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A twistor framework for foundations of physics is outlined, in which (1) spacetime is secondary, derived from twistor constructions; (2) the primary physical agents are described by elements of sheaf cohomology defined on twistor space, which encode all information about the spacetime entities (fields and particles); (3) the dynamics of spacetime entities is encoded in the holomorphic structures of twistor space and revealed through Penrose transform and its generalizations (twistor diagrams); (4) the nonlinearity of gravity and nonabelian gauge interactions is a manifestation of the deformation of twistor space effected by the self-interacting twistor agents. A consistent framework of quantum gravity is attainable within twistor formalism because not only can gravity be derived from the primary physical agents of the formalism but also, and most importantly, the formalism itself can be proven intrinsically quantal without the necessity of imposing a quantum postulate upon it extrinsically through a quantization procedure. Indeed, the Planck constant and the whole quantum edifice built thereupon can be derived from formalism. Thus, the conflicts between the principles of general relativity and quantum theory are dissolved because both theories are consequences of twistor formalism, and the century-long debate concerning the possibility of a consistent framework of quantum gravity is closed. Many technical and conceptual issues must be clarified before the twistor formalism can be turned into a practical research framework. But the foundational pillars of the framework provided by twistor formalism are rock solid.

Assuming the validity of the equivalence principle in the quantum regime, we argue that one of the assumptions of the usual definition of quantum mechanics, namely separation between the “classical” detector and the “quantum” system, must be relaxed. We argue, therefore, that if both the equivalence principle and quantum mechanics continue to survive experimental tests, then this favors “informational” interpretations of quantum mechanics (where formalism is built around relations between observables, defined as information that can be accessed by an observer of the system. In particular “collapse” is understood as a change of relative information as the detector interacts with the system) over “ontic ones” (assuming the physical reality of states and wave functions, which are assumed to be more than informational objects. In particular collapse is understood as a physical process). In particular, we show that relational-type interpretations can readily accommodate the equivalence principle via a minor modification of the assumptions used to justify the formalism. We qualitatively speculate what a full generally covariant quantum dynamics could look like, and comment on experimental investigations.

A black hole is a mysterious cosmic object that cannot be observed directly, and research on its structure continues with increasing interest today. Investigating the effects on the motion of a half-spin relativistic particle around a black hole may provide us with interesting information. Research on the particle scale requires considering quantum contributions, but a well-established quantum gravity theory has not yet been constructed in literature. In our research, the interaction between the spin of a Dirac particle and the gravitational field is discussed within the scope of the rainbow formalism, which is one of the most studied quantum gravity approaches.

We study the state-sum models of quantum gravity based on a representation 2-category of the Poincaré 2-group. We call them spin-cube models, since they are categorical generalizations of spin-foam models. A spin-cube state sum can be considered as a path integral for a constrained 2-BF theory, and depending on how the constraints are imposed, a spin-cube state sum can be reduced to a path integral for the area-Regge model with the edge-length constraints, or to a path integral for the Regge model. We also show that the effective actions for these spin-cube models have the correct classical limit.

Two singularity theorems can be proven if one attempts to let a Lorentzian cobordism interpolate between two topologically distinct manifolds. On the other hand, Cartier and DeWitt-Morette have given a rigorous definition for quantum field theories (QFTs) by means of path integrals. This paper uses their results to study whether QFTs can be made compatible with topology changes. We show that path integrals over metrics need a finite norm for the latter and for degenerate metrics, this problem can sometimes be resolved with tetrads. We prove that already in the neighborhood of some cuspidal singularities, difficulties can arise to define certain QFTs. On the other hand, we show that simple QFTs can be defined around conical singularities that result from a topology change in a simple setup. We argue that the ground state of many theories of quantum gravity will imply a small cosmological constant and, during the expansion of the universe, will cause frequent topology changes. Unfortunately, it is difficult to describe the transition amplitudes consistently due to the aforementioned problems. We argue that one needs to describe QFTs by stochastic differential equations, and in the case of gravity, by Regge calculus in order to resolve this problem.

