Space-time Point Research Articles

Asymptotic properties of the minimum contrast estimators for projections of inhomogeneous space-time shot-noise Cox processes

We consider a flexible class of space-time point process models—inhomogeneous shot-noise Cox point processes. They are suitable for modelling clustering phenomena, e.g. in epidemiology, seismology, etc. The particular structure of the model enables the use of projections to the spatial and temporal domain. They are used to formulate a stepwise estimation method to estimate different parts of the model separately. In the first step, the Poisson likelihood approach is used to estimate the inhomogeneity parameters. In the second and third steps, the minimum contrast estimation based on K-functions of the projected processes is used to estimate the interaction parameters. We study the asymptotic properties of the resulting estimators and formulate a set of conditions sufficient for establishing consistency and asymptotic normality of the estimators under the increasing domain asymptotics.

Causal Poisson bracket via deformation quantization

Starting with the well-defined product of quantum fields at two spacetime points, we explore an associated Poisson structure for classical field theories within the deformation quantization formalism. We realize that the induced star-product is naturally related to the standard Moyal product through an appropriate causal Green’s functions connecting points in the space of classical solutions to the equations of motion. Our results resemble the Peierls–DeWitt bracket that has been analyzed in the multisymplectic context. Once our star-product is defined, we are able to apply the Wigner–Weyl map in order to introduce a generalized version of Wick’s theorem. Finally, we include some examples to explicitly test our method: the real scalar field, the bosonic string and a physically motivated nonlinear particle model. For the field theoretic models, we have encountered causal generalizations of the creation/annihilation relations, and also a causal generalization of the Virasoro algebra for the bosonic string. For the nonlinear particle case, we use the approximate solution in terms of the Green’s function, in order to construct a well-behaved causal bracket.

Variants of Geometric RSK, Geometric PNG, and the Multipoint Distribution of the Log-Gamma Polymer

We show that the reformulation of the geometric Robinson–Schensted–Knuth correspondence via local moves, introduced in [29], can be extended to cases where the input matrix is replaced by more general polygonal, Young-diagram-like, arrays of the form ⁠. We also show that a rearrangement of the sequence of the local moves gives rise to a geometric version of the polynuclear growth model. These reformulations are used to obtain integral formulae for the Laplace transform of the joint distribution of the point-to-point partition functions of the log-gamma polymer at different space–time points. In the case of two points at equal time |$N$| and space at distance of order |$N^{2/3}$|⁠, we show formally that the joint law of the partition functions, scaled by |$N^{1/3}$|⁠, converges to the two-point function of the Airy process.

The Growing-Block: just one thing after another?

In this article, we consider two independently appealing theories—the Growing-Block view and Humean Supervenience—and argue that at least one is false. The Growing-Block view is a theory about the nature of time. It says that (a) past and present things exist, while future things do not, and (b) the passage of time consists in new things coming into existence. Humean Supervenience is a theory about the nature of entities like laws, nomological possibility, counterfactuals, dispositions, causation, and chance. It says that none of these entities are fundamental, since if they were, this would entail the existence of irreducible necessary connections between matters of fact. Instead, these entities supervene on a fundamental, non-nomological ‘Humean mosaic’ of property instances at spacetime points. We will further explain and motivate the Growing-Block view and Humean Supervenience in Sects. 2 and 3, but first, we turn to our master argument.

Minimal dissipation hybrid bicompact schemes for hyperbolic equations

New monotonicity-preserving hybrid schemes are proposed for multidimensional hyperbolic equations. They are convex combinations of high-order accurate central bicompact schemes and upwind schemes of first-order accuracy in time and space. The weighting coefficients in these combinations depend on the local difference between the solutions produced by the high- and low-order accurate schemes at the current space-time point. The bicompact schemes are third-order accurate in time, while having the fourth order of accuracy and the first difference order in space. At every time level, they can be solved by marching in each spatial variable without using spatial splitting. The upwind schemes have minimal dissipation among all monotone schemes constructed on a minimum space-time stencil. The hybrid schemes constructed has been successfully tested as applied to a number of two-dimensional gas dynamics benchmark problems.

A complete convergence theorem for stationary regularly varying multivariate time series

For a class of stationary regularly varying and weakly dependent time series, we prove the so-called complete convergence result for the corresponding space-time point processes. As an application of our main theorem, we give a simple proof of the invariance principle for the corresponding partial maximum process.

