Space-time Discretization Research Articles

Derivation of the deformed Heisenberg algebra from discrete spacetime

Abstract Although the deformation of the Heisenberg algebra by a minimal length has become a central tool in quantum gravity phenomenology, it has never been rigorously obtained and is often derived using heuristic reasoning. In this study, we move beyond the heuristic derivation of the deformed Heisenberg algebra and explicitly derive it using a model of discrete spacetime, which is motivated by quantum gravity. Initially, we investigate the effects of the leading order Planckian lattice corrections and demonstrate that they precisely match those suggested by the heuristic arguments commonly used in quantum gravity phenomenology. Furthermore, we rigorously obtain deformations from the higher order Planckian lattice corrections. In contrast to the leading order corrections, these higher order corrections are model-dependent. We select a specific model that breaks the rotational symmetry, as the importance of such rotational symmetry breaking lies in the relationship between CMB anisotropies and quantum gravitational effects. Based on the mathematical similarity of the Planckian lattice used here with the graphene lattice, we propose that graphene can serve as an analogue system for the study of quantum gravity. Finally, we examine the deformation of the covariant form of the Heisenberg algebra using a four-dimensional Euclidean lattice.

Towards a Unitary Formulation of Quantum Field Theory in Curved Spacetime: The Case of de Sitter Spacetime

Before we ask what the quantum gravity theory is, there is a legitimate quest to formulate a robust quantum field theory in curved spacetime (QFTCS). Several conceptual problems, especially unitarity loss (pure states evolving into mixed states), have raised concerns over several decades. In this paper, acknowledging the fact that time is a parameter in quantum theory, which is different from its status in the context of General Relativity (GR), we start with a “quantum first approach” and propose a new formulation for QFTCS based on the discrete spacetime transformations which offer a way to achieve unitarity. We rewrite the QFT in Minkowski spacetime with a direct-sum Fock space structure based on the discrete spacetime transformations and geometric superselection rules. Applying this framework to QFTCS, in the context of de Sitter (dS) spacetime, we elucidate how this approach to quantization complies with unitarity and the observer complementarity principle. We then comment on understanding the scattering of states in de Sitter spacetime. Furthermore, we discuss briefly the implications of our QFTCS approach to future research in quantum gravity.

The Continuity or Discreteness of Space-time

General relativity and quantum field theory have always held opposite views on the continuum and the dispersion of space-time. Scientists have debated this for nearly a century. This paper reviews the basis of general relativity and quantum field theory on the continuity and discreteness of space-time in a unified viewpoint, describing what both stand for in this discussion point and their respective experimental evidence to illustrate their plausibility. By using mathematical reasoning and figures to help describe and apply string theory's ideas, this paper describes the universe as a wave. Based on this, a new conjecture of the unified field theory is obtained: space-time is described as a piecewise function, which presents different states under different conditions. In the normal state, the continuous wave appears, and in the energy exchange with the outside world, it is transformed into a discrete particle state. Space-time is similar to the waves described in string theory, but it is more similar to electromagnetic waves, taking on an ever-changing form.

Counterintuitive Scenarios in Discrete Gravity Without Quantum Effects or Causality Violations

In certain established approaches to quantum gravity, such as causal set theory and causal dynamical triangulations, discrete spacetime structure is taken to be a primary feature, not a secondary effect of “quantizing” a pre-existing classical continuum-based theory, as is done in approaches such as string theory and loop quantum gravity. For a priori discrete models, the full quantum theory is often obtained via some version of Feynman’s sum-over-histories approach, in which each “history” is a discrete object viewed as a classical spacetime. Counterintuitive physical scenarios such as Schrödinger’s cat or the grandfather paradox are typically associated with either quantum effects or causality violations, but we demonstrate that equally bizarre scenarios can arise at a purely classical level in the discrete causal context due to symmetry considerations. In particular, the graph-theoretic phenomenon of pseudosimilarity leads to situations in which alternative events occurring at physically distinguishable locations in the universe can cause different parts of the universe to “swap identities” in a fugue-like manner alien to continuum-based theories. This phenomenon is perhaps best understood as an extension of the relativity principle, which we call relativity of identity.

