We consider the manner in which the spacetime manifold emerges from a quantum substratum through the transactional process, in which spacetime events and their connections are established. In this account, there is no background spacetime as is generally assumed in physical theorizing. Instead, the usual notion of a background spacetime is replaced by the quantum substratum, comprising quantum systems with nonvanishing rest mass. Rest mass corresponds to internal periodicities that function as internal clocks defining proper times, and in turn, inertial frames that are not themselves aspects of the spacetime manifold, but are pre-spacetime reference structures. Specific processes in the quantum substratum serve to distinguish absolute from relative motion.
This paper discusses how the transactional interpretation of quantum mechanics can provide for a natural account of the emergence of spacetime events from a quantum substratum. In this account, spacetime is not a substantive manifold that becomes occupied with events; rather, spacetime itself exists only in virtue of specific actualized events. This implies that spacetime is discrete rather than continuous, and that properties attributed to spacetime based on the notion of a continuum are idealizations that do not apply to the real physical world. It is further noted that the transactional pic…
The causal set program as well as the Wolfram physics project leave open the problem of how a graph that is a (3+1)-dimensional Minkowski-spacetime according to its simple geodesic distances, could be generated solely from simple deterministic rules. This paper provides a solution by describing simple rules that characterize discrete Lorentz boosts between 4D lattice graphs, which combine further to form Wigner rotations that produce isotropy and lead to the emergence of the continuous Lorentz group and the (3+1)-dimensional Minkowski-spacetime. On such graphs, the speed of light, the proper t…
This is a chapter of the planned monograph "Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and Christian Wüthrich and under contract with Oxford University Press. (More information at https://beyondspacetime.net/.) This chapter introduces the problem of emergence of spacetime in quantum gravity. It introduces the main philosophical challenge to spacetime emergence and sketches our preferred solution to it.
We introduce the set of constraints the wave function of the Universe has to satisfy in order to describe an Universe undergoing through the process of spontaneous breaking of supersymmetry and discuss the way this may lead to the emergence of classical spacetime.
Using the coherent-state representation we show that the classical Einstein equation for the FRW cosmological model with a general minimal scalar field can be derived from the semiclassical quantum Einstein equation.
This is a chapter of the planned monograph "Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and Christian Wüthrich and under contract with Oxford University Press. (More information at www.beyondspacetime.net.) This chapter analyses the nature and derivation of spacetime topology and geometry according to string theory.
This is a chapter of the planned monograph "Out of Nowhere: The Emergence of Spacetime in Quantum Theories of Gravity", co-authored by Nick Huggett and Christian Wüthrich and under contract with Oxford University Press. (More information at www.beyondspacetime.net.) This chapter sketches how spacetime emerges in causal set theory and demonstrates how this question is deeply entangled with genuinely philosophical concerns.
Spacetime emergence refers to the notion that classical spacetime "emerges" as an approximate macroscopic entity from a non-spatio-temporal structure present in a more complete theory of interacting fundamental constituents. In this article, we propose a novel mechanism involving the "soldering" of internal and external spaces for the emergence of spacetime and the twin transmutation of general covariance. In the context of string theory, this mechanism points to a critical four dimensional spacetime background.
We present and discuss an asynchronous coordinate system covering de Sitter spacetime, notably in a complete way in 1+1 dimensions. The new coordinates have several interesting cosmological properties: the worldlines of comoving ($x^i=const$) observers are geodesics, cosmic time is finite in the past, and the coordinates asymptotically tend to that of a flat Robertson & Walker model at large times. This analysis also provides an argument in favor of the natural emergence of an equation of state of the type $p=-ρ$ in the context of the standard cosmological model.
