Geometric constructions for repulsive gravity and quantization

Theories of Quantum Gravity predict a minimum measurable length and a corresponding modification of the Heisenberg Uncertainty Principle to the so-called Generalized Uncertainty Principle (GUP). However, this modification is non-relativistic, making it unclear whether the minimum length is Lorentz invariant. We formulate a Relativistic Generalized Uncertainty Principle, resulting in a Lorentz invariant minimum measurable length and the resolution of the composition law problem. This proved to be an important step in the formulation of Quantum Field Theory with minimum length. We derived the Lagrangians consistent with the existence of minimal length and describing the behaviour of scalar, spinor, and $U(1)$ gauge fields. We calculated the Feynman rules (propagators and vertices) associated with these Lagrangians. Furthermore, we calculated the Quantum Gravity corrected scattering cross-sections for a lepton-lepton scattering. Finally, we compared our results with current experiments, which allowed us to improve the bounds on scale at which quantum gravity phenomena will become relevant.

Doubly special relativity (DSR) is usually regarded as a low-energy limit of a quantum gravity theory with testable predictions. On the other hand, non-local quantum field theories have been presented as a solution to the inconsistencies arising when quantizing gravity. Here, we present a new formulation of quantum field theories in DSR with non-local behavior. Our construction restricts the models to those showing linear Lorentz invariance. We derive the deformed Klein–Gordon, Dirac, and electromagnetic Lagrangians, as well as the deformed Maxwell equations. We also discuss the electric potential of a point charge. Finally, we analyze the connection between the nonlocality of field theories and DSR.

Observational missions have provided us with a reliable model of the evolution of the universe starting from the last scattering surface all the way to future infinity. Furthermore given a specific model of inflation, using quantum field theory on curved space-times this history can be pushed back in time to the epoch when space-time curvature was some 1062 times that at the horizon of a solar mass black hole! However, to extend the history further back to the Planck regime requires input from quantum gravity. An important aspect of this input is the choice of the background quantum geometry and of the Heisenberg state of cosmological perturbations thereon, motivated by Planck scale physics. This paper introduces first steps in that direction. Specifically we propose two principles that link quantum geometry and Heisenberg uncertainties in the Planck epoch with late time physics and explore in detail the observational consequences of the initial conditions they select. We find that the predicted temperature–temperature (T–T) correlations for scalar modes are indistinguishable from standard inflation at small angular scales even though the initial conditions are now set in the deep Planck regime. However, there is a specific power suppression at large angular scales. As a result, the predicted spectrum provides a better fit to the PLANCK mission data than standard inflation, where the initial conditions are set in the general relativity regime. Thus, our proposal brings out a deep interplay between the ultraviolet and the infrared. Finally, the proposal also leads to specific predictions for power suppression at large angular scales also for the (T–E and E–E) correlations involving electric polarization3. The PLANCK team is expected to release this data in the coming year.

In theoretical physics, important new results have often been found by realising that two different concepts are related to each other at a deep and fundamental level. Examples of such relations are dualities which relate two seemingly different quantum theories to each other by stating that the theories are in fact equivalent. In particular, the Hilbert spaces and the dynamics of the two theories agree. From a mathematical point of view, this means that the theories are identical. However, from a physical point of view, their descriptions may differ, for instance there may be different Lagrangians for the two theories. The duality examples mentioned in box 5.1 either relate quantum field theories together, or they relate string theories together. The Anti-de Sitter/Conformal Field Theory correspondence (AdS/CFT), however, is a new type of duality which relates a quantum field theory on flat spacetime to a string theory. This is particularly remarkable since string theory is a very promising candidate for a consistent theory of quantum gravity. Naively, quantum field theory on flat spacetime does not appear to be a theory of quantum gravity. However, the AdS/CFT correspondence, being a duality, implies that the two theories are equivalent. This explains why many scientists think that the AdS/CFT correspondence, discovered by Maldacena in 1997, is one of the most exciting discoveries in modern theoretical physics in the last two decades. Moreover, the AdS/CFT correspondence is an important realisation of the holographic principle . This principle states that in a gravitational theory, the number of degrees of freedom in a given volume V scales as the surface area ∂ V of that volume, as described in box 5.2. The theory of quantum gravity involved in the AdS/CFT correspondence is defined on a manifold of the form AdS × X , where AdS is the Anti-de Sitter space and X is a compact space. The quantum field theory may be thought of as being defined on the conformal boundary of this Anti-de Sitter space.