Discrete spacetime crystal: Intertwined spacetime symmetry breaking in a driven-dissipative spin system

Discrete spacetime crystal: Intertwined spacetime symmetry breaking in a driven-dissipative spin system

A low-cost space-time finite element for free-surface flows under total Lagrangian description

In this work, we propose a position-based space-time finite element formulation for incompressible Newtonian flows under a total Lagrangian description. This formulation is suggested within the context of finite strain free-surface flows and differs from the traditional finite element approach for fluid dynamics by utilizing current nodal positions as the main variables instead of nodal velocities. In contrast to time-marching methods, space-time formulations involve applying the finite element technique not only to the spatial domain but also to the temporal domain. The proposed approach employs a finite element discretization that can be unstructured or structured in space, but is always structured in time. Thus, the space-time shape functions are expressed as a tensor product of linear shape functions in the spatial direction and specially designed quadratic shape functions in the temporal direction. Consequently, the space-time domain is divided into time slabs that are solved progressively, with the final velocities and positions from the previous time slab serving as initial conditions for the current one, thereby reducing the dimension of the discrete system of equations. The proposed shape functions in the temporal direction yield a system of equations of the same size as standard time-marching methods, but with advantages stemming from the space-time discretization: different stability and high-frequency dissipation can be achieved based on the selection of the time test functions. To solve the incompressible problem stably, we employ mixed equal-order position-pressure finite elements with Petrov-Galerkin/pressure stabilization (PSPG). This formulation possesses several significant features that justify its development: 1) the space-time formulation facilitates dynamic re-meshing by permitting the spatial discretization at the end of one time slab to differ entirely from the spatial discretization at its beginning, thus extending its applicability to flows with undefined distortion and topological changes; 2) by considering positions as variational parameters, it becomes straightforward to couple with Lagrangian hyper-elastic solid solvers, which may also be formulated in terms of positions or displacements in a monolithic way. Numerical examples conducted to validate the formulation demonstrate its robustness and efficiency for finite strain free surface flows, including phenomena such as dam breaks with smooth surfaces and sloshing.

Dispersive vacuum as a decoherence amplifier of an Unruh–DeWitt detector

Abstract Recently, interest has been growing in studies on discrete or "pixelated'' space-time that, through modifications in the dispersion relation, can treat the vacuum as a dispersive medium. Discrete spacetime considers that spacetime has a cellular structure on the order of the Planck length, and if this is true we should certainly have observable effects. In this paper, we investigated the effects caused by the dispersive vacuum on the decoherence process of an Unruh-DeWitt detector, our setup consists of a uniformly accelerated detector, initially in a qubit state, which interacts with a massless scalar field during a time interval finite. We use dispersion relations drawn from doubly special relativity and Hořava-Lifshitz gravity, with these modifications the vacuum becomes dispersive and has a corresponding refractive index. We calculate the probability transition rates, the probability of finding the detector in the ground state, and the quantum coherence variation. Our results indicate that the decoherence process occurs more quickly in cases with changes in the dispersion relation in the regime of high accelerations and interaction time. Additionally, the decoherence increases as the vacuum becomes more dispersive due to the increase in the order of modification in the dispersion relation, and this happens because the dispersive vacuum amplifies the effects of quantum fluctuations that are captured by the detector when interacting with the field.

Adjoint-based inversion for stress and frictional parameters in earthquake modeling

We present an adjoint-based optimization method to invert for stress and frictional parameters used in earthquake modeling. The forward problem is linear elastodynamics with nonlinear rate-and-state frictional faults. The misfit functional quantifies the difference between simulated and measured particle displacements or velocities at receiver locations. The misfit may include windowing or filtering operators. We derive the corresponding adjoint problem, which is linear elasticity with linearized rate-and-state friction and, for forward problems involving fault normal stress changes, nonzero fault opening, with time-dependent coefficients derived from the forward solution. The gradient of the misfit is efficiently computed by convolving forward and adjoint variables on the fault. The method thus extends the framework of full-waveform inversion to include frictional faults with rate-and-state friction. In addition, we present a space-time dual-consistent discretization of a dynamic rupture problem with a rough fault in antiplane shear, using high-order accurate summation-by-parts finite differences in combination with explicit Runge–Kutta time integration. The dual consistency of the discretization ensures that the discrete adjoint-based gradient is the exact gradient of the discrete misfit functional as well as a consistent approximation of the continuous gradient. Our theoretical results are corroborated by inversions with synthetic data. We anticipate that adjoint-based inversion of seismic and/or geodetic data will be a powerful tool for studying earthquake source processes; it can also be used to interpret laboratory friction experiments.