Based on a formal analogy between space-time quantum fluctuations and classical Kolmogorov fluid turbulence, we suggest that the dynamic growth of the Universe from Planckian to macroscopic scales should be characterized by the presence of a fluctuating volume-flux (FVF) invariant. The existence of such an invariant could be tested in numerical simulations of quantum gravity, and may also stimulate the development of a new class of hierarchical models of quantum foam, similar to those currently employed in modern phenomenological research on fluid turbulence. The use of such models shows that the simple analogy with Kolmogorov turbulence is not compatible with a fine-scale fractal structure of quantum space-time. Hence, should such theories prove correct, they would imply that the scaling properties of quantum fluctuations of space-time are subtler than those described by the simple Kolmogorov analogy.

A space-time collocation method (STCM) using asymptotically-constant basis functions is proposed and applied to the quantum Hamiltonian constraint for a loop-quantized treatment of the Schwarzschild interior. Canonically, these descriptions take the form of a partial difference equation (PDE). The space-time collocation approach presents a computationally efficient, convergent, and easily parallelizable method for solving this class of equations, which is the main novelty of this study. Results of the numerical simulations will demonstrate the benefit from a parallel computing approach; and show general flexibility of the framework to handle arbitrarily-sized domains. Computed solutions will be compared, when applicable, to a solution computed in the conventional method via iteratively stepping through a predefined grid of discrete values, computing the solution via a recursive relationship.

The problem of constructing a quantum theory of gravity is considered from a novel viewpoint. It is argued that any consistent theory of gravity should incorporate a relational character between the matter constituents of the theory. In particular, the traditional approach of quantizing a space–time metric is criticized and two possible avenues for constructing a satisfactory theory are put forward.

We discuss the conditions under which one can expect to have the usual identification of black hole entropy with the area of the horizon. We then construct an example in which the actual presence of the event horizon on a given hypersurface depends on a quantum event in which a certain quantum variable acquires a value and which occurs in the future of the given hypersurface. This situation indicates that there is something fundamental that is missing in the most popular of the current approaches towards the construction of a theory of quantum gravity, or, alternatively, that there is something fundamental that we do not understand about entropy in general, or at least in its association with black holes.

It has been argued by several authors that the quantum mechanical spectrum of black hole horizon area must be discrete. This has been confirmed in different formalisms, using different approaches. Here we concentrate on two approaches, the one involving quantization on a reduced phase space of collective coordinates of a Black Hole and the algebraic approach of Bekenstein. We show that for non-rotating, neutral black holes in any spacetime dimension, the approaches are equivalent. We introduce a primary set of operators sufficient for expressing the dynamical variables of both, thus mapping the observables in the two formalisms onto each other. The mapping predicts a Planck size remnant for the black hole.

Black hole entropy and its relation to the horizon area are considered. More precisely, the conditions and specifications that are expected to be required for the assignment of entropy, and the consequences that these expectations have when applied to a black hole are explored. In particular, the following questions are addressed: When do we expect to assign an entropy?; when are entropy and area proportional? and, what is the nature of the horizon? It is concluded that our present understanding of black hole entropy is somewhat incomplete, and some of the relevant issues that should be addressed in pursuing these questions are pointed out.

We evaluate the local contribution g_μνL of coherent matter with Lagrangian density L to the vacuum energy density. Focusing on the case of superconductors obeying the Ginzburg–Landau equation, we express the relativistic invariant density L in terms of low-energy quantities containing the pairs density. We discuss under which physical conditions the sign of the local contribution of the collective wave function to the vacuum energy density is positive or negative. Effects of this kind can play an important role in bringing the local changes in the amplitude of gravitational vacuum fluctuations — a phenomenon reminiscent of the Casimir effect in QED.

A class of metrics g_ab(xⁱ) describing spacetimes with horizons (and associated thermodynamics) can be thought of as a limiting case of a family of metrics g_ab(xⁱ;λ)without horizons when λ→0. We construct specific examples in which the curvature corresponding g_ab(xⁱ;λ) becomes a Dirac delta function and gets concentrated on the horizon when the limit λ→0 is taken, but the action remains finite. When the horizon is interpreted in this manner, one needs to remove the corresponding surface from the Euclidean sector, leading to winding numbers and thermal behavior. In particular, the Rindler spacetime can be thought of as the limiting case of (horizon-free) metrics of the form [g₀₀=ε²+a²x²; g_μν=-δ_μν] or [g₀₀=-g^xx=(ε²+4a²x²)^1/2, g_yy=g_zz=-1] when ε→0. In the Euclidean sector, the curvature gets concentrated on the origin of t_E-x plane in a manner analogous to Aharanov–Bohm effect (in which the vector potential is a pure gauge everywhere except at the origin) and the curvature at the origin leads to nontrivial topological features and winding number.