Black holes provide a window into the microscopic structure of spacetime in quantum gravity. Recently the quantum information contained in Hawking radiation has been calculated, verifying a key aspect of the consistency of black hole evaporation with quantum mechanical unitarity. This calculation relied crucially on recent progress in understanding the emergence of bulk spacetime from a boundary holographic description. Spacetime wormholes have played an important role in understanding the underpinnings of this result, and the precision study of such wormholes, in this and other contexts, has …
The Wightman function i.e. vacuum two-point function, for a massless free scalar field in Fock quantization is inversely proportional to the invariant distance squared between the corresponding spacetime points. Naturally it diverges when these two points are taken to be infinitesimally close to each other. Using a combination of analytical and numerical methods, we show that the Wightman function is bounded from above in polymer quantization of scalar field. The bounded value of the Wightman function is governed by the polymer scale and its bounded nature can be viewed as if the spacetime has…
We present the case for a fundamentally discrete quantum spacetime and for Group Field Theories as a candidate consistent description of it, briefly reviewing the key properties of the GFT formalism. We then argue that the outstanding problem of the emergence of a continuum spacetime and of General Relativity from fundamentally discrete quantum structures should be tackled from a condensed matter perspective and using purely QFT methods, adapted to the GFT context. We outline the picture of continuum spacetime as a condensed phase of a GFT and a research programme aimed at realizing this pictu…
This paper examines two cosmological models of quantum gravity (from string theory and loop quantum gravity) to investigate the foundational and conceptual issues arising from quantum treatments of the big bang. While the classical singularity is erased, the quantum evolution that replaces it may not correspond to classical spacetime: it may instead be a non-spatiotemporal region, which somehow transitions to a spatiotemporal state. The different kinds of transition involved are partially characterized, the concept of a physical transition without time is investigated, and the problem of empir…
Gravity is not only able to be mimicked in flat spacetimes, but also in curved spacetimes. We study analogue gravity models in curved spacetime by considering the relativistic Gross-Pitaevskii theory and Yang-Mills theory in the fixed background spacetime geometry. The results show that acoustic metrics can be emergent from curved spacetimes yielding a Hadamard product of a real metric-tensor and an analogue metric-tensor. Taking \emph{quantum vortices} as \emph{test particles}, we evaluate their released energy ratio during the "gravitational binding". The $2+1$-dimensional flat Minkowski met…
We summarize the main ideas behind TGFT condensate cosmology and sketch the technical steps that bring from the fundamental theory to the effective cosmological dynamics. This framework is presented as an explicit illustration of (and possibly a general template for) the emergence of spacetime from non-spatiotemporal quantum entities in quantum gravity, and the many aspects involved in it.
We explore the issue of spacetime emergence in quantum gravity, by articulating several levels at which this can be intended. These levels correspond to the reconstruction moves that are needed to recover the classical and continuum notion of space and time, which are progressively lost in a progressively deeper sense in the more fundamental quantum gravity description. They can also be understood as successive steps in a process of widening of the perspective, revealing new details and new questions at each step. Each level carries indeed new technical issues and opportunities, and raises new…
We study the wave function of a tensor model in the canonical formalism by Hamiltonian Monte Carlo method for Lie group symmetric or nearby values for the argument of the wave function, and show that there emerge Lie-group symmetric semi-classical spacetimes. More precisely, we consider some $SO(n+1)\ (n=1,2,3)$ symmetric values for the tensor argument of the wave function, and show that there emerge discrete $n$-dimensional spheres. A key fact is that there exist two phases, the classical phase and the quantum phase, depending on the values of the argument of the wave function, and emergence …
We present a novel derivation of Einstein equations from the balance between Clausius entropy crossing the boundary of a local causal diamond and entanglement entropy associated with its horizon. Comparing this derivation with the entanglement equilibrium approach developed by Jacobson, we are able to argue for the equivalence of matter entanglement and Clausius entropy in the semiclassical regime. We also provide a direct comparison of both entropies for conformal matter, showing their equivalence without appealing to gravitational dynamics. Furthermore, we determine that gravitational dynami…
Using a novel, string theory-inspired formalism based on a Hamiltonian constraint, we obtain a conformal mechanical system for the spatially flat four-dimensional Robertson-Walker Universe. Depending on parameter choices, this system describes either a relativistic particle in the Robertson-Walker background, or metric fluctuations of the Robertson-Walker geometry. Moreover we derive a tree-level M-theory matrix model in this time-dependent background. Imposing the Hamiltonian constraint forces the spacetime geometry to be fuzzy near the big bang, while the classical Robertson-Walker geometry …
To understand spacetime dynamics in the canonical tensor model of quantum gravity for the positive cosmological constant case, we analytically and numerically study the phase profile of its exact wave function in a coordinate representation, instead of the momentum representation analyzed so far. A saddle point analysis shows that Lie group symmetric spacetimes are strongly favored due to abundance of continuously existing saddle points, giving an emergent fluid picture. The phase profile suggests that spatial sizes grow in "time", where sizes are measured by the tensor-geometry correspondence…
We have shown how the quantization of two-dimensional quantum gravity with an action which contains only a positive cosmological constant and boundary cosmological constants leads to the emergence of a spacetime which can be described as a constant negative curvature spacetime with superimposed quantum fluctuations.
