General Covariance Research Articles

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.

Gravity as a gapless phase and biform symmetries

We study effective field theories (EFTs) enjoying (maximal) biform symmetries. These are defined by the presence of a conserved (electric) current that has the symmetries of a Young tableau with two columns of equal length. When these theories also have a topological (magnetic) biform current, its conservation law is anomalous. We go on to show that this mixed anomaly uniquely fixes the two-point function between the electric and magnetic currents. We then perform a Källén-Lehmann spectral decomposition of the current-current correlator, proving that there is a massless mode in the spectrum, whose masslessness is protected by the anomaly. Furthermore, the anomaly gives rise to a universal form of the EFT whose most relevant term — which resembles the linear Einstein action — dominates the infrared physics. As applications of this general formalism, we study the theories of a Galileon superfluid and linearized gravity. Thus, one can view the masslessness of the graviton as being protected by the anomalous biform symmetries. The associated EFT provides an organizing principle for gravity at low energies in terms of physical symmetries, and allows interactions consistent with linearized diffeomorphism invariance. These theories are not ultraviolet-complete — the relevant symmetries can be viewed as emergent — nor do they include the nonlinearities necessary to make them fully diffeomorphism invariant, so there is no contradiction with the expectation that quantum gravity cannot have any global symmetries.

Sojourn functionals for spatiotemporal Gaussian random fields with long memory

AbstractThis paper addresses the asymptotic analysis of sojourn functionals of spatiotemporal Gaussian random fields with long-range dependence (LRD) in time, also known as long memory. Specifically, reduction theorems are derived for local functionals of nonlinear transformation of such fields, with Hermite rank $m\geq 1,$ under general covariance structures. These results are proven to hold, in particular, for a family of nonseparable covariance structures belonging to the Gneiting class. For $m=2,$ under separability of the spatiotemporal covariance function in space and time, the properly normalized Minkowski functional, involving the modulus of a Gaussian random field, converges in distribution to the Rosenblatt-type limiting distribution for a suitable range of values of the long-memory parameter.

Berry-Esseen bounds for the multivariate ℬ-free CLT and operator-valued matrices

We provide bounds of Berry-Esseen type for fundamental limit theorems in operator-valued free probability theory such as the operator-valued free Central Limit Theorem and the asymptotic behaviour of distributions of operator-valued matrices. Our estimates are on the level of operator-valued Cauchy transforms and the Lévy distance. We address the single-variable as well as the multivariate setting for which we consider linear matrix pencils and noncommutative polynomials as test functions. The estimates are in terms of operator-valued moments and yield the first quantitative bounds on the Lévy distance for the operator-valued free Central Limit Theorem. Our results also yield quantitative estimates on joint noncommutative distributions of operator-valued matrices having a general covariance profile. In the scalar-valued multivariate case, these estimates could be passed to explicit bounds on the order of convergence under the Kolmogorov distance.

Localized scalar structures around static black holes

In this work we address a way to capture scalar field solutions on static spacetimes by using BPS formalism and relaxing the general covariance condition. We focus on configurations where the background geometry describes topological black holes and present both analytical and numerical solutions, in addition to discussing the use of conserved charges associated to such field configurations. The obtained solutions are radially stable and the zero-mode arising from the stability equation can be written analytically.

On-Shell Covariance of Quantum Field Theory Amplitudes

Scattering amplitudes in quantum field theory are independent of the field parametrization, which has a natural geometric interpretation as a form of "coordinate invariance." Amplitudes can be expressed in terms of Riemannian curvature tensors, which makes the covariance of amplitudes under nonderivative field redefinitions manifest. We present a generalized geometric framework that extends this manifest covariance to all allowed field redefinitions. Amplitudes satisfy a recursion relation to all orders in perturbation theory that closely resembles the application of covariant derivatives to increase the rank of a tensor. This allows us to argue that tree-level amplitudes possess a notion of "on-shell covariance," in that they transform as a tensor under any allowed field redefinition up to a set of terms that vanish when the equations of motion and on-shell momentum constraints are imposed. We highlight a variety of immediate applications to effective field theories.

Cross-collection latent Beta-Liouville allocation model training with privacy protection and applications.