The study of BTZ blackhole physics and the cosmological horizon of 3D de Sitter spaces are carried out in unified way using the connections to the Chern Simons theory on three manifolds with boundary. The relations to CFT on the boundary is exploited to construct exact partition functions and obtain logarithmic corrections to Bekenstein formula in the asymptotic regime. Comments are made on the dS/CFT correspondence frising from these studies.

The conventional group of four-dimensional diffeomorphisms is not realizeable as a canonical transformation group in phase space. Yet there is a larger field-dependent symmetry transformation group which does faithfully reproduce 4-D diffeomorphism symmetries. Some properties of this group were first explored by Bergmann and Komar. More recently the group has been analyzed from the perspective of projectability under the Legendre map. Time translation is not a realizeable symmetry, and is therefore distinct from diffeomorphism-induced symmetries. This issue is explored further in this paper. It is shown that time is not "frozen". Indeed, time-like diffeomorphism invariants must be time-dependent. Intrinsic coordinates of the type proposed by Bergmann and Komar are used to construct invariants. Lapse and shift variables are retained as canonical variables in this approach, and therefore will be subject to quantum fluctuations in an eventual quantum theory. Concepts and constructions are illustrated using the relativistic classical and quantum free particle. In this example concrete time-dependent invariants are displayed and fluctuation in proper time is manifest.

We use the framework of noncommutative geometry to define a discrete model for fluctuating geometry. Instead of considering ordinary geometry and its metric fluctuations, we consider generalized geometries where topology and dimension can also fluctuate. The model describes the geometry of spaces with a countable number n of points. The spectral principle of Connes and Chamseddine is used to define dynamics. We show that this simple model has two phases. The expectation value 〈n〉, the average number of points in the universe, is finite in one phase and diverges in the other. Moreover, the dimension δ is a dynamical observable in our model, and plays the role of an order parameter. The computation of 〈δ〉 is discussed and an upper bound is found, 〈δ〉 < 2. We also address another discrete model defined on a fixed d = 1 dimension, where topology fluctuates. We comment on a possible spontaneous localization of topology.

We use exact results in a new approach to quantum gravity to study the effect of quantum loop corrections on the behavior of the metric of spacetime near the Schwarzschild radius of a massive point particle in the standard model. We show that the classical conclusion that such a system is a black hole is obviated. Phenomenological implications are discussed.

The brick wall model and thin film model are most representative models in black hole entropy calculation. However, each of them must have a cutoff in order to avoid the divergence, and the divergence itself cannot be explained satisfactorily. Li Xiang⁶ substituted the classical uncertainty relation with the generalized uncertainty relation, the divergence was removed, consequently the cutoff was also removed. But due to the complex expression, he did not give the final solution. He only drew a conclusion that the upper bound of a black hole entropy is in proportional to its horizon area. The method using the generalized uncertainty relation to brick wall model is studied in depth. It is finally found out that the black hole entropy itself is also proportional to its horizon area instead of the upper bound.

We propose a multi-graviton theory with non-nearest-neighbor couplings in the theory space. The resulting four-dimensional discrete mass spectrum reflects the structure of a latticed extra dimension. For a plausible mass spectrum motivated by the discretized Randall–Sundrum brane-world, the induced cosmological constant turns out to be positive and may serve as a quite simple model for the dark energy of our accelerating universe.

In this work we present a discussion of the existing links between the procedures of endowing the quantum gravity with a real time and of including in the theory a physical reference frame.

More precisely, as a first step, we develop the canonical quantum dynamics, starting from the Einstein equations in presence of a dust fluid and arrive at a Schrödinger evolution. Then, by fixing the lapse function in the path-integral of gravity, we get a Schrödinger quantum dynamics, of which eigenvalues problem provides the appearance of a dust fluid in the classical limit.

The main issue of our analysis is to claim that a theory, in which the time displacement invariance, on a quantum level, is broken, is indistinguishable from a theory for which this symmetry holds, but a real reference fluid is included.