It is well known that phase transitions arise if the interaction among particles embodies an attractive as well as a repulsive contribution. In this work it will be shown that the breakdown of Lorentz symmetry, characterized through a deformation in the relation dispersion, plus the bosonic statistics predict the emergence of critical points. In other words, in some quantum gravity models the structure of spacetime implies the emergence of critical points even when no interaction among the particle has been considered.
We show that the uncertainty in distance and time measurements found by the heuristic combination of quantum mechanics and general relativity is reproduced in a purely classical and flat multi-fractal spacetime whose geometry changes with the probed scale (dimensional flow) and has non-zero imaginary dimension, corresponding to a discrete scale invariance at short distances. Thus, dimensional flow can manifest itself as an intrinsic measurement uncertainty and, conversely, measurement-uncertainty estimates are generally valid because they rely on this universal property of quantum geometries. …
Impact of supersymmetry on the dynamical emergence of the spacetime in the type IIB matrix model with the Lorentz symmetry "gauge fixed"
National Technical University（国立技术大学）；Setsunan University（摂南大学）；University of Milano-Bicocca（米兰比可卡大学）；High Energy Accelerator Research Organization（高能加速器研究机构）；Graduate University for Advanced Studies (SOKENDAI)（综合研究大学院大学（SOKENDAI））；Shizuoka University（静冈大学）
We discuss the hints for the disappearance of continuum space and time at microscopic scale. These include arguments for a discrete nature of them or for a fundamental non-locality, in a quantum theory of gravity. We discuss how these ideas are realized in specific quantum gravity approaches. Turning then the problem around, we consider the emergence of continuum space and time from the collective behaviour of discrete, pre-geometric atoms of quantum space, and for understanding spacetime as a kind of "condensate", and we present the case for this emergence process being the result of a phase …
Is there an approach to quantum gravity which is conceptually simple, relies on very few fundamental physical principles and ingredients, emphasizes geometric (as opposed to algebraic) properties, comes with a definite numerical approximation scheme, and produces robust results, which go beyond showing mere internal consistency of the formalism? The answer is a resounding yes: it is the attempt to construct a nonperturbative theory of quantum gravity, valid on all scales, with the technique of so-called Causal Dynamical Triangulations. Despite its conceptual simplicity, the results obtained up…
In a recent paper [arXiv:1206.4916] by T. Padmanabhan, it was argued that our universe provides an ideal setup to stress the issue that cosmic space is emergent as cosmic time progresses and that the expansion of the universe is due to the difference between the number of degrees of freedom on a holographic surface and the one in the emerged bulk. In this note following this proposal we obtain the Friedmann equation of a higher dimensional Friedmann-Robertson-Walker universe. By properly modifying the volume increase and the number of degrees of freedom on the holographic surface from the entr…
Research in quantum gravity strongly suggests that our world in not fundamentally spatiotemporal, but that spacetime may only emerge in some sense from a non-spatiotemporal structure, as this paper illustrates in the case of causal set theory and loop quantum gravity. This would raise philosophical concerns regarding the empirical coherence and general adequacy of theories in quantum gravity. If it can be established, however, that spacetime emerges in the appropriate circumstances and how all its relevant aspects are explained in fundamental non-spatiotemporal terms, then the challenge is ful…
We show that holographic RG flow can be defined precisely such that it corresponds to emergence of spacetime. We consider the case of pure Einstein's gravity with a negative cosmological constant in the dual hydrodynamic regime. The holographic RG flow is a system of first order differential equations for radial evolution of the energy-momentum tensor and the variables which parametrize it's phenomenological form on hypersurfaces in a foliation. The RG flow can be constructed without explicit knowledge of the bulk metric provided the hypersurface foliation is of a special kind. The bulk metric…
