Quantum gravity, minimum length and holography

The Karolyhazy uncertainty relation is the statement that if a device is used to measure a length $l$, there will be a minimum uncertainty $\delta l$ in the measurement, given by $(\delta l)^3 \sim L_P^2\; l$. This is a consequence of combining the principles of quantum mechanics and general relativity. In this note we show how this relation arises in our approach to quantum gravity, in a bottom-up fashion, from the matrix dynamics of atoms of space-time-matter. We use this relation to define a space-time-matter foam at the Planck scale, and to argue that our theory is holographic. By coarse graining over time scales larger than Planck time, one obtains the laws of quantum gravity. Quantum gravity is not a Planck scale phenomenon; rather it comes into play whenever no classical space-time background is available to describe a quantum system. Space-time and classical general relativity arise from spontaneous localisation in a highly entangled quantum gravitational system. The Karolyhazy relation continues to hold in the emergent theory. An experimental confirmation of this relation will constitute a definitive test of the quantum nature of gravity.

We recall a classical theory of torsion gravity with an asymmetric metric, sourced by a Nambu–Goto + Kalb–Ramond string [R. T. Hammond, Rep. Prog. Phys. 65, 599 (2002)]. We explain why this is a significant gravitational theory and in what sense classical general relativity is an approximation to it. We propose that a noncommutative generalisation of this theory (in the sense of Connes’ noncommutative geometry and Adler’s trace dynamics) is a “quantum theory of gravity.” The theory is in fact a classical matrix dynamics with only two fundamental constants – the square of the Planck length and the speed of light, along with the two string tensions as parameters. The guiding symmetry principle is that the theory should be covariant under general coordinate transformations of noncommuting coordinates. The action for this noncommutative torsion gravity can be elegantly expressed as an invariant area integral and represents an atom of space–time–matter. The statistical thermodynamics of a large number of such atoms yields the laws of quantum gravity and quantum field theory, at thermodynamic equilibrium. Spontaneous localisation caused by large fluctuations away from equilibrium is responsible for the emergence of classical space–time and the field equations of classical general relativity. The resolution of the quantum measurement problem by spontaneous collapse is an inevitable consequence of this process. Quantum theory and general relativity are both seen as emergent phenomena, resulting from coarse graining of the underlying noncommutative geometry. We explain the profound role played by entanglement in this theory: entanglement describes interaction between the atoms of space–time–matter, and indeed entanglement appears to be more fundamental than quantum theory or space–time. We also comment on possible implications for black hole entropy and evaporation and for cosmology. We list the intermediate mathematical analysis that remains to be done to complete this programme.

RETRACTED: On the same origin of quantum physics and general relativity from Riemannian geometry and Planck scale formalism

Quantum Gravity (Cambridge Monographs on Mathematical Physics)

Einstein’s theory of general relativity has passed all precision tests to date. At some length scale, however, general relativity (GR) must break down and be reconciled with quantum mechanics in a quantum theory of gravity (a beyond-GR theory). Binary black hole mergers probe the non-linear, highly dynamical regime of gravity, and gravitational waves from these systems may contain signatures of such a theory. In this thesis, we seek to make gravitational wave predictions for binary black hole mergers in a beyond-GR theory. These predictions can then be used to perform model-dependent tests of GR with gravitational wave detections. We make predictions using numerical relativity, the practice of precisely numerically solving the equations governing spacetime. This allows us to probe the behavior of a binary black hole system through full inspiral, merger, and ringdown. We choose to work in dynamical Chern-Simons gravity (dCS), a higher-curvature beyond-GR effective field theory that couples spacetime curvature to a scalar field, and has motivations in string theory and loop quantum gravity. In order to obtain a well-posed initial value formalism, we perturb this theory around GR. We compute the leading-order behavior of the dCS scalar field in a binary black hole merger, as well as the leading-order dCS correction to the spacetime metric and hence gravitational radiation. We produce the first numerical relativity beyond-GR waveforms in a higher-curvature theory of gravity. This thesis contains additional results, all of which harness the power of numerical relativity to test GR. We compute black hole shadows in dCS gravity, numerically prove the leading-order stability of rotating black holes in dCS gravity, and lay out a formalism for determining the start time of binary black hole ringdown using information from the strong-field region of a binary black hole simulation.