Cross-collection topic models extend previous single-collection topic models, such as Latent Dirichlet Allocation (LDA), to multiple collections. The purpose of cross-collection topic modeling is to model document-topic representations and reveal similarities between each topic and differences among groups. However, the restriction of Dirichlet prior and the significant privacy risk have hampered those models' performance and utility. Training those cross-collection topic models may, in particular, leak sensitive information from the training dataset. To address the two issues mentioned above, we propose a novel model, cross-collection latent Beta-Liouville allocation (ccLBLA), which operates a more powerful prior, Beta-Liouville distribution with a more general covariance structure that enhances topic correlation analysis. To provide privacy protection for the ccLBLA model, we leverage the inherent differential privacy guarantee of the Collapsed Gibbs Sampling (CGS) inference scheme and then propose a hybrid privacy protection algorithm for the ccLBLA model (HPP-ccLBLA) that prevents inferring data from intermediate statistics during the CGS training process without sacrificing its utility. More crucially, our technique is the first attempt to use the cross-collection topic model in image classification applications and investigate the cross-collection topic model's capabilities beyond text analysis. The experimental results for comparative text mining and image classification will show the merits of our proposed approach.

Proposal for D0-brane anti-D0-brane action

We propose an action for the D0-brane anti-D0-brane system that has its exact solution corresponding to the marginal tachyon profile defined for arbitrary constant separation of these D-branes. We also find its covariant formulation and generalization to higher dimensional Dp-branes.

3D Conformational Generative Models for Biological Structures Using Graph Information-Embedded Relative Coordinates.

Developing molecular generative models for directly generating 3D conformation has recently become a hot research area. Here, an autoencoder based generative model was proposed for molecular conformation generation. A unique feature of our method is that the graph information embedded relative coordinate (GIE-RC), satisfying translation and rotation invariance, was proposed as a novel way for encoding molecular three-dimensional structure. Compared with commonly used Cartesian coordinate and internal coordinate, GIE-RC is less sensitive on errors when decoding latent variables to 3D coordinates. By using this method, a complex 3D generation task can be turned into a graph node feature generation problem. Examples were shown that the GIE-RC based autoencoder model can be used for both ligand and peptide conformation generation. Additionally, this model was used as an efficient conformation sampling method to augment conformation data needed in the construction of neural network-based force field.

Scalar fields and Lifshitz black holes from Derrick’s theorem evasion

We study classical scalar fields in asymptotically Lifshitz spacetimes. By evading Derrick's theorem requiring the scalar potential to explicitly depend on the background coordinates, we induce a diffeomorphism invariance breaking and settle a formalism to find spatially localized solutions in the probe field limit. By inserting a backreaction equipped with a cosmological constant and a Maxwell term coupled to a dielectric function in the model, we find a Lifshitz black hole solution.

Background independence and field redefinitions in quantum gravity

It is generally believed that a full-fledged theory of quantum gravity should exhibit background independence and diffeomorphism invariance. In its most general form, the latter comprises field redefinitions, which are diffeomorphisms in configuration space. We show that any path-integral approach to quantum gravity leads to a tension between these properties, such that they cannot hold simultaneously.

NEUROFILAMENT LIGHT CHAIN (NFL) AND GENERAL COGNITIVE ABILITY IN ADULTS APPROACHING MIDLIFE

Abstract Neurofilament light chain (NfL) is a biomarker indexing axonal integrity where small NfL variations may be associated with cognitive performance in early adulthood and high values associated with neurodegenerative disorders such as Alzheimer’s disease. In the Colorado Adoption/Twin Study of Lifespan behavioral development and cognitive aging (CATSLife1) individuals were tested at 28–49 years (M=33.1, SD=4.9). Quanterix Simoa assays of plasma NfL (pNfL) were measured in duplicate, and we included values for 1159 individuals where 1098 had available general cognitive ability scores and sociodemographic covariates. Unadjusted NfL values were consistent with other studies of early-mid adulthood (M = 5.9, SD = 3.1, range = 1.14 – 40.1 pg/mL) and 6% showed values outside expected normal reference limits (>10 pg/mL). After adjusting for technical covariates and skew, higher natural log-transformed pNfL was associated with age (r = 0.27) and female sex (r = 0.07). Moreover, adjusting for sociodemographic covariates, higher pNfL was associated with lower general cognitive ability (GCA) (r = -.06), where associations were more pronounced above the mean pNfL value (r = -.08). Multi-level regression analyses suggested that GCA-NfL associations were modified by age, whereby the worse performance was observed at higher ages and pNfL values (p <= 0.03), accounting for sibling relatedness and sociodemographic covariates. We observed small negative associations of higher plasma NfL and lower cognitive performance, where associations may become magnified with increasing age in early- to mid-adulthood.