Effects of a scalar scaling field on quantum mechanics

This paper describes the effects of a complex scalar scaling field on quantum mechanics. The field origin is an extension of the gauge freedom for basis choice in gauge theories to the underlying scalar field. The extension is based on the idea that the value of a number at one space time point does not determine the value at another point. This, combined with the description of mathematical systems as structures of different types, results in the presence of separate number fields and vector spaces, both as structures, at different space time locations. Complex number structures and vector spaces at each location, are scaled by a complex space time dependent scaling factor. The effect of this scaling factor on several physical and geometric quantities has been described in other work. Here the emphasis is on quantum mechanics of one and two particles, their states and properties. Multiparticle states are also briefly described. The effect shows as a complex, nonunitary, scalar field connection on a fiber bundle description of nonrelativistic quantum mechanics. The lack of physical evidence for the presence of this field so far means that the coupling constant of this field to fermions is very small. It also means that the gradient of the field must be very small in a local region of cosmological space and time. Outside this region there are no restrictions on the field gradient.

Generalized Einstein’s Equations from Wald Entropy

We derive the gravitational equations of motion of general theories of gravity from thermodynamics applied to a local Rindler horizon through any point in spacetime. Specifically, for a given theory of gravity, we substitute the corresponding Wald entropy into the Clausius relation. Our approach works for all diffeomorphism-invariant theories of gravity in which the Lagrangian is a polynomial in the Riemann tensor.

Modelling spatio-temporal air pollution data from a mobile monitoring station

ABSTRACTEnvironmental data is typically indexed in space and time. This work deals with modelling spatio-temporal air quality data, when multiple measurements are available for each space-time point. Typically this situation arises when different measurements referring to several response variables are observed in each space-time point, for example, different pollutants or size resolved data on particular matter. Nonetheless, such a kind of data also arises when using a mobile monitoring station moving along a path for a certain period of time. In this case, each spatio-temporal point has a number of measurements referring to the response variable observed several times over different locations in a close neighbourhood of the space-time point. We deal with this type of data within a hierarchical Bayesian framework, in which observed measurements are modelled in the first stage of the hierarchy, while the unobserved spatio-temporal process is considered in the following stages. The final model is very flexible and includes autoregressive terms in time, different structures for the variance-covariance matrix of the errors, and can manage covariates available at different space-time resolutions. This approach is motivated by the availability of data on urban pollution dynamics: fast measures of gases and size resolved particulate matter have been collected using an Optical Particle Counter located on a cabin of a public conveyance that moves on a monorail on a line transect of a town. Urban microclimate information is also available and included in the model. Simulation studies are conducted to evaluate the performance of the proposed model over existing alternatives that do not model data over the first stage of the hierarchy.

Toward a quantum theory of tachyon fields

We construct momentum space expansions for the wave functions that solve the Klein–Gordon and Dirac equations for tachyons, recognizing that the mass shell for such fields is very different from what we are used to for ordinary (slower than light) particles. We find that we can postulate commutation or anticommutation rules for the operators that lead to physically sensible results: causality, for tachyon fields, means that there is no connection between space–time points separated by a timelike interval. Calculating the conserved charge and four-momentum for these fields allows us to interpret the number operators for particles and antiparticles in a consistent manner; and we see that helicity plays a critical role for the spinor field. Some questions about Lorentz invariance are addressed and some remain unresolved; and we show how to handle the group representation for tachyon spinors.

Summoning information in spacetime, or where and when can a qubit be?

One of the most important properties of quantum information, and the one ultimately responsible for its cryptographic applications, is that it can not be copied. That statement, however, is not completely accurate. While the no-cloning theorem of quantum mechanics prevents quantum information from being copied in space, the reversibility of microscopic physics actually requires that the information be copied in time. In spacetime as a whole, therefore, quantum information is widely replicated but in a restricted fashion. We fully characterize which regions of spacetime can all hold the same quantum information. Because quantum information can be delocalized through quantum error correction and teleportation, it need not follow well-defined trajectories. Instead, replication of the information in any configuration of spacetime regions not leading to violations of causality or the no-cloning principle is allowed. To demonstrate this, we answer the operational question of exactly when the information can be summoned to a set of spacetime points, showing how to do so efficiently using a combination of teleportation and codeword-stabilized quantum codes. This provides a simple and complete description of where and when a qubit can be located in spacetime, revealing a remarkable variety of possibilities.