Students who are interested in quantum gravity usually face the difficulty of working through a large amount of prerequisite material before being able to deal with actual quantum gravity. A First Course in Loop Quantum Gravity by Rodolfo Gambini and Jorge Pullin, aimed at undergraduate students, marvellously succeeds in starting from the basics of special relativity and covering basic topics in Hamiltonian dynamics, Yang Mills theory, general relativity and quantum field theory, ending with a tour on current (loop) quantum gravity research. This is all done in a short 173 pages!As such the authors cannot cover any of the subjects in depth and indeed this book should be seen more as a motivation and orientation guide so that students can go on to follow the hints for further reading. Also, as there are many subjects to cover beforehand, slightly more than half of the book is concerned with more general subjects (special and general relativity, Hamiltonian dynamics, constrained systems, quantization) before the starting point for loop quantum gravity, the Ashtekar variables, are introduced.The approach taken by the authors is heuristic and uses simplifying examples in many places. However they take care in motivating all the main steps and succeed in presenting the material pedagogically. Problem sets are provided throughout and references for further reading are given. Despite the shortness of space, alternative viewpoints are mentioned and the reader is also referred to experimental results and bounds.In the second half of the book the reader gets a ride through loop quantum gravity; the material covers geometric operators and their spectra, the Hamiltonian constraints, loop quantum cosmology and, more broadly, black hole thermodynamics. A glimpse of recent developments and open problems is given, for instance a discussion on experimental predictions, where the authors carefully point out the very preliminary nature of the results. The authors close with an 'open issues and controversies' section, addressing some of the criticism of loop quantum gravity and pointing to weak points of the theory. Again, readers aiming at starting research in loop quantum gravity should take this as a guide and motivation for further study, as many technicalities are naturally left out.In summary this book fully reaches the aim set by the authors – to introduce the topic in a way that is widely accessible to undergraduates – and as such is highly recommended.

The construction of a continuum limit for the dynamics of loop quantum gravity is unavoidable to complete the theory. We explain that such a construction is equivalent to obtaining the continuum physical Hilbert space, which encodes the solutions of the theory. We present iterative coarse graining methods to construct physical states in a truncation scheme and explain in which sense this scheme represents a renormalization flow. We comment on the role of diffeomorphism symmetry as an indicator for the continuum limit.

About a century ago, Albert Einstein realized how the Theory of Newton's Universal Gravitation was inadequate to describe nature and needed to be revised in order to be neatly incorporated into the relativistic scheme with its undesirable instantaneous force shortcomings. Since then we have come to adopt the relativistic high energy tensor theory of General Relativity, which saw its inception into mainstream physics in the year 1916. General Relativity has stood firm against varied tests, such as: gravitational lensing, perihelion of mercury, gravitational time dilation, PSR1913+16 etc, but with the steady advancement of Quantum Mechanics, and its overwhelming experimental success, we are yet again faced with the same problem that was faced a century ago, but this time, the problem is much worse, and the elephant in the room is General Relativity. The fact that General relativity and Quantum mechanics, share different views in the description of space and time or space-time, with quantum mechanics in favor of the latter and general relativity of the former; does not allow one to have a smooth transition from one representation to the other. This automatically implies that for scenarios where both quantum mechanics and gravity / general relativity are required, then that scenario is unsolvable with our current model of physics (hence either General Relativity or Quantum Mechanics is flawed). It becomes clear from the microscopic realm of quantum mechanics and the cosmological problems associated with General Relativity, such as dark energy, dark matter, the inflation force... perhaps a bit closer to home for general relativity being the singularities inherent in its own equations, and the very nature of General Relativity, allowing mostly approximate solutions (reason why most physicist, even today still cling to Newtonian dynamics), because of this and more a new theory of gravity is required – the much anticipated “Quantum Gravity” – just as it was required those many years ago. In the theory that I propose, we will see how from the most basic concepts of quantum mechanics, a theory of gravity can emerge that has the power to not only be mathematically simpler than General Relativity, explain dark energy, dark matter and inflation without any ad-hoc mechanisms, but also have the power to unify all the forces of nature into a single unified theory of everything.