Relativistic generalized uncertainty principle for a test particle in four-dimensional spacetime

The generalized uncertainty principle provides a promising phenomenological approach to reconciling the fundamentals of general relativity with quantum mechanics. As a result, noncommutativity, measurement uncertainty, and the fundamental theory of quantum mechanics are subject to finite gravitational fields. The generalized uncertainty principle (RGUP) on curved spacetime allows for the imposition of quantum-induced elements on general relativity (GR) in four dimensions. In the relativistic regimes, the determination of a test particle’s spacetime coordinates, [Formula: see text], becomes uncertain. There exists a specific range of coordinates where the accessibility to [Formula: see text] is notably limited. Consequently, the spacetime coordinates lack both smoothness and continuity. The quantum-mechanical calculations of the spacetime coordinates are directly linked to their measurement through the expectation value [Formula: see text]. This expectation value is dependent on [Formula: see text], which is itself limited by a minimum measurable length [Formula: see text]. A crucial finding presented in this script is the existence of a lower bound for the Hamiltonian, which implies the stability of the quantum nature of spacetime. The direct correlation between [Formula: see text] and [Formula: see text] is a significant discovery, suggesting a proportional relationship. The conjecture is made that the primary metric g carries all essential information regarding spacetime curvature and serves a role akin to the Jacobian determinant in general relativity. Moreover, the linear relationship between [Formula: see text] and the Planck length [Formula: see text] is established with a proportionality factor of [Formula: see text], where [Formula: see text] denotes the RGUP parameter. The discretization of spacetime coordinates results in the discontinuity of the test particle’s wavefunction [Formula: see text], leading to an unrealistic [Formula: see text]. It is also noted that the lower limit of [Formula: see text] is directly proportional to the fundamental tensor.

A class of space–time discretizations for the stochastic [formula omitted]-Stokes system

The main objective of the present paper is to construct a new class of space–time discretizations for the stochastic p-Stokes system and analyze its stability and convergence properties. We derive regularity results for the approximation that are similar to the natural regularity of solutions. One of the key arguments relies on discrete extrapolation that allows us to relate lower moments of discrete maximal processes. We show that, if the generic spatial discretization is constraint conforming, then the velocity approximation satisfies a best-approximation property in the natural distance. Moreover, we present an example such that the resulting velocity approximation converges with rate 1/2 in time and 1 in space towards the (unknown) target velocity with respect to the natural distance. The theory is corroborated by numerical experiments.

Exploring discrete space–time models for information transfer: Analogies from mycelial networks to the cosmic web

Fungal mycelium networks are large scale biological networks along which nutrients, metabolites flow. Recently, we discovered a rich spectrum of electrical activity in mycelium networks, including action-potential spikes and trains of spikes. Reasoning by analogy with animals and plants, where travelling patterns of electrical activity perform integrative and communicative mechanisms, we speculated that waves of electrical activity transfer information in mycelium networks. Using a new discrete space–time model with emergent radial spanning-tree topology, hypothetically comparable mycelial morphology and physically comparable information transfer, we provide physical arguments for the use of such a model, and by considering growing mycelium network by analogy with growing network of matter in the cosmic web, we develop mathematical models and theoretical concepts to characterise the parameters of the information transfer.

Mereological Models of Spacetime Emergence

AbstractRecent work in quantum gravity has prompted a re‐evaluation of the fundamental nature of spacetime. Spacetime is potentially emergent from non‐spatiotemporal entities posited by a theory of quantum gravity. Recent efforts have sought to interpret the relationship between spacetime and the fundamental entities through a mereological framework. These frameworks propose that spacetime can be conceived as either having non‐spatiotemporal entities as its constituents or being a constituent part of a non‐spatiotemporal structure. I present a roadmap for those interested in exploring the role of composition in understanding the emergence of spacetime. I establish a taxonomy based on four crucial parameters that should be considered when constructing potential mereological models. Subsequently, I connect these models to a range of perspectives found in current literature, with the aim of pinpointing areas that require further exploration. Finally, I identify three potential challenges facing mereological models of spacetime emergence, rooted in issues of mereological harmony, the distinction between continuous and discrete spacetime, and the implications of quantum superposition.

