General Covariance Research Articles

Energy cost of localization of relational quantum information

Entanglement of spatially separated quantum states is usually defined with respect to a reference frame provided by some external observer. Thus, if one wishes to localize the quantum information within a spatially separated entangled state, one must enact an entanglement extraction protocol also defined with respect to that external frame. Entanglement extraction for Gaussian ground states in such an external frame construction has been shown to require a minimum energy and is hence an interesting process for gravitational physics, where examinations of localization vs energy cost have a long history. General covariance, however, precludes dependence on external frames. In order to enact an extraction protocol in a generally covariant theory, dependence on the external reference frame must first be removed and the states made relational. We examine the implementation of an extraction protocol for Gaussian states, whose center of mass and relational degrees of freedom are entangled, in a relational toy model where translation invariance stands in for full diffeomorphism invariance. Constructing fully relational states and the corresponding extraction/localization can, in principle, be done in two ways. External frame position information can be removed through $G$-twirling over translations or one can spontaneously break the translation symmetry via the gradient of an auxiliary field, or $Z$-model. We determine the energetics of quantum information localization after the states have been made fully relational via both the $G$-twirl and $Z$-model. We also show one can obtain the $G$-twirl construction from a $Z$-model as a limit of positive operator valued measurements.

Holography of information in massive gravity using Dirac brackets

The principle of holography of information states that in massless gravity, it is possible to extract bulk information using asymptotic boundary operators. In our work, we study this principle in a linearized setting about empty flat space and formulate it using Dirac brackets between boundary Hamiltonian and bulk operators. We then address whether the storage of bulk information in flat space linearized massive gravity resembles that of massless gravity. For linearized massless gravity, using Dirac brackets, we recover the necessary criteria for the holography of information. In contrast, we show that the Dirac bracket of the relevant boundary observable with bulk operators vanishes for massive gravity. We use this important distinction to outline the canonical Hilbert space. This leads to split states, and consequently, one cannot use asymptotic boundary observables to extract bulk information in massive gravity. We also argue the split property directly without an explicit reference to the Hilbert space. The result reflects that we can construct local bulk operators in massive gravity about the vacuum, which are obscured from boundary observables due to the lack of diffeomorphism invariance. Our analysis sheds some light on evaporating black holes in the context of the islands proposal.

Emergent Diffeomorphism Invariance in Toy Models

AbstractConceptual difficulties in semiclassical and quantum gravity arise from diffeomorphism invariance of classical general relativity. With a motivation to shed some light on these difficulties, a class of toy models for which one‐dimensional diffeomorphism invariance, namely time‐reparametrization invariance, emerges at the classical level from energy conservation are studied. An attempt to quantize the models while taking the invariance seriously leads to toy versions of the problem of time in quantum gravity, of the cosmological constant problem, and of the black hole firewall problem. Nevertheless, all these problems are easily resolved by taking into account that the invariance emerges only at the classical level, while the fundamental theory that needs to be quantized is not diffeomorphism invariant.

Infrared (in)sensitivity of relativistic effects in cosmological observable statistics

The relativistic effects in cosmological observables contain critical information about the initial conditions and gravity on large scales. Compared to the matter density fluctuation, some of these relativistic contributions scale with negative powers of comoving wave number, implying a growing sensitivity to infrared modes. However, this can be inconsistent with the equivalence principle and can also lead to infrared divergences in the observable N-point statistics. Recent perturbative calculations have shown that this infrared sensitivity is indeed spurious due to subtle cancellations in the cosmological observables that have been missed in the bulk of the literature. Here we demonstrate that the cosmological observable statistics are infrared-insensitive in a general and fully non-linear way, assuming diffeomorphism invariance and adiabatic fluctuations on large scales.