Many of the most exciting open problems in high-energy physics are related to the behavior and ultimate nature of gravity and spacetime. In this dissertation, several categories of new results in quantum and classical gravity are presented, with applications to our understanding of both quantum field theory and cosmology. A fundamental open question in quantum field theory is related to ultraviolet completion: Which low-energy effective field theories can be consistently combined with quantum gravity? A celebrated example of the swampland program---the investigation of this question---is the weak gravity conjecture, which mandates, for a U(1) gauge field coupled consistently to gravity, the existence of a state with charge-to-mass ratio greater than unity. In this thesis, we demonstrate the tension between the weak gravity conjecture and the naturalness principle in quantum field theory, generalize the weak gravity conjecture to multiple gauge fields, and exhibit a model in which the weak gravity conjecture solves the standard model hierarchy problem. Next, we demonstrate that gravitational effective field theories can be constrained by infrared physics principles alone, namely, analyticity, unitarity, and causality. In particular, we derive bounds related to the weak gravity conjecture by placing such infrared constraints on higher-dimension operators in a photon-graviton effective theory. We furthermore place bounds on higher-curvature corrections to the Einstein equations, first using analyticity of graviton scattering amplitudes and later using unitarity of an arbitrary tree-level completion, as well as constrain the couplings in models of massive gravity. Completing our treatment of perturbative quantum gravity, outside of the swampland program, we also reformulate graviton perturbation theory itself, finding a field redefinition and gauge-fixing of the Einstein-Hilbert action that drastically simplifies the Feynman diagram expansion. Furthermore, our reformulation also exhibits a hidden symmetry of general relativity that corresponds to the double copy relations equating gravity amplitudes to sums of squares of gluon amplitudes in Yang-Mills theory, a surprising correspondence that yields insights into the structure of quantum field theories. Moving beyond perturbation theory into nonperturbative questions in quantum gravity, we consider the deep relation between spacetime geometry and properties of the quantum state. In the context of holography and the anti-de Sitter/conformal field theory correspondence, we test the proposed ER=EPR correspondence equating quantum entanglement with wormholes in spacetime. In particular, we demonstrate that the no-cloning theorem in quantum mechanics and the no-go theorem for topology change of spacetime are dual under the ER=EPR correspondence. Furthermore, we prove that the presence of a wormhole is not an observable in quantum gravity, rescuing ER=EPR from potential violation of linearity of quantum mechanics. Excitingly, we also prove a new area theorem within classical general relativity for arbitrary dynamics of two collections of wormholes and black holes; this area theorem is the ER=EPR analogue of entanglement conservation. We next turn our attention to the emergence of spacetime itself, placing consistency conditions on the proposed correspondence between anti-de Sitter space and the Multiscale Entanglement Renormalization Ansatz, a special tensor network that constitutes a computational tool for finding the ground state of certain quantum systems. Further examining the role of quantum entanglement entropy in the emergence of general relativity, we ask whether there is a consistent microscopic formulation of the entropy in theories of entropic gravity; we find that our results weaken equation-of-state proposals for entropic gravity while strengthening those more akin to holography, guiding future investigation of theories of emergent gravity. Finally, we examine the consequences of the Hamiltonian constraint in classical gravity for the early universe. The Hamiltonian constraint allows for the Liouville measure on the phase space of cosmological parameters for homogeneous, isotropic universes to be converted into a probability distribution on trajectories, or equivalently, on initial conditions. However, this measure diverges on the set of spacetimes that are spatially flat, like the observable universe. In this thesis, we derive the unique, classical, Hamiltonian-conserved measure for the subset of flat universes. This result allows for distinction between different models of cosmic inflation with similar observable predictions; for example, we find that the measure favors models of large-scale inflation, as such potentials more naturally produce the number of e-folds necessary to match cosmological observations.