Deep Reinforcement Learning Technique for Traffic Metering in Connected Urban Street Networks

Proper metering can improve traffic operations in congested urban street networks. The available approaches either (a) use macroscopic fundamental diagrams to model traffic dynamics or (b) use numerical time–space discretization of the hydrodynamic traffic flow model with high computational requirements. Therefore, they either (a) do not represent traffic dynamics accurately or (b) are not suitable for online applications. This study introduces a deep reinforcement learning (DRL) methodology to capture traffic dynamics on a micro-level scale with the capability of capturing detailed traffic dynamics with low computational time. The DRL methodology employs two neural networks that map the location of connected vehicles in a network to traffic metering signal indications and estimate the objective function of the traffic metering problem. The outputs of the neural networks are used to construct a loss function, whose optimization provides the optimal parameters for the neural networks. Because of the complexity of the loss function, the gradient descent optimization technique with Monte Carlo simulation is used to optimize the loss function. The proposed methodology was tested on a simulated case study network in Vissim software with 20 intersections. The numerical results showed that the methodology increased throughput by 41.2% and 21.3% and reduced the total travel time of vehicles by 3.4% and 15.5% compared to a no-metering strategy. Comparing the computational time of the proposed methodology with one of the existing traffic metering approaches also showed the potential of the methodology for online applications.

Mathematical modeling and numerical simulation of the N-component Cahn-Hilliard model on evolving surfaces

Establishing multi-component phase field models on dynamic surfaces, and exploring the similarities and differences in phase field evolution between evolving and static surfaces, are interesting studies in multi-phase flow problems. This paper focuses on mathematical modeling and numerical simulation of the N-component Cahn-Hilliard model on evolving surfaces. In the modeling process, the evolution of the energy functional, componential mass conservation, and point-wise mass conservation (hyperplane link condition) are considered. On an evolving surface, the energy dissipation law does not hold because the surface velocity can be viewed as an external force in the system. Meanwhile, due to the velocity, the surface area may change, the componential mass conservation property and the hyperplane link condition can not be satisfied simultaneously. From the point of view of preserving these two physical properties of mass conservation, three types of N-component Cahn-Hilliard model are established on the evolving surface: componential mass conservation, point-wise mass conservation, and both componential and point-wise mass conservation. For the numerical simulation, the evolving surface finite element method is considered for the space-time discretization of the proposed model. In order to obtain a linear, decoupled, high-accurate, and stable scheme for long time numerical simulations, the stabilized semi-implicit method is integrated into the framework of the evolving surface finite element method. Through several numerical examples, the rationality of the model and the efficiency of the numerical method are shown. Additionally, three- and four-component phase separation phenomena are shown to investigate the N-component Cahn-Hilliard dynamic on various evolving surfaces.

An ultra-weak space-time variational formulation for the Schrödinger equation

We present a well-posed ultra-weak space-time variational formulation for the time-dependent version of the linear Schrödinger equation with an instationary Hamiltonian. We prove optimal inf-sup stability and introduce a space-time Petrov-Galerkin discretization with optimal discrete inf-sup stability.We show norm-preservation of the ultra-weak formulation. The inf-sup optimal Petrov-Galerkin discretization is shown to be asymptotically norm-preserving, where the deviation is shown to be in the order of the discretization. In addition, we introduce a Galerkin discretization, which has suboptimal inf-sup stability but exact norm-preservation.Numerical experiments underline the performance of the ultra-weak space-time variational formulation, especially for non-smooth initial data.

Goal-oriented adaptive space-time finite element methods for regularized parabolic p-Laplace problems

We consider goal-oriented adaptive space-time finite-element discretizations of the regularized parabolic p-Laplace problem on completely unstructured simplicial space-time meshes. The adaptivity is driven by the dual-weighted residual (DWR) method since we are interested in an accurate computation of some possibly nonlinear functionals at the solution. Such functionals represent goals in which engineers are often more interested than the solution itself. The DWR method requires the numerical solution of a linear adjoint problem that provides the sensitivities for the mesh refinement. This can be done by means of the same full space-time finite element discretization as used for the primal non-linear problems. The numerical experiments presented demonstrate that this goal-oriented, full space-time finite element solver efficiently provides accurate numerical results for different functionals.