Fully conservative f(R,T) gravity and Solar System constraints

The $f(R,T)$ gravity is a model whose action contains an arbitrary function of the Ricci scalar $R$ and the trace of the energy-momentum tensor $T$. We consider the separable model $f (R, T ) = \chi(R) + \varphi(T )$ and shown that, for perfect fluids, the dynamical equations are sufficient to determine how $\varphi$ depends on $T$, independently of the matter state equation and the geometry of space-time. Imposing the energy-momentum tensor conservation we obtain that $\varphi$ must be linear in $T$. However, the $T$ dependence is severely constrained using the full Will-Nordtvedt version of the parameterized post-Newtonian (PPN) formalism. The result of the PPN analysis is discussed and in addition it is shown that the diffeomorphism invariance of the matter action imposes strong constraints on conservative versions of $f(R,T)$ gravity.

Asymptotic normality for eigenvalue statistics of a general sample covariance matrix when p/n→∞ and applications

Asymptotic normality for eigenvalue statistics of a general sample covariance matrix when p/n→∞ and applications

Free‐Fall Non‐Universality in Quantum Theory

AbstractThe author shows by embodying the Einstein equivalence principle—local Poincaré invariance—and general covariance in quantum theory that wave‐function spreading rules out the universality of free fall, that is, the free‐fall trajectory of a quantum (test) particle depends on its internal properties. The author provides a quantitative estimate of the free‐fall non‐universality in terms of the Eötvös parameter, which turns out to be measurable in atom interferometry.

A General Modeling Framework for Network Autoregressive Processes

ABSTRACT A general flexible framework for Network Autoregressive Processes (NAR) is developed, wherein the response of each node in the network linearly depends on its past values, a prespecified linear combination of neighboring nodes and a set of node-specific covariates. The corresponding coefficients are node-specific, and the framework can accommodate heavier than Gaussian errors with spatial-autoregressive, factor-based, or in certain settings general covariance structures. We provide a sufficient condition that ensures the stability (stationarity) of the underlying NAR that is significantly weaker than its counterparts in previous work in the literature. Further, we develop ordinary and (estimated) generalized least squares estimators for both fixed, as well as diverging numbers of network nodes, and also provide their ridge regularized counterparts that exhibit better performance in large network settings, together with their asymptotic distributions. We derive their asymptotic distributions that can be used for testing various hypotheses of interest to practitioners. We also address the issue of misspecifying the network connectivity and its impact on the aforementioned asymptotic distributions of the various NAR parameter estimators. The framework is illustrated on both synthetic and real air pollution data.

Bayesian inverse problems with heterogeneous variance

AbstractWe consider inverse problems in Hilbert spaces under correlated Gaussian noise, and use a Bayesian approach to find their regularized solution. We focus on mildly ill‐posed inverse problems with fractional noise, using a novel wavelet‐based vaguelette–vaguelette approach. It allows us to apply sequence space methods without assuming that all operators are simultaneously diagonalizable. The results are proved for more general bases and covariance operators. Our primary aim is to study posterior contraction rate in such inverse problems over Sobolev classes and compare it to the derived minimax rate. Secondly, we study effect of plugging in a consistent estimator of variances in sequence space on the posterior contraction rate. This result is applied to the problem with error in forward operator. Thirdly, we show that empirical Bayes posterior distribution with a plugged‐in maximum marginal likelihood estimator of the prior scale contracts at the optimal rate, adaptively, in the minimax sense.

The RISE study protocol: resilience impacted by positive stressful events for people with cystic fibrosis.

For people with cystic fibrosis (CF), gaining access to elexacaftor/tezacaftor/ivacaftor (ETI) therapy, a new modulator drug combination, is perceived as a positive life event. ETI leads to a strong improvement of disease symptoms. However, some people with CF experience a deterioration in mental wellbeing after starting ETI therapy. The primary objective of this study is to investigate if and in which direction mental wellbeing of people with CF changes after starting ETI therapy. Our secondary objectives include, among others, investigation of underlying biological and psychosocial factors associated with a change in mental wellbeing of people with CF after starting ETI therapy. The Resilience lmpacted by Positive Stressful Events (RISE) study is a single-arm, observational, prospective longitudinal cohort. It has a timeframe of 60 weeks: 12 weeks before, 12 weeks after, 24 weeks after and 48 weeks after the start of ETI therapy. The primary outcome is mental well-being, measured at each of these four time points. Patients aged ≥12 years at the University Medical Center Utrecht qualifying for ETI therapy based on their CF mutation are eligible. Data will be analysed using a covariance pattern model with a general variance covariance matrix. The RISE study was classified by the institutional review board as exempt from the Medical Research Involving Human Subjects Act. Informed consent was obtained by both the children (12-16 years) and their caregivers, or only provided by the participants themselves when aged ≥16 years.