Maxwell theory of fractons

We show that the main properties of the fracton quasiparticles can be derived from a generalized covariant Maxwell-like action. Starting from a rank-2 symmetric tensor field ${A}_{\ensuremath{\mu}\ensuremath{\nu}}(x)$, we build a partially symmetric rank-3 tensor field strength ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$ which obeys a kind of Bianchi identity. The most general action invariant under the covariant ``fracton'' transformation ${\ensuremath{\delta}}_{\text{fract}}{A}_{\ensuremath{\mu}\ensuremath{\nu}}(x)={\ensuremath{\partial}}_{\ensuremath{\mu}}{\ensuremath{\partial}}_{\ensuremath{\nu}}\mathrm{\ensuremath{\Lambda}}(x)$ consists of two independent terms: one describing linearized gravity (LG) and the other referable to fractons. The whole action can be written in terms of ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$, and the fracton part of the invariant Lagrangian writes as ${F}^{2}(x)$, in analogy with Maxwell theory. The canonical momentum derived from the fracton Lagrangian coincides with the tensor electric field appearing in the fracton literature, and the field equations of motion, which have the same form as the covariant Maxwell equations (${\ensuremath{\partial}}^{\ensuremath{\mu}}{F}_{\ensuremath{\alpha}\ensuremath{\beta}\ensuremath{\mu}}(x)=0$), can be written in terms of the generalized electric and magnetic fields and yield two of the four Maxwell equations (generalized electric Gauss and Amp\`ere laws), while the other two (generalized magnetic Gauss and Faraday laws) are consequences of the ``Bianchi identity'' for the tensor ${F}_{\ensuremath{\mu}\ensuremath{\nu}\ensuremath{\rho}}(x)$, as in Maxwell theory. In the covariant generalization of the fracton theory, the equations describing the fracton limited mobility, i.e., the charge and dipole conservation, are not external constraints, but rather consequences of the field equations of motion, hence of the invariant action and, ultimately, of the fracton covariant symmetry. Finally, we increase the known analogies between LG and fracton theory by noting that both satisfy the generalized Gauss constraint which underlies the limited mobility property, which one would not expect in LG.

The metaphysics of fibre bundles

Recently, Dewar (2019) has suggested that one can apply the strategy of ‘sophistication’—as exemplified by sophisticated substantivalism as a response to the diffeomorphism invariance of General Relativity—to gauge theories such as electrodynamics. This requires a shift to the formalism of fibre bundles. In this paper, I develop and defend this suggestion. Where my approach differs from previous discussions is that I focus on the metaphysical picture underlying the fibre bundle formalism. In particular, I aim to affirm the physical reality of gauge properties. I argue that this allows for a local and separable explanation of the Aharonov-Bohm effect. Its puzzling features are explained by a form of holism inherent to fibre bundles.

Pole Skipping in Holographic Theories with Bosonic Fields.

We study pole skipping in holographic conformal field theories dual to diffeomorphism invariant theories containing an arbitrary number of bosonic fields in the large N limit. Defining a weight to organize the bulk equations of motion, a set of general pole skipping conditions are derived. In particular, the frequencies simply follow from general covariance and weight matching. In the presence of higher-spin fields, we find that the imaginary frequency for the highest-weight pole skipping point equals the higher-spin Lyapunov exponent which lies outside of the chaos bound. Without higher-spin fields, we show that the energy density Green's function has its highest-weight pole skipping happening at a location related to the out-of-time-order correlator for arbitrary higher-derivative gravity, with a Lyapunov exponent saturating the chaos bound and a butterfly velocity matching that extracted from a shockwave calculation. We also suggest an explanation for this matching at the metric level by obtaining the on-shell shockwave solution from a regularized limit of the metric perturbation at the skipped pole.

Holography from Singular Supertranslations on a Black Hole Horizon.

We investigate the standard and dual Bondi-Metzner-Sachs (BMS) supertranslation generators on a black hole horizon and draw some conclusions about black hole physics. Recently, it has been shown that in addition to conventional BMS supertranslation symmetries, there exists an additional infinite set of magnetic asymptotic symmetries, dual BMS supertranslations, again parametrized by a function on the two-sphere. We show that the Dirac bracket between these generators exhibits an anomalous central term when one parameter function exhibits a singularity in the complex stereographical coordinates on the sphere. In order to preserve general coordinate invariance, we demonstrate that this central term can be removed by postulating a holographic gravitational Chern-Simons theory on the horizon. This indicates that for an anomaly-free theory of quantum gravity in the presence of a black hole, one should include a boundary theory on the horizon.