This is a reference book for researchers working in the field of general relativity, quantum mechanics andquantum gravity. A major part of the book deals with the formulation of special relativistic mechanics, special relativistic fluid dynamics and its generalization to general relativity where the gravitational field is described by a metric tensor. Emphasis is laid on the fact that the general theory of relativity is of tensorial character under all dieomorphisms of space-time and hence its field equations, namely the Einstein field equations for gravitation, the Maxwell equations in a curved space-time geometry and the fluid dynamical equations in curved space time are all valid for all observers in the universe. The emphasis throughout is on the fact that matter generates a gravitational field described by a metric that has a non-vanishing curvature tensor and hence such space-times are inherently curved, ie, cannot be transformed into Minkowsian form. There is a final section on quantum mechanics and quantum field theory which introduces supersymmetry and quantum gravity to the reader. The reader after going through this book will be sufficiently well equipped to start research in quantum gravity, i.e, background independent physics which is as yet an unsolved problem owing to renormalization problems.Note: T& F does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka.

Finite element approximation in quantum field theory

Holography, Locality and Symmetries of The Universe Xiao Xiao It is an interesting question that, with a well tested duality between the quantum gravity in anti de Sitter space and a quantum field theory in one lower dimension, whether quantum gravity in a cosmological background has a well defined dual description. In large N limit, this duality could be a correspondence between an approximately local gravity theory describing cosmology and a quantum field theory. In dS/CFT, the quantum field theory is a Euclidean CFT living at the conformal boundary of de Sitter space, in large N limit, we should expect the local observables in de Sitter cosmology be recovered from the CFT. We explicitly develop this construction for scalar fields and derive the operator map at lowest order of 1 N expansion. Having addressed the fundamental question of how local fields in de Sitter cosmology arise via holography, we focus on the theory of cosmological perturbations that is described in terms of local field theory. The curvature perturbations during inflation, which originated from quantum fluctuations of inflaton and which induced the CMB inhomogeneity we see today, as well as the large scale structure, can be described as Goldstone boson fields which nonlinearly realize a subset of general coordinate transformations as residual symmetries. This fact puts strong constraints on the behavior of the cosmological correlation functions, and a series of consistency relations constraining the soft limits of these correlation functions can be derived as Ward identities.

It was shown in the early Seventies that, in Local Quantum Theory (that is the most general formulation of Quantum Field Theory, if we leave out only the unknown scenario of Quantum Gravity) the notion of Statistics can be grounded solely on the local observable quantities (without assuming neither the commutation relations nor even the existence of unobservable charged field operators); one finds that only the well known (para)statistics of Bose/Fermi type are allowed by the key principle of local commutativity of observables. In this frame it was possible to formulate and prove the Spin and Statistics Theorem purely on the basis of First Principles. In a subsequent stage it has been possible to prove the existence of a unique, canonical algebra of local field operators obeying ordinary Bose/Fermi commutation relations at spacelike separations. In this general guise the Spin - Statistics Theorem applies to Theories (on the four dimensional Minkowski space) where only massive particles with finite mass degeneracy can occur. Here we describe the underlying simple basic ideas, and briefly mention the subsequent generalisations; eventually we comment on the possible validity of the Spin - Statistics Theorem in presence of massless particles, or of violations of locality as expected in Quantum Gravity.