Efficient spectral implementation of ODE wall model and the extension of integral wall model to unstructured LES solvers

While wall modeling has become an indispensable tool for scale-resolving simulation of high Reynolds numbers wall-bounded turbulent flows, discussion of efficient implementation of wall models in parallel unstructured-grid solvers is lacking. In this paper, we discuss physical and numerical improvements to two zonal wall models based on integral form of the boundary layer differential equations within an unstructured-grid cell-centered finite-volume LES solver. The first model is a novel implementation of the ODE equilibrium wall model, where the velocity profile is expressed in the integral form using the constant shear-stress layer assumption and the integral is evaluated using a spectral quadrature method, resulting in a local and algebraic (grid-free) formulation. The second model is the modified form of the integral wall model of Yang et al. (2015) [14], which is based on the vertically-integrated thin-boundary-layer PDE along with a prescribed composite velocity profile in the wall-modeled region. A modification to the inner layer velocity profile which fixes the coordinate invariance issue of the original model is proposed. Additionally, several numerical challenges unique to the implementation of these integral models in unstructured mesh environments, such as the exchange of wall quantities between wall faces and LES cells, and the computation of surface gradients, are identified and possible remedies are proposed. The computational performance of the wall models is assessed both in a priori and a posteriori settings against the traditional finite-volume based ODE equilibrium wall model, showing a comparable computational cost for the integral wall model, and superior cost scaling for the spectral implementation over the finite-volume based approach. Load imbalance among the processors in parallel simulations seems to severely degrade the parallel efficiency of finite-volume based ODE wall model, whereas the spectral implementation is remarkably agnostic to these effects. The integral wall model, however, is even more prone to parallel efficiency degradation than the finite-volume based ODE wall model due to communication overhead among processors in highly parallelized settings.

Gravidynamics, spinodynamics and electrodynamics within the framework of gravitational quantum field theory

By noticing the fact that the charged leptons and quarks in the standard model are chirality-based Dirac spinors since their weak interaction violates maximally parity symmetry though they behave as Dirac fermions in electromagnetic interaction, we show that such a chirality-based Dirac spinor possesses not only electric charge gauge symmetry U(1) but also inhomogeneous spin gauge symmetry WS(1,3) = SP(1,3)$\rtimes$W$^{1,3}$, which reveals the nature of gravity and spacetime. The gravitational force and spin gauge force are governed by the gauge symmetries $W^{1,3}$ and SP(1,3), respectively, and a biframe spacetime with globally flat Minkowski spacetime as base spacetime and locally flat gravigauge spacetime as a fiber is described by the gravigauge field through emergent non-commutative geometry. The gauge-geometry duality and renormalizability in gravitational quantum field theory (GQFT) are carefully discussed. A detailed analysis and systematic investigation on gravidynamics and spinodynamics as well as electrodynamics are carried out within the framework of GQFT. A full discussion on the generalized Dirac equation and Maxwell equation as well as Einstein equation and spin gauge equation is made in biframe spacetime. New effects of gravidynamics as extension of general relativity are particularly analyzed. All dynamic equations of basic fields are demonstrated to preserve the spin gauge covariance and general coordinate covariance due to the spin gauge symmetry and emergent general linear group symmetry GL(1,3,R), so they hold naturally in any spinning reference frame and motional reference frame.