Discussion of “Spatial statistics: Marks, maps and shapes”

A. Stein from the Department of Earth Observation and Geoinformation Science, University of Twente, Enschede, The Netherlands discusses spatial statistics and its relationships with marks, maps and shapes. The first and foremost issue is the recent development of space-time procedures. It is a straightforward extension of spatial statistics domain, and it has currently received so much attention that its mentioning is important. Space-time point patterns have been addressed by Gabriel and Diggle in 2009. Another issue that is being debated in spatial statistics concerns integrated nested Laplace approximations (INLAs), a relatively new development that has found a place in spatial statistics.

“Regularity singularities” and the scattering of gravity waves in approximate locally inertial frames

Abstract. It is an open question whether solutions of the EinsteinEuler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether “regularity singularities” can exist at such points. The term regularity singularity was proposed by the authors as a point in spacetime where the gravitational metric tensor is Lipschitz continuous (C), but no smoother, in any coordinate system of the C atlas. An existence theory for shock wave solutions in C admitting arbitrary interactions has been proven for the Einstein-Euler equations in spherically symmetric spacetimes, but C is the requisite smoothness required for space-time to be locally flat. Thus the open problem of regularity singularities is the problem as to whether locally inertial coordinate systems exist at shock waves, within the larger C atlas. Our purpose here is to clarify and motivate the open problem of regularity singularities, and to prove that if locally inertial coordinates do not lie within the C atlas, then the scattering of gravitational radiation by a regularity singularity produces quantifiable physical e↵ects analogous to non-removable Coriolis forces.

Propagation of Massless Spin-1 Particle between Two Media

In general explanation for propagation of electromagnetic wave at a plane between two media of different dielectric properties. The description of this phenomenon be based on the actual physical process of the well know interpretation of the classical rays with the polarization vectors. In this research, we investigate this situation by using the knowledge of the amplitude of emission and absorption of massless spin-1 particle (photon) excitation by the emitter and detector, respectively. By using time evolution process based on vacuum-to-vacuum transition amplitude in quantum (field) theory QED formalism. This process shows that the propagation of photon excitation by the external sources is given with an explanation of the amplitude of propagation between different spacetime points from an emitter to a detector, which exclude with the Feynman propagator. The purpose of this research is developing a formalism based on the actual physical process of photons propagate from an emitter to a detector. It shows the consistency to the actual physical process of interpretation of the polarization vectors in the classical rays.

Contact interactions between particle worldlines

We construct contact interactions for bosonic and fermionic point particles. We first relate the resulting theories to classical electrostatics by taking functional averages over worldlines whose endpoints are fixed to charged particles. Counting those paths which pass through a space-time point $x^{\mu}$ gives the static electric field at that point, provided we take the limit where the length measured along the worldlines is large. We also investigate corrections to the classical field that arise beyond leading order in this limit before constructing a theory of point particles that interact when their worldlines intersect. We quantise this theory and show that the partition function contains propagator couplings between the endpoints of the particles before discussing how this is related to the worldline formalism of quantum field theory and general action at a distance theories.

Holism and Reductionism: How to Get the Balance Right

This is an extensive study of the conditions that allow or impede reductionism. After an overview of the different anti-reductionist conditions in approaches like holism, downward causation, emergence, supervenience, and several forms of uncertainty and sensitive dependence, a more detailed analysis of downward causation and holism is presented. For the latter, strong and numerous arguments could be identified why the whole is more than the sum of its parts. However, the examination of the case of official big bang theory, specifically a finite universe based on ultimate particles, allows the formulation of a potential reductionist theory in principle. The key notions for the further argumentation are different forms of infinity: the infinitely small, the infinitely big and the infinity of points in a continuous space-time, even if this is finite in its extension. Several cases are presented in which systems are at the same time holistic and reductionist if conditions of infinity are applied. This is identified as a natural ontological property of such systems as soon as infinity intervenes. The final conclusion, however, is that the certainties of reductionism are methodologically only useful on very small objects. In the macroscopic world the only way to influence things is by controlling or directing the processes by downward causation, which is based on the elimination or management of uncertainties at higher non-microscopic levels.