The main findings of the generalized uncertainty principle (GUP), the phenomenological approach, for instance, the emergence of a minimal measurable length uncertainty, are obtained in various versions from theories of quantum gravity, such as string theory, loop quantum gravity, doubly special relativity and black hole physics. GUP counts for impacts of relativistic energies and finite gravitational fields on the fundamental theories of quantum mechanics (QM), the noncommutation and measurement uncertainty. Utilizing GUP in reconciling principles of general relativity (GR) and QM, thereby enables to draw convincing conclusions about quantum gravity. To resolve the shortcuts reported with the nonrelativistic three-dimensional GUP, namely, violation of Lorentz covariance, dependence on frame of reference, and violation of the linear additional law of momenta, we introduce relativistic four-dimensional generalized uncertainty principle (RGUP) to curved spacetime. To unify GR and QM, we apply the Born reciprocity principle (BRP), distance-momentum duality symmetry and RGUP to estimate the fundamental tensor in discretized curved spacetime. To this end, we generalize Riemann geometry. The Finsler geometry, which is characterized by manifold and Finsler structure, allows to directly apply RGUP to the Finsler structure of a free particle so that [Formula: see text] can be expressed as [Formula: see text], from which the metric tensor in discretized Riemann spacetime could be deduced. We conclude that [Formula: see text] is homogeneous with degree [Formula: see text] in [Formula: see text], while [Formula: see text] is [Formula: see text]-homogeneous resulting in [Formula: see text]. Despite, the astonishing similarity with the conformal transformation, know as Weyl tensor, this study suggests that principles of QMs could be unambiguously imposed on the resulting fundamental tensor. Also, we conclude that the features of Finsler geometry assumed in this study are likely the ones of the duel Hamilton geometry.

Attempts to formulate a quantum theory of gravitation are collectively known as {\it quantum gravity}. Various approaches to quantum gravity such as string theory and loop quantum gravity, as well as black hole physics and doubly special relativity theories predict a minimum measurable length, or a maximum observable momentum, and related modifications of the Heisenberg Uncertainty Principle to a so-called generalized uncertainty principle (GUP). We have proposed a GUP consistent with string theory, black hole physics and doubly special relativity theories and have showed that this modifies all quantum mechanical Hamiltonians. When applied to an elementary particle, it suggests that the space that confines it must be quantized, and in fact that all measurable lengths are quantized in units of a fundamental length (which can be the Planck length). On the one hand, this may signal the breakdown of the spacetime continuum picture near that scale, and on the other hand, it can predict an upper bound on the quantum gravity parameter in the GUP, from current observations. Furthermore, such fundamental discreteness of space may have observable consequences at length scales much larger than the Planck scale. Because this influences all the quantum Hamiltonians in an universal way, it predicts quantum gravity corrections to various quantum phenomena. Therefore, in the present work we compute these corrections to the Lamb shift, simple harmonic oscillator, Landau levels, and the tunneling current in a scanning tunneling microscope.

A number of approaches to fundamental physics can lead to the violation of Lorentz and CPT symmetry. This talk discusses the low-energy phenomenology associated with such eects and reviews various sample experiments within this c ontext. Introduction.—The Standard Model (SM) and General Relativity (GR) provide and excel- lent phenomenological description of nature. However, from a theoretical viewpoint these two theories leave unanswered a variety of key conceptual questions. It is therefore believed that the SM and GR merge into a single unified theory at high energies that resolves these issues. One possibility for experimental research in this field is to increase the energy in experiments and hope to excite new degrees of freedom, which can give insight into such a unified theory. A complementary experimental approach is characterized by tests at comparatively low or moderate energies, but with ultra-high precision. Various eort s along these lines, such as searches for axions, axion-like particles, weakly interacting massive particles, and weakly interacting sub-eV particles, have already been discussed at this meeting. This presentation is focused on another class of precision experiments, namely tests of Lorentz and CPT symmetry. The special theory of relativity and its underlying Lorentz symmetry have been established over a century ago. Since that time, Lorentz symmetry has been subjected to numerous tests, but no credible experimental evidence for departures from Lorentz symmetry has been found. In fact, special relativity has matured into one of the most important cornerstones of physics. It provides not only the basis for present-day physics, but it is also the starting point for most theoretical approaches to new physics beyond the SM and GR. In recent years, however, it has been realized that various of these approaches to new physics (although built on Lorentz invariance) can accommodate mild, minuscule deviations from this symmetry in the ground state (1). Examples of candidate underlying models with the possibility of Lorentz violation are strings, loop quantum gravity, cosmologically varying scalars, non-commutative geometry, and multiverses (2). A further motivation for Lorentz and CPT tests is provided by the fundamental character of these symmetries: they should be backed by experimental evidence of steadily increasing quality. At energy regimes below the Planck scale, such departures from Lorentz and CPT symmetry can be described in great generality by the Standard-Model Extension (SME) (3). The SME is an eective field theory that contains both the usual SM and GR. Th e remaining terms in the SME Lagrangian control the extent of Lorentz and CPT breakdown; they are constructed to involve all operators for Lorentz and CPT violation that are scalars under coordinate changes. This broad scope guarantees widest applicability: it eliminates the association to a particular underlying theory and ensures that practically all present and near-future experiments can be