We give a short account of the quantisation of the Szekeres spacetime by considering the symmetries of a reduced action principle. This is an alternative approach than the one followed in the literature for the study of inhomogeneities, which is usually based on perturbations of an inhomogeneous field on the spacetime background. Here, we examine the emergence of classicality with an exact inhomogeneous solution. We check whether the two criteria for classicality are satisfied, that is the correlations on the phase space and decoherence. We verify that these two properties indeed hold, thus th…
We overview a new mechanism whereby classical Riemannian geometry emerges out of the differential structure on quantum spacetime, as extension data for the classical algebra of differential forms. Outcomes for physics include a new formula for the standard Levi-Civita connection, a new point of view of the cosmological constant as a very small mass for the graviton of around $10^{-33}$ev, and a weakening of metric-compatibility in the presence of torsion. The same mechanism also provides a new construction for quantum bimodule connections on quantum spacetimes and a new approach to the quantum…
We analyze a wave function of a tensor model in the canonical formalism, when the argument of the wave function takes Lie group invariant or nearby values. Numerical computations show that there are two phases, which we call the quantum and the classical phases, respectively. In the classical phase, fluctuations are suppressed, and there emerge configurations which are discretizations of the classical geometric spaces invariant under the Lie group symmetries. This is explicitly demonstrated for the emergence of $S^n\ (n=1,2,3)$ for $SO(n+1)$ symmetries by checking the topological and the geome…
The loop quantum dynamics of Kantowski-Sachs spacetime and the interior of higher genus black hole spacetimes with a cosmological constant has some peculiar features not shared by various other spacetimes in loop quantum cosmology. As in the other cases, though the quantum geometric effects resolve the physical singularity and result in a non-singular bounce, after the bounce a spacetime with small spacetime curvature does not emerge in either the subsequent backward or the forward evolution. Rather, in the asymptotic limit the spacetime manifold is a product of two constant curvature spaces. …
A new kind of duality between the deep structures of spacetime and matter is proposed here, considering two partial orders which incorporate causality, extensity, and discreteness. This may have surprising consequences for the emergence of quantum mechanics, which are discussed.
The spectral dimension has proven to be a very informative observable to understand the properties of quantum geometries in approaches to quantum gravity. In loop quantum gravity and its spin foam description, it has not been possible so far to calculate the spectral dimension of spacetime. As a first step towards this goal, here we determine the spacetime spectral dimension in the simplified spin foam model restricted to hypercuboids. Using Monte Carlo methods we compute the spectral dimension for state sums over periodic spin foam configurations on infinite lattices. For given periodicity, i…
The idea of a graph theoretical approach to modeling the emergence of a quantized geometry and consequently spacetime, has been proposed previously, but not well studied. In most approaches the focus has been upon how to generate a spacetime that possesses properties that would be desirable at the continuum limit, and the question of how to model matter and its dynamics has not been directly addressed. Recent advances in network science have yielded new approaches to the mechanism by which spacetime can emerge as the ground state of a simple Hamiltonian, based upon a multi-dimensional Ising mo…
We study the {\em propagation of electromagnetic waves} in a spacetime devoid of a metric but equipped with a {\em linear} electromagnetic spacetime relation $H\simχ\cdot F$. Here $H$ is the electromagnetic excitation $({\cal D},{\cal H})$ and $F$ the field strength $(E,B)$, whereas $χ$ (36 independent components) characterizes the electromagnetic permittivity/permeability of spacetime. We derive analytically the corresponding Fresnel equation and show that it is always quartic in the wave covectors. We study the `Fresnel tensor density' ${\cal G}^{ijkl}$ as (cubic) function of $χ$ and identif…