Semisimple four‐dimensional topological field theories cannot detect exotic smooth structure

AbstractWe prove that semisimple four‐dimensional oriented topological field theories lead to stable diffeomorphism invariants and can therefore not distinguish homeomorphic closed oriented smooth four‐manifolds and homotopy equivalent simply connected closed oriented smooth four‐manifolds. We show that all currently known four‐dimensional field theories are semisimple, including unitary field theories, and once‐extended field theories which assign algebras or linear categories to 2‐manifolds. As an application, we compute the value of a semisimple field theory on a simply connected closed oriented 4‐manifold in terms of its Euler characteristic and signature. Moreover, we show that a semisimple four‐dimensional field theory is invariant under ‐stable diffeomorphisms if and only if the Gluck twist acts trivially. This may be interpreted as the absence of fermions amongst the ‘point particles’ of the field theory. Such fermion‐free field theories cannot distinguish homotopy equivalent 4‐manifolds. Throughout, we illustrate our results with the Crane–Yetter–Kauffman field theory associated to a ribbon fusion category, settling in the negative the question of whether it is sensitive to smooth structure. As a purely algebraic corollary of our results applied to this field theory, we show that a ribbon fusion category contains a fermionic object if and only if its Gauss sums vanish.

Spectral Efficiency of Time-Variant Massive MIMO Using Wiener Prediction

Massive multiple-input multiple-output (MIMO) serves as technological pillar in current 5G deployments. The spectral efficiency (SE) reduction due to time-variance of the wireless channel, and respective mitigation strategies, are still not sufficiently understood. In this paper, we propose a multi-step Wiener predictor with a general temporal covariance function to quantify the impact of time-variance on the SE for different numbers of consecutive pilot symbols and prediction horizons. We provide asymptotic expressions of the SE for maximum ratio combining (MRC) beam-forming that show excellent agreement to numerical simulations. We investigate the channel hardening capability of massive MIMO systems in time-variant propagation channels and show that it is largely independent of the prediction horizon. The numerical evaluations show that channel prediction allows to double SE utilizing four instead of one uplink pilot symbol for a prediction horizon of 0.3 λ for both MRC and regularized zero-forcing (RZF) beam-forming.

Highly Efficient Estimators with High Breakdown Point for Linear Models with Structured Covariance Matrices

A unified approach is provided for a method of estimation of the regression parameter in balanced linear models with a structured covariance matrix that combines a high breakdown point with high asymptotic efficiency at models with multivariate normal errors. Of main interest are linear mixed effects models, but our approach also includes several other standard multivariate models, such as multiple regression, multivariate regression, and multivariate location and scatter. Sufficient conditions are provided for the existence of the estimators and corresponding functionals, strong consistency and asymptotic normality is established, and robustness properties are derived in terms of breakdown point and influence function. All the results are obtained for general identifiable covariance structures and are established under mild conditions on the distribution of the observations, which goes far beyond models with elliptically contoured densities. Some results are new and others are more general than existing ones in the literature. In this way, results on high breakdown estimation with high efficiency in a wide variety of multivariate models are completed and improved.

Learning Feature-Sparse Principal Subspace.

The principal subspace estimation is directly connected to dimension reduction and is important when there is more than one principal component of interest. In this article, we introduce two new algorithms to solve the feature-sparsity constrained PCA problem (FSPCA) for the principal subspace estimation task, which performs feature selection and PCA simultaneously. Existing optimization methods for FSPCA require data distribution assumptions and are lack of global convergence guarantee. Though the general FSPCA problem is NP-hard, we show that, for a low-rank covariance, FSPCA can be solved globally (Algorithm 1). Then, we propose another strategy (Algorithm 2) to solve FSPCA for the general covariance by iteratively building a carefully designed proxy. We prove (data-dependent) approximation bound and regular stationary convergence guarantees for the new algorithms. For the spectrum of covariance with exponential/Zipf's distribution, we provide exponential/posynomial approximation bounds. Constructive examples and numerical results are provided to demonstrate the tightness of our results. Experimental results show the promising performance and efficiency of the new algorithms compared with the state-of-the-arts on both synthetic and real-world datasets.

