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The Laplace-Belmatri operator in Equation 60 and the following sentence should be represented by a tombstone symbol □ instead of a W. The second expression after the sentence \"Our metric equation takes the form of the Schwarzschild solution...\" in the Discussion section also erroneously contains the ϑ symbol. Download corrected item. https://doi.org/10.1371/annotation/41868a14-4583-48e7-8fde-499055b2ec69.s001.cn Citation: Feldman MR (2013) Correction: Re-Examination of Globally Flat Space-Time.

Spacetime functionalism is the view that spacetime is a functional structure implemented by a more fundamental ontology. Lam and Wüthrich have recently argued that spacetime functionalism helps to solve the epistemological problem of empirical coherence in quantum gravity and suggested that it also (dis)solves the hard problem of spacetime, namely the problem of offering a picture consistent with the emergence of spacetime from a non-spatio-temporal structure. First, I will deny that spacetime functionalism solves the hard problem by showing that it comes in various species, each entailing a different attitude towards, or answer to, the hard problem. Second, I will argue that the existence of an explanatory gap, which grounds the hard problem, has not been correctly taken into account in the literature.

Objects of more than three dimensions can be used to model geographic phenomena that occur in space, time and scale. For instance, a single 4D object can be used to represent the changes in a 3D object's shape across time or all its optimal representations at various levels of detail. In this paper, we look at how such higher-dimensional space-time and space-scale objects can be visualised as projections from R4 to R3 . We present three projections that we believe are particularly intuitive for this purpose: (i) a simple `long axis' projection that puts 3D objects side by side; (ii) the well-known orthographic and perspective projections; and (iii) a projection to a 3-sphere (S3 ) followed by a stereographic projection to R3 , which results in an inwards-outwards fourth axis. Our focus is in using these projections from R4 to R3 , but they are formulated from Rn to Rn−1so as to be easily extensible and to incorporate other non-spatial characteristics. We present a prototype interactive visualiser that applies these projections from 4D to 3D in real-time using the programmable pipeline and compute shaders of the Metal graphics API.

The primordial CMB anisotropies generated at the last scattering surface are distorted by late time effects as the path of the CMB photons is perturbed by the intervening spacetime. One of these, weak gravitational lensing, is caused by transverse deflections to the photon trajectories by the mass overdensities of the large-scale structure of the universe. The effect on the CMB anisotropies is usually calculated with a remapping approach where higher order anisotropies are reconstructed from solutions to the per-turbed geodesic equation. In this thesis we study an alternative approach: solving the Boltzmann equation directly. This allows one to see explicitly which physical effects are involved, how they couple with weak lensing, and what assumptions must be made to derive the remapping approach. We review in detail how both the geodesic equation and the Boltzmann equation can be used to derive the weak lensing effect. By comparing the two methods we make the approximations made in the standard geodesic equation approach explicit and contrast them with those of the Boltzmann equation method. We use both methods to calculate the effect of weak lensing on the CMB anisotropies up to fourth order and then prove their equivalence up to fourth order. In second order calculations of the weak lensing effect the \"Born approximation\" is used: integrals are evaluated along the unperturbed photon path. In previous work it has been argued that extending the remapping approach to higher orders necessitates relaxing the Born approximation, resulting in terms that are due to \"post-Born\" physical effects. By comparing equivalent results derived using the two methods, and making use of diagrams that track the coupling of sources and lenses at different points in spacetime, we clarify the physical interpretation of the results and show that these \"post-Born\" effects of weak lensing at fourth order are in fact equivalent to lens-lens couplings in the Born approximation.

A Duistermaat-Guillemin-Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch [Adv. Math. \\textbf{376}, 107434 (2021)] to a vector bundle setting. We have analysed the spectrum of the Lie derivative with respect to a global timelike Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the time evolution operator has singularities at the periods of induced Killing flow on the manifold of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period. A pivotal technical ingredient to prove these results is the Feynman propagator. In order to obtain a Fourier integral description of this propagator, we have generalised the classic work of Duistermaat and H\"{o}rmander [Acta Math. \\textbf{128}, 183 (1972)] on distinguished parametrices for normally hyperbolic operators on a globally hyperbolic spacetime by propounding their microlocalisation theorem to a bundle setting. As a by-product of these analyses, another proof of the existence of Hadamard bisolutions for a normally hyperbolic operator (resp. Dirac-type operator) is reported.