Coalescing Brownian flows: A new approach

The coalescing Brownian flow on $\mathbb{R}$ is a process which was introduced by Arratia [Coalescing Brownian motions on the line (1979) Univ. Wisconsin, Madison] and T\'{o}th and Werner [Probab. Theory Related Fields 111 (1998) 375-452], and which formally corresponds to starting coalescing Brownian motions from every space-time point. We provide a new state space and topology for this process and obtain an invariance principle for coalescing random walks. This result holds under a finite variance assumption and is thus optimal. In previous works by Fontes et al. [Ann. Probab. 32 (2004) 2857-2883], Newman et al. [Electron. J. Probab. 10 (2005) 21-60], the topology and state-space required a moment of order $3-\varepsilon$ for this convergence to hold. The proof relies crucially on recent work of Schramm and Smirnov on scaling limits of critical percolation in the plane. Our approach is sufficiently simple that we can handle substantially more complicated coalescing flows with little extra work - in particular similar results are obtained in the case of coalescing Brownian motions on the Sierpinski gasket. This is the first such result where the limiting paths do not enjoy the noncrossing property.

Quantum gravity as an information network self-organization of a 4D universe

I propose a quantum gravity model in which the fundamental degrees of freedom are information bits for both discrete space-time points and links connecting them. The Hamiltonian is a very simple network model consisting of a ferromagnetic Ising model for space-time vertices and an antiferromagnetic Ising model for the links. As a result of the frustration between these two terms, the ground state self-organizes as a new type of low-clustering graph with finite Hausdorff dimension 4. The spectral dimension is lower than the Hausdorff dimension: it coincides with the Hausdorff dimension 4 at a first quantum phase transition corresponding to an IR fixed point while at a second quantum phase transition describing small scales space-time dissolves into disordered information bits. The large-scale dimension 4 of the universe is related to the upper critical dimension 4 of the Ising model. At finite temperatures the universe graph emerges without big bang and without singularities from a ferromagnetic phase transition in which space-time itself forms out of a hot soup of information bits. When the temperature is lowered the universe graph unfolds and expands by lowering its connectivity, a mechanism I have called topological expansion. The model admits topological black hole excitations corresponding to graphs containing holes with no space-time inside and with "Schwarzschild-like" horizons with a lower spectral dimension.

Shimony–Wolf states and hidden coherences in classical light

The classical theory of polarisation coherence is briefly summarised and then extended. The extension is motivated by the recognition that the traditional theory of two-point coherence provides only what we identify as ‘diagonal’ correlation functions and their associated two-point coherence matrices. It is pointed out that a wider focus is possible when taking account of the three-sector vector space underlying all two-point coherences in classical optics. This reveals the possibility of observing a new type of ‘off-diagonal’ correlations that arise when the correlation functions under investigation are associated with points in two distinct vector spaces, pairs of points that are not analogous to the pairs of space points or time points that underlie traditional measures of spatial and temporal coherence. Quantum theory has experience with correlations engaging such ‘cross-sector’ coherences, for example in tests of Bell inequalities, and the quantum formulations are shown to be easily adopted by classical theory without incorporating quantum features in the optical signals. The familiar theory of classical coherence that is associated with the pioneering work of Emil Wolf is extended in conformance with three criteria advanced by Abner Shimony to obtain formulas for correlation functions and for the Bell measure of coherence. Values of greater than the standard upper limit are predicted for certain classical Shimony–Wolf fields, indicating strong cross-sector coherence, but only when standard measures of coherence such as degree of polarisation are minimised. Experimental results confirming the predictions for cross-sector coherence are exhibited.

Entanglement Entropy in Two-Dimensional String Theory.

To understand an emergent spacetime is to understand the emergence of locality. Entanglement entropy is a powerful diagnostic of locality, because locality leads to a large amount of short distance entanglement. Two-dimensional string theory is among the very simplest instances of an emergent spatial dimension. We compute the entanglement entropy in the large-N matrix quantum mechanics dual to two-dimensional string theory in the semiclassical limit of weak string coupling. We isolate a logarithmically large, but finite, contribution that corresponds to the short distance entanglement of the tachyon field in the emergent spacetime. From the spacetime point of view, the entanglement is regulated by a nonperturbative "graininess" of space.