On the breakdown of space–time in general relativity

Based on the canonical quantization of d > 2-dimensional general relativity (GR) via the Dirac constraint formalism (also termed as ‘constraint quantization’), we propose the loss of covariance as a fundamental property of the theory. This breakdown occurs for the first-order Einstein–Hilbert action, whereby the loss of diffeomorphism invariance, besides first-class constraints, second-class constraints also exist leading to non-standard ghost fields that render the path integral non-covariant. We also attempt, for the first time, the canonical quantization via calculation of the path integral for the equivalent Hamiltonian formulation of GR, for which only first-class constraints exist. However, loss of covariance still occurs in this action due to loss of diffeomorphism invariance and structures arising from non-covariant constraints in the path integral. In contrast, we find that covariance as a symmetry is restored and quantization with perturbative calculations is possible in the weak limit of the gravitational field of these actions. Hence, we first establish, for the first time, that the breakdown in space–time is a property of GR itself as its limitation, indicating it to be an effective field theory (EFT). We further propose that the breakdown of space–time occurs as a non-perturbative feature of GR in the strong field limit of the theory. Besides GR, we also note that covariance is preserved when constraint quantization is conducted for non-Abelian gauge theories, such as the Yang–Mills theory. These findings are novel from a canonical gravity formalism and EFT approach, and are consistent with GR singularity theorems, yet are general and extend them, as the singularity theorems indicate breakdown at a strong field limit of GR in black holes. Our findings are in contrast to the asymptotic safety program. They support emergent theories of space–time and gravity, though are unique, as they do not require thermodynamics such as the entropic gravity program. From an EFT view, these indicate that new degrees of freedom and(or) principles in the non-perturbative sector of the full theory are a requirement, whereby covariance as a symmetry is broken in the high-energy (strong field) sector of GR.

General covariance from the viewpoint of stacks

General covariance from the viewpoint of stacks

Toward quantization of inhomogeneous field theory

We explore the quantization of a $$(1+1)$$ -dimensional inhomogeneous scalar field theory in which Poincaré symmetry is explicitly broken. We show the ‘classical equivalence’ between a scalar field theory on curved spacetime background and its corresponding inhomogeneous scalar field theory. This implies that a hidden connection may exist among some inhomogeneous field theories, which corresponds to general covariance in field theory on curved spacetime. Based on the classical equivalence, we propose how to quantize a specific field theory with broken Poincaré symmetry inspired by standard field theoretic approaches, canonical and algebraic methods, on curved spacetime. Consequently, we show that the Unruh effect can be realized in inhomogeneous field theory and propose that it may be tested by a condensed matter experiment. We suggest that an algebraic approach is appropriate for the quantization of a generic inhomogeneous field theory.

Schrödinger from Wheeler–DeWitt: The issues of time and inner product in canonical quantum gravity

The wave-function in quantum gravity is supposed to obey the Wheeler–DeWitt (WDW) equation, however there is neither a satisfactory probability interpretation nor a successful solution to the problem of time in the WDW framework. To gain some insight on these issues we compare quantization of ordinary systems, first in the usual way having the Schrödinger equation and second by promoting them as parametrized theories by introducing embedding coordinate fields, which yields first class constraints and the WDW equation. We observe that the time evolution in the WDW framework can be described with respect to the embedding coordinates, where the WDW equation becomes Schrödinger like, i.e. it involves first order timelike functional derivatives. Moreover, the equivalence with the ordinary quantization procedure determines a suitable Hilbert space with a viable probability interpretation. We then apply the same construction to general relativity by adding embedding fields without any prior coordinate choice. The reparametrized general relativity has two different types of diffeomorphism invariance, which arises from world-volume and target-space reparametrizations. As in the case of ordinary systems, the time evolution can be described with respect to the embedding fields and the WDW equation becomes Schrödinger like; the construction is almost identical to an ordinary parametrized field theory in terms of time evolution and Hilbert space structure. However, this time, the constraint algebra enforces the wave-function to be in a subspace of states annihilated by an operator that can be identified as the Hamiltonian. The implications of these results for the canonical quantization program, and in particular for the minisuperspace quantum cosmology, are discussed.