Many quantum lattice models have an emergent relativistic description in their continuum limit. The celebrated example is graphene, whose continuum limit is described by the Dirac equation on a Minkowski spacetime. Not only does the continuum limit provide us with a dictionary of geometric observables to describe the models with, but it also allows one to solve models that were otherwise analytically intractable. In this thesis, we investigate novel features of this relativistic description for a range of quantum lattice models. In particular, we demonstrate how to generate emergent curved spacetimes and identify observables at the lattice level which reveal this emergent behaviour, allowing one to simulate relativistic effects in the laboratory. We first study carbon nanotubes, a system with an edge, which allows us to test the interesting feature of the Dirac equation that it allows for bulk states with support on the edges of the system. We then study Kitaev's honeycomb model which has a continuum limit describing Majorana spinors on a Minkowski spacetime. We show how to generate a non-trivial metric in the continuum limit of this model and how to observe the effects of this metric and its corresponding curvature in the lattice observables, such as Majorana correlators, Majorana zero modes and the spin densities. We also discuss how lattice defects and Z₂ gauge fields at the lattice level can generate chiral gauge fields in the continuum limit and we reveal their adiabatic equivalence. Finally, we discuss a chiral modification of the 1D XY model which makes the model interacting and introduces a non-trivial phase diagram. We see that this generates a black hole metric in the continuum limit, where the inside and outside of the black hole are in different phases. We then demonstrate that by quenching this model we can simulate Hawking radiation.

We study the regularity properties of cylindrical Lévy processes and Lévy space-time white noises, by examining their embeddings on the one hand in the spaces of general and tempered (Schwartz) distributions, and on the other hand in weighted Besov spaces. In this manner we analyse when the embedded Lévy object possesses a regularised version in the sense of Itô and Nawata. Lévy space-time white noises are defined as independently scattered random measures and cylindrical Lévy processes are defined by means of the theory of cylindrical processes. It is shown that Lévy space-time white noises correspond to an entire subclass of cylindrical Lévy processes, which is completely characterised by the characteristic functions of its members. We embed the Lévy space-time white noise, or the corresponding cylindrical Lévy process, in the space of general and tempered distributions and establish that in each case the embedded cylindrical processes are induced by (genuine) Lévy processes in the corresponding space. We use wavelet analysis to characterise the Lévy measures in weighted Besov spaces. Then we characterise the ranges of such Besov spaces in which L²(R^d) is or is not embedded continuously and the embedding is or is not Radonifying. We apply these results, given a cylindrical Lévy process L in L²(R^d), to characterise when L is and is not induced by a Lévy process in a given Besov space. These results are then applied to give sharp Besov regularity analysis to two important classes of cylindrical Lévy processes, the canonical stable cylindrical process, and 'hedgehog' processes constructed as a P-a.s. weakly convergent infinite random sum.

We study a toy model of the Kerr/CFT correspondence using string theory on AdS3 × S3. We propose a single trace irrelevant deformation of the dual CFT generated by a vertex operator with spacetime dimensions (2, 1). This operator shares the same quantum numbers as the integrable TJ¯\\[ T\\overline{J} \\] deformation of two-dimensional CFTs where J¯\\[ \\overline{J} \\] is a chiral U(1) current. We show that the deformation is marginal on the worldsheet and that the target spacetime is deformed to null warped AdS3 upon dimensional reduction. We also calculate the spectrum of the deformed theory on the cylinder and compare it to the field theory analysis of TJ¯\\[ T\\overline{J} \\]-deformed CFTs.

This thesis addresses the development of efficient finite element methods and their analysis for nonlocal problems, with particular focus on the integral fractional Laplacian. The specific topics addressed in this thesis are: regularity theory of solutions in polygonal domains, graded and hp versions of the finite element method, operator preconditioning, space-time adaptive methods for variational inequalities, and interface problems. The numerical analysis is supplemented by applications to biological and robotic systems. The precise regularity theory of the solutions in polygonal domains is first addressed as a basis for the numerical analysis. It is shown that the solution admits an asymptotic expansion with a tensor product decomposition, which leads to the optimal rate of convergence for finite element discretisations on graded meshes and for the hp-version on quasi-uniform meshes. An operator preconditioner for general elliptic pseudodifferential equations in a domain is then presented, based on a classical formula by Boggio. The thesis also considers the a priori and a posteriori analysis of a large class of space and space-time variational inequalities associated with the fractional Laplacian. The resulting space-time adaptive methods are studied in numerical experiments. Two further chapters of this thesis study applications to biological and robotic systems. Analysis and numerical experiments of the resulting continuum nonlocal equations allow for efficient quantitative characterisation of relevant quantities.

In prior work, we have argued that spacetime functionalism provides tools for clarifying the conceptual difficulties specifically linked to the emergence of spacetime in certain approaches to quantum gravity. We argue in this article that spacetime functionalism in quantum gravity is radically different from other functionalist approaches that have been suggested in quantum mechanics and general relativity: in contrast to these latter cases, it does not compete with purely interpretative alternatives, but is rather intertwined with the physical theorizing itself at the level of quantum gravity. Spacetime functionalism allows one to articulate a coherent realist perspective in the context of quantum gravity, and to relate it to a straightforward realist understanding of general relativity.