<p><strong>The future of theoretical physics is unclear. Two large areas that fall under the umbrella of theoretical physics are cosmology and quantum gravity. Modern cosmology is relatively a much younger field than quantum gravity, and both of these fields require further developments of general relativity. In this thesis we do not hope to resolve the problems facing modern cosmology or theories of quantum gravity. Rather, we will conduct original research into aspects of general relativity that may be used in the future to aid the development and testing of theories of cosmology and quantum gravity.</strong></p><p>It is our view that the largest problem facing astrophysics and cosmology stem from the existence of the dark sector of the Universe. The implication here being that more than ninety percent of the energy density of the Universe is “missing in action” and seemingly consists of dark energy and dark matter. Furthermore, it is apparent that there exist conceptual flaws in our understanding of observational concepts such as expansion versus motion and observer biases. To this end, we investigate the standard spacetime metric used in cosmology, the Friedmann–Lemaˆıtre-Roberston–Walker (FLRW) metric in a peculiar coordinate system — the Painlev´e–Gullstrand coordinates. In this coordinate system (slicing), space is no longer expanding, rather, the galaxies are receding from each other. We hope this will aid in the understanding of expansion, motion, curvature, and observer bias with future work. We further investigate the possibility of black holes in cosmology being directly coupled to the accelerated expansion of the Universe — in other words, black holes as a source for dark energy. However, we show that this is highly implausible.</p><p>Relatively recently it has been postulated that the near black hole horizon limit may be a regime where quantum gravity effects become relevant i.e., quantum gravity may not be restricted to near the Planck scale. We investigate a curious model of black and white holes that shows how one may transition into the other over a finite period of time. This is research conducted in the near horizon limit of the Schwarzschild black hole. We introduce a time dependent function into the usual Schwarzschild black hole spacetime (leaving this new spacetime not a simple coordinate transformed version of the original). This function allows the black hole to transition into a white hole. Importantly, the action for this transition can be shown to be zero, meaning it can be added to the Feynman path integral at no cost.</p><p>Finally, we move to investigating the black hole memory effect. During the last decade, there has been an interesting connection made between the Bondi– Metzner–Sachs (BMS) group — an infinite dimensional group of symmetries found at null infinity — and the gravitational memory effect. In particular, it was shown that the passage of a gravitational wave that alters a Schwarzschild black hole is seen as a supertranslation of the spacetime at null infinity. We extend these calculations to the Kerr and Kerr–Newman black holes. Hence, showing that there may be a way to verify the abstract mathematical ideas predicated on the BMS group by detection of the memory effect in future observations.</p><p>It is our hope that when future gravitational wave detectors such as the laser-interferometer-space-antenna (LISA) are launched, research conducted in this thesis may shed light on how the memory may relate to black holes in their asymptotic & near horizon limits to aid our understanding of the nature of quantum gravity.</p>

f (R) theories are the simplest generalisations of General Relativity, Einstein’s theory of gravity. f (R) theories, and modified gravity theories more broadly, have been invoked to explain the accelerated expansion of the universe. A number of “viable” models have been proposed which satisfy cosmological tests and get around present day laboratory and solar system constraints by invoking the chameleon mechanism. It is important to determine whether such alternative models can be distinguished by precision experiments at the laboratory scale. This thesis focuses on recently proposed f (R) models in the context of the upcoming neutron quantum bouncer experiment GRANIT [1], which has the potential to directly observe violations of Einstein gravity. This experiment works by precisely measuring the energy levels of ultra-cold neutrons trapped in the Earth’s gravitational field above a mirror. Neutrons in these states have energies of the order of 10−12 eV (peV) and wavefunctions with spatial extents of ∼ 10 μm. The size of the neutron wavefunctions gives gravitational spectroscopy sensitivity to any change of the Newtonian potential above the mirror surface on a scale of tens of microns. These experiments have a unique opportunity to study gravity and quantum mechanics in an unprecedented regime. GRANIT claims an energy sensitivity of 0.01 peV with an improvement of one or two orders of magnitude expected over the next several years. The theoretical limit to the sensitivity is 10−7 peV. Previous work by Brax and Pignol [2] has shown that scalar-tensor theories, which are closely related to f (R) theories, can give rise to detectable shifts in the energy levels from their Newtonian values. Here the energy levels shifts are examined for three viable f (R) models: the exponential gravity model of Linder [3], the model of Hu and Sawicki [4] and the “new” exponential gravity model of Xu and Chen [5]. It is found that exponential gravity predicts energy level shifts of the order exp ( −10 ) mNc , where mN is the neutron mass. This is unobservable in any experiment lasting less than the age of the universe. The Hu-Sawicki model predicts much larger shifts which depend on the model parameters n and c2, where n, c2 > 0. It is found that the shifts are below the theoretical sensitivity limit for n ≥ 1 2 , with shifts at n = 1 2 of the order of 10 −19 peV. It is unlikely, though not yet ruled out, that n 1 models could be observed at neutron bouncer experiments operating under extreme conditions. The computation of the shifts for n 1 models is hampered by convergence issues. This may be immaterial, however, because Hu-Sawicki models with n 2 Hu-Sawicki model at laboratory scales, hence it is unobservable at a neutron bouncer experiment. The reason for the difference between these results with those of Brax and Pignol is shown to be due to the special form of the scalar potential and matter coupling, which are free in scalar-tensor theory and can be chosen to give a large effect, but are constrained in f (R) gravity. Because of these special features the neutron bouncer experiment can distinguish between f (R) and non-f (R) modified gravity, but not between the f (R) models examined in this thesis. Note the original date of this thesis was 22 November 2012. This version, with several copyrighted figures removed but nothing else modified apart from this notice, was created 21 October 2014.

