. The particle, even while at rest, partakes in relative recessional and peculiar motion of connected particles in 3 dimensions. A geometrical argument due to Berkeley suggests that the nonlocal mass of recessional energy associated with the particle is 3 times its Newtonian mass. It follows that nonlocal recessional and peculiar energy of the Universe are equal, and match Misner-Sharp energy within t…
Cosmological correlation functions contain valuable information about the primordial Universe, with possible signatures of new massive particles at very high energies. Recent developments, including the cosmological bootstrap, bring new perspectives and powerful tools to study these observables. In this paper, we systematically classify inflationary three-point correlators of scalar perturbations using the bootstrap method. For the first time, we derive a complete set of single-exchange cosmological collider bispectra with new shapes and potentially detectable signals. Specifically, we focus o…
The problem of time emerging in the canonical quantization procedure of gravity signals a necessity to properly define a relational time parameter. Previous approaches, which are here briefly discussed, make use of the dependence of the quantum system on semiclassical gravitational variables in order to define time. We show that such paths, despite the following studies, lead to a non-unitary evolution. We propose a different model for the quantization of the gravity-matter system, where the time parameter is defined via an additional term, i.e. the kinematical action, which acts as a clock fo…
Galaxy imaging surveys provide us with information on both the galaxy distribution and their shapes. In this paper, we systematically investigate the sensitivity of galaxy shapes to new physics in the initial conditions. For this purpose, we decompose the galaxy shape function into spin components, and compute the contributions to each spin component from both intrinsic alignment and weak lensing. We then consider the angular-dependent primordial non-Gaussianity, which is generated by a non-zero integer spin particle when active during inflation, and show that a galaxy imaging survey essential…