The mutual conceptual incompatibility between General Relativity and Quantum Mechanics / Quantum Field Theory is generally seen as the most essential motivation for the development of a theory of Quantum Gravity. It leads to the insight that, if gravity is a fundamental interaction and Quantum Mechanics is universally valid, the gravitational field will have to be quantized, not at least because of the inconsistency of semi-classical theories of gravity. The objective of a theory of Quantum Gravity would then be to identify the quantum properties and the quantum dynamics of the gravitational field. If this means to quantize General Relativity, the general-relativistic identification of the gravitational field with the spacetime metric has to be taken into account. The quantization has to be conceptually adequate, which means in particular that the resulting quantum theory has to be background-independent. This can not be achieved by means of quantum field theoretical procedures. More sophisticated strategies, like those of Loop Quantum Gravity, have to be applied. One of the basic requirements for such a quantization strategy is that the resulting quantum theory has a classical limit that is (at least approximately, and up to the known phenomenology) identical to General Relativity. However, should gravity not be a fundamental, but an induced, residual, emergent interaction, it could very well be an intrinsically classical phenomenon. Should Quantum Mechanics be nonetheless universally valid, we had to assume a quantum substrate from which gravity would result as an emergent classical phenomenon. And there would be no conflict with the arguments against semi-classical theories, because there would be no gravity at all on the substrate level. The gravitational field would not have any quantum properties to be captured by a theory of Quantum Gravity, and a quantization of General Relativity would not lead to any fundamental theory. The objective of a theory of 'Quantum Gravity' would instead be the identification of the quantum substrate from which gravity results. The requirement that the substrate theory has General Relativity as a classical limit – that it reproduces at least the known phenomenology – would remain. The paper tries to give an overview over the main options for theory construction in the field of Quantum Gravity. Because of the still unclear status of gravity and spacetime, it pleads for the necessity of a plurality of conceptually different approaches to Quantum Gravity.

Since the publication of the general theory of relativity (GTR), gravity has been described by classical field equations. Using this approach the GTR has resolved a number of problems in gravity, but is associated with some other difficulties. Mathematically GTR results in the formation of infinite density singularities in black holes, it challenges simultaneity and causality, and it is generally incompatible with quantum mechanics. A separate problem is the presence of “dark energy,” the energy inherent in space time. GTR helps explain this energy by the addition of a separate cosmological constant. However, what is required are formulas that treat the energy in space time as an integral part of quantum gravity. This space-time energy is treated as integral in the quintessence model, and may be resolvable by the use of a minimum energy scale. In this paper we use the standard minimal energy scale, Planck's constant, and in turn define a new quintessence. Using this string quintessence, we obtain advanced quantum gravity (AQG), which technically agrees exactly with the GTR, in the range where the GTR has been widely tested. Additionally, the principle of relativity is also maintained, and advanced in order to restore simultaneity and causality. Moreover, using string quintessence, AQG resolves the difficulties related to singularities, and in turn explains the apparent presence of dark matter. The separate presence of dark energy can also be explained and is based directly on Planck's constant and the minimal distance scale, the Planck length. Overall, in this paper, gravitation is taken to the next level, black holes and in turn dark matter are explained, and dark energy, the presence of space-time energy, becomes integral to the equations of AQG.

First we contemplate the operational definition of space–time in four dimensions in light of basic principles of quantum mechanics and general relativity and consider some of its phenomenological consequences. The quantum gravitational fluctuations of the background metric that comes through the operational definition of space–time are controlled by the Planck scale and are therefore strongly suppressed. Then we extend our analysis to the braneworld setup with low fundamental scale of gravity. It is observed that in this case the quantum gravitational fluctuations on the brane may become unacceptably large. The magnification of fluctuations is not linked directly to the low quantum gravity scale but rather to the higher-dimensional modification of Newton's inverse square law at relatively large distances. For models with compact extra dimensions the shape modulus of extra space can be used as a most natural and safe stabilization mechanism against these fluctuations.