Quantum systems governed by non-Hermitian Hamiltonians with symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that symmetry may also be important and present at the level of Hermitian quantum field theories because of the process of renormalisation. In some quantum field theories renormalisation leads to -symmetric effective Lagrangians. We show how symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework. From the study of examples -symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the norm that occurs in Hamiltonian quantum theory. We discuss examples where -symmetric field theories emerge from Hermitian field theories due to effects of renormalisation. We also consider the effects of renormalisation on field theories that are non-Hermitian but -symmetric from the start.

Rules of quantization and equations of motion for a finite-dimensional formulation of quantum field theory are proposed which fulfill the following properties: (a) Both the rules of quantization and the equations of motion are covariant; (b) the equations of evolution are second order in derivatives and first order in derivatives of the spacetime coordinates; and (c) these rules of quantization and equations of motion lead to the usual (canonical) rules of quantization and the (Schrödinger) equation of motion of quantum mechanics in the particular case of mechanical systems. We also comment briefly on further steps to fully develop a satisfactory quantum field theory and the difficuties which may be encountered when doing so.

We propose a foundational reformulation of quantum gravity and quantum field theory by replacing the continuous spacetime manifold with a causal lattice framework based on sedenionic algebra and displacement operators. This formulation enforces microcausality and saturation principles, allowing physical laws to emerge from non-commutative operator dynamics. Gravitational interactions are represented by commutators of symmetric and anti-symmetric sedenionic spinor fields, giving rise to both attractive Yukawa-type corrections and a repulsive component interpretable as dark energy. Notably, we demonstrate that flat galactic rotation curves can be accurately modeled without invoking dark matter halos or MOND modifications, using a stretched exponential mass distribution and an emergent Yukawa potential. The framework naturally generates the internal gauge symmetries of the Standard Model from the decomposition of sedenions and resolves quantum divergences without renormalization. This theory suggests that gravity, dark energy, and the stability of galactic structures arise from deeper algebraic geometry rather than ad hoc hypotheses, potentially marking a paradigm shift in fundamental physics.

When quantum fields are studied on manifolds with boundary, the corresponding one-loop quantum theory for bosonic gauge fields with linear covariant gauges needs the assignment of suitable boundary conditions for elliptic differential operators of Laplace type. There are however deep reasons to modify such a scheme and allow for pseudo-differential boundary-value problems. When the boundary operator is allowed to be pseudo-differential while remaining a projector, the conditions on its kernel leading to strong ellipticity of the boundary-value problem are studied in detail. This makes it possible to develop a theory of one-loop quantum gravity from first principles only, i.e. the physical principle of invariance under infinitesimal diffeomorphisms and the mathematical requirement of a strongly elliptic theory. It therefore seems that a non-local formulation of quantum field theory has some attractive features which deserve further investigation.

The holographic principle implies that quantum field theory (QFT) overcounts the number of independent degrees of freedom in quantum gravity. An argument due to Cohen, Kaplan, and Nelson (CKN) suggests that the number of degrees of freedom well described by QFT is even smaller than required by holographic bounds, and CKN interpreted this result as indicative of a correlation between the UV and IR cutoffs on QFT. Here, we consider an alternative interpretation in which the QFT degrees of freedom are depleted as a function of scale. We use a simple recipe to estimate the impact of depleted densities of states on precision observables, including the Lamb shift and lepton $g\ensuremath{-}2$. Although these observables are not sensitive to the level of depletion motivated by gravitational considerations, the phenomenological exercises also provide an interesting test of quantum field theory that is independent of underlying quantum gravity assumptions. A depleted density of states can also render the QFT vacuum energy UV insensitive, reconciling the success of QFT in describing ordinary particle physics processes and its apparent failure in predicting the cosmological constant.