We solve Einstein vacuum equations in a spacetime region up to the "center" of gravitational collapse. Within this region, we construct a sequence of marginally outer trapped surfaces (MOTS) with areas going to zero. These MOTS form a marginally outer trapped tube (apparent horizon). It emerges from a point and is smooth (except at that point) and spacelike. In the proof we employ a scale critical trapped surface formation criterion established by An and Luk and a new type of quasilinear elliptic equation is studied. The main conclusion of this paper proves a conjecture of Ashtekar on black ho…
The emergence of (3+1)-dimensional expanding spacetime from complex Langevin simulations of the Lorentzian type IIB matrix model with deformations
National Technical University of Athens（雅典国立技术大学）；Setsunan University（摂南大学）；Università degli Studi di Milano-Bicocca（米兰比可卡大学）；Istituto Nazionale di Fisica Nucleare (INFN)（意大利国家核物理研究所）；High Energy Accelerator Research Organization（高能加速器研究机构）；SOKENDAI（日本综合研究大学院大学）；Shizuoka University（静冈大学）
Emergence of Dark Energy from topology and chiral spinors
Universidad de La Frontera（弗雷罗纳大学）；Universidad de La Serena（拉塞雷纳大学）；Lanzhou University（兰州大学）；Instituto de Astrofísica e Ciências do Espaço, Universidade de Lisboa（里斯本大学空间天体物理研究所）；Université de Paris-Cité（巴黎西岱大学）
A very interesting new proposal on the origin of the cosmic expansion was recently suggested by Padmanabhan [arXiv:1206.4916]. He argued that the difference between the surface degrees of freedom and the bulk degrees of freedom in a region of space drives the accelerated expansion of the universe, as well as the standard Friedmann equation through relation $% \triangle V=\triangle t(N_{\mathrm{sur}}-N_{\mathrm{bulk}})$. In this paper, we first present the general expression for the number of degrees of freedom on the holographic surface, $N_{\mathrm{sur}}$, using the general entropy corrected …
We show that under certain conditions an axisymmetric rotating spacetime contains a ring of points in the equatorial plane, where a particle at rest with respect to an asymptotic static observer remains at rest in a static orbit. We illustrate the emergence of such orbits for boson stars. Further examples are wormholes, hairy black holes and Kerr-Newman solutions.
Non-perturbative theories of quantum gravity inevitably include configurations that fail to resemble physically reasonable spacetimes at large scales. Often, these configurations are entropically dominant and pose an obstacle to obtaining the desired classical limit. We examine this "entropy problem" in a model of causal set quantum gravity corresponding to a discretisation of 2D spacetimes. Using results from the theory of partial orders we show that, in the large volume or continuum limit, its partition function is dominated by causal sets which approximate to a region of 2D Minkowski space.…
There are numerous indications that a discrete substratum underlies continuum spacetime. Any fundamentally discrete approach to quantum gravity must provide some prescription for how continuum properties emerge from the underlying discreteness. The causal set approach, in which the fundamental relation is based upon causality, finds it easy to reproduce timelike distances, but has a more difficult time with spatial distance, due to the unique combination of Lorentz invariance and discreteness within that approach. We describe a method to deduce spatial distances from a causal set. In addition,…
The Causal Dynamical Triangulation (CDT) approach to quantum gravity is a lattice approximation to the gravitational path integral. Developed by Ambjørn, Jurkiewicz and Loll, it has yielded some important results, notably the emergence of classical spacetime and short scale dimensional reduction. However, virtually all the results reported so far have been based on a single computer code. In this paper we present the first completely independent verification of the CDT algorithm, and report the successful reproduction of the emergence of classical spacetime and smooth reduction in the spectral…