Space-time error estimates for approximations of linear parabolic problems with generalized time boundary conditions

Abstract We first give a general error estimate for the nonconforming approximation of a problem for which a Banach–Nečas–Babuška (BNB) inequality holds. This framework covers parabolic problems with general conditions in time (initial value problems as well as periodic problems) under minimal regularity assumptions. We consider approximations by two types of space-time discretizations, both based on a conforming Galerkin method in space. The first one is the Euler $\theta -$scheme. In this case, we show that the BNB inequality is always satisfied, and may require an extra condition on the time step for $\theta \le \frac 1 2$. The second one is the time discontinuous Galerkin method, where the BNB condition holds without any additional condition.

Space-time CutFEM on overlapping meshes II: simple discontinuous mesh evolution

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that evolves inside/“on top” of it. Here the overlapping mesh is prescribed by a simple discontinuous evolution, meaning that its location, size, and shape as functions of time are discontinuous and piecewise constant. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche’s method. The simple discontinuous mesh evolution results in a space-time discretization with a slabwise product structure between space and time which allows for existing analysis methodologies to be applied with only minor modifications. We follow the analysis methodology presented by Eriksson and Johnson (SIAM J Numer Anal 28(1):43–77, 1991; SIAM J Numer Anal 32(3):706–740, 1995). The greatest modification is the introduction of a Ritz-like “shift operator” that is used to obtain the discrete strong stability needed for the error analysis. The shift operator generalizes the original analysis to some methods for which the discrete subspace at one time does not lie in the space of the stiffness form at the subsequent time. The error analysis consists of an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.

Space-time CutFEM on overlapping meshes I: simple continuous mesh motion

We present a cut finite element method for the heat equation on two overlapping meshes: a stationary background mesh and an overlapping mesh that moves around inside/“on top” of it. Here the overlapping mesh is prescribed by a simple continuous motion, meaning that its location as a function of time is continuous and piecewise linear. For the discrete function space, we use continuous Galerkin in space and discontinuous Galerkin in time, with the addition of a discontinuity on the boundary between the two meshes. The finite element formulation is based on Nitsche’s method and also includes an integral term over the space-time boundary between the two meshes that mimics the standard discontinuous Galerkin time-jump term. The simple continuous mesh motion results in a space-time discretization for which standard analysis methodologies either fail or are unsuitable. We therefore employ what seems to be a relatively uncommon energy analysis framework for finite element methods for parabolic problems that is general and robust enough to be applicable to the current setting. The energy analysis consists of a stability estimate that is slightly stronger than the standard basic one and an a priori error estimate that is of optimal order with respect to both time step and mesh size. We also present numerical results for a problem in one spatial dimension that verify the analytic error convergence orders.

Dynamic large strain formulation for nematic liquid crystal elastomers

Liquid crystal elastomers (LCEs) are a class of materials which exhibit an anisotropic behavior in their nematic state due to the main orientation of their rod-like molecules called mesogens. The reorientation of mesogens leads to the well-known actuation properties of LCEs, i.e. exceptionally large deformations as a consequence of particular external stimuli, such as temperature increase. Another key feature of nematic LCEs is the capability to undergo deformation by constant stresses while being stretched in a direction perpendicular to the orientation of mesogens. During this plateau stage, the mesogens rotate towards the stretching direction. Such characteristic is defined as semisoft elastic response of nematic LCEs. We aim at modeling the semisoft behavior in a dynamic finite element method based on a variational-based mixed finite element formulation. The reorientation process of the rigid mesogens relative to the continuum rotation is introduced by micropolar drilling degrees of freedom. Responsible for the above-mentioned characteristics is an appropriate free energy function. Starting from an isothermal free energy function based on the small strain theory, we aim to widen it into the framework of large strains by identifying tensor invariants. In this work, we analyze the isothermal influence of the tensor invariants on the mechanical response of the finite element formulation and show that its space-time discretization preserves mechanical balance laws in the discrete setting.