BRST formalism of Weyl conformal gravity

We present the BRST formalism of a Weyl conformal gravity in Weyl geometry. Choosing the extended de Donder gauge-fixing condition (or harmonic gauge condition) for the general coordinate invariance and the new scalar gauge-fixing for the Weyl invariance we find that there is a Poincar${\rm{\acute{e}}}$-like $\IOSp(10|10)$ supersymmetry as in a Weyl invariant scalar-tensor gravity in Riemann geometry. We also point out that there is a gravitational conformal symmetry in quantum gravity although there is a massive Weyl gauge field as a result of spontaneous symmetry breakdown of Weyl gauge symmetry and account for how the gravitational conformal symmetry is spontaneously broken to the Poincar\'e symmetry. The corresponding massless Nambu-Goldstone bosons are the graviton and the dilaton. We also prove the unitarity of the physical S-matrix on the basis of the BRST quartet mechanism.

Introducing block-Toeplitz covariance matrices to remaster linear discriminant analysis for event-related potential brain–computer interfaces

Objective. Covariance matrices of noisy multichannel electroencephalogram (EEG) time series data provide essential information for the decoding of brain signals using machine learning methods. However, small datasets and high dimensionality make it hard to estimate these matrices. In brain–computer interfaces (BCI) based on event-related potentials (ERP) and a linear discriminant analysis (LDA) classifier, the state of the art covariance estimation uses shrinkage regularization. As this is a general covariance regularization approach, we aim at improving LDA further by better exploiting the domain-specific characteristics of the EEG to regularize the covariance estimates. Approach. We propose to enforce a block-Toeplitz structure for the covariance matrix of the LDA, which implements an assumption of signal stationarity in short time windows for each channel. Main results. An offline re-analysis of data collected from 213 subjects under 13 different event-related potential BCI protocols showed a significantly increased binary classification performance of this ‘ToeplitzLDA’ compared to shrinkage regularized LDA (up to 6 AUC points, p < 0.001) and Riemannian classification approaches (up to 2 AUC points, p < 0.001). In an unsupervised visual speller application, this improvement would translate to a relative reduction of spelling errors by 81% on average for 25 subjects. Additionally, aside from lower memory and reduced time complexity for LDA training, ToeplitzLDA proves to be robust against drastic increases of the number of temporal features. Significance. The proposed covariance estimation allows BCI researchers to improve classification rates and reduce calibration times of BCI protocols using event-related potentials and thus support the usability of corresponding applications. Its lower computational and memory needs could make it a valuable algorithm especially for mobile BCIs.

Wald’s entropy in Coincident General Relativity

The equivalence principle and its universality enables the geometrical formulation of gravity. In the standard formulation of General Relativity (GR) á la Einstein, the gravitational interaction is geometrized in terms of the spacetime curvature. However, if we embrace the geometrical character of gravity, two alternative, though equivalent, formulations of GR emerge in flat spacetimes, in which gravity is fully ascribed either to torsion or to non-metricity. The latter allows a much simpler formulation of GR oblivious to the affine spacetime structure, the Coincident General Relativity (CGR). The entropy of a black hole can be computed using the Euclidean path integral approach, which strongly relies on the addition of boundary or regulating terms in the standard formulation of GR. A more fundamental derivation can be performed using Wald’s formula, in which the entropy is directly related to Noether charges and is applicable to general theories with diffeomorphism invariance. In this work we extend Wald’s Noether charge method for calculating black hole entropy to spacetimes endowed with non-metricity. Using this method, we show that CGR with an improved action principle gives the same entropy as the well-known entropy in standard GR. Furthermore the first law of black hole thermodynamics holds and an explicit expression for the energy appearing in the first law is obtained.

Tracy-Widom limit for the largest eigenvalue of high-dimensional covariance matrices in elliptical distributions

Let X be an M×N random matrix consisting of independent M-variate elliptically distributed column vectors x1,…,xN with general population covariance matrix Σ. In the literature, the quantity XX∗ is referred to as the sample covariance matrix after scaling, where X∗ is the transpose of X. In this article, we prove that the limiting behavior of the scaled largest eigenvalue of XX∗ is universal for a wide class of elliptical distributions, namely, the scaled largest eigenvalue converges weakly to the same limit regardless of the distributions that x1,…,xN follow as M,N→∞ with M∕N→ϕ0>0 if the weak fourth moment of the radius of x1 exists. In particular, via comparing the Green function with that of the sample covariance matrix of multivariate normally distributed data, we conclude that the limiting distribution of the scaled largest eigenvalue is the celebrated Tracy-Widom law.