Lorentzian Signature Research Articles

Relational Lorentzian Asymptotically Safe Quantum Gravity: Showcase Model

In a recent contribution, we identified possible points of contact between the asymptotically safe and canonical approaches to quantum gravity. The idea is to start from the reduced phase space (often called relational) formulation of canonical quantum gravity, which provides a reduced (or physical) Hamiltonian for the true (observable) degrees of freedom. The resulting reduced phase space is then canonically quantized, and one can construct the generating functional of time-ordered Wightman (i.e., Feynman) or Schwinger distributions, respectively, from the corresponding time-translation unitary group or contraction semigroup, respectively, as a path integral. For the unitary choice, that path integral can be rewritten in terms of the Lorentzian Einstein–Hilbert action plus observable matter action and a ghost action. The ghost action depends on the Hilbert space representation chosen for the canonical quantization and a reduction term that encodes the reduction of the full phase space to the phase space of observables. This path integral can then be treated with the methods of asymptotically safe quantum gravity in its Lorentzian version. We also exemplified the procedure using a concrete, minimalistic example, namely Einstein–Klein–Gordon theory, with as many neutral and massless scalar fields as there are spacetime dimensions. However, no explicit calculations were performed. In this paper, we fill in the missing steps. Particular care is needed due to the necessary switch to Lorentzian signature, which has a strong impact on the convergence of “heat” kernel time integrals in the heat kernel expansion of the trace involved in the Wetterich equation and which requires different cut-off functions than in the Euclidian version. As usual we truncate at relatively low order and derive and solve the resulting flow equations in that approximation.

Non-degenerate metrics, hypersurface deformation algebra, non-anomalous representations and density weights in quantum gravity

Classical General Relativity is a dynamical theory of spacetime metrics of Lorentzian signature. In particular the classical metric field is nowhere degenerate in spacetime. In its initial value formulation with respect to a Cauchy surface the induced metric is of Euclidian signature and nowhere degenerate on it. It is only under this assumption of non-degeneracy of the induced metric that one can derive the hypersurface deformation algebra between the initial value constraints which is absolutely transparent from the fact that the inverse of the induced metric is needed to close the algebra. This statement is independent of the density weight that one may want to equip the spatial metric with. Accordingly, the very definition of a non-anomalous representation of the hypersurface deformation algebra in quantum gravity has to address the issue of non-degeneracy of the induced metric that is needed in the classical theory. In the Hilbert space representation employed in Loop Quantum Gravity (LQG) most emphasis has been laid to define an inverse metric operator on the dense domain of spin network states although they represent induced quantum geometries which are degenerate almost everywhere. It is no surprise that demonstration of closure of the constraint algebra on this domain meets difficulties because it is a sector of the quantum theory which is classically forbidden and which lies outside the domain of definition of the classical hypersurface deformation algebra. Various suggestions for addressing the issue such as non-standard operator topologies, dual spaces (habitats) and density weights have been proposed to address this issue with respect to the quantum dynamics of LQG. In this article we summarise these developments and argue that insisting on a dense domain of non-degenerate states within the LQG representation may provide a natural resolution of the issue thereby possibly avoiding the above mentioned non-standard constructions.

Matter coupled to 3d quantum gravity: one-loop unitarity

Abstract We expect quantum field theories for matter to acquire intricate corrections due to their coupling to quantum fluctuations of the gravitational field. This can be precisely worked out in 3d quantum gravity: after integrating out quantum gravity, matter fields are effectively described as noncommutative quantum field theories, with quantum-deformed Lorentz symmetries. An open question remains: Are such theories unitary or not? On the one hand, since these are effective field theories obtained after integrating out high energy degrees of freedom, we may expect the loss of unitarity. On the other hand, as rigorously defined field theories built with Lorentz symmetries and standing on their own, we naturally expect the conservation of unitarity. In an effort to settle this issue, we explicitly check unitarity for a scalar field at one-loop level in both Euclidean and Lorentzian space-time signatures. We find that unitarity requires adding an extra-term to the propagator of the noncommutative theory, corresponding to a massless mode and given by a representation with vanishing Plancherel measure, thus usually ignored in spinfoam path integrals for quantum gravity. This indicates that the inclusion of matter in spinfoam models, and more generally in quantum gravity, might be more subtle than previously thought.

Nonperturbative phase diagram of two-dimensional N=(2, 2) super-Yang-Mills theory

We consider two-dimensional N=(2,2) Yang-Mills theory with gauge group SU(N) in Euclidean signature compactified on a torus with thermal fermion boundary conditions imposed on one cycle. We perform nonperturbative lattice analyses of this theory for large 12≤N≤20. Although no holographic dual of this theory is yet known, we conduct numerical investigations to check for features similar to the two-dimensional N=(8,8) Yang–Mills theory, which has a well-defined gravity dual. We perform lattice field theory calculations to determine the phase diagram, observing a spatial deconfinement transition similar to the maximally supersymmetric case. However, the transition does not continue to low temperature, implying the absence of a topology-changing transition between black hole geometries in any holographic dual for this four-supercharge theory. Published by the American Physical Society 2024

The gap phenomenon for conformally related Einstein metrics

AbstractWe determine the submaximal dimensions of the spaces of almost Einstein scales and normal conformal Killing fields for connected conformal manifolds. The results depend on the signature and dimension of the conformally nonflat conformal manifold. In definite signature, these two dimensions are at most and , respectively. In Lorentzian signature, these two dimensions are at most and , respectively. In the remaining signatures, these two dimensions are at most and , respectively. This upper bound is sharp and to realize examples of submaximal dimensions, we first provide them directly in dimension 4. In higher dimensions, we construct the submaximal examples as the (warped) product of the (pseudo)‐Euclidean base of dimension with one of the 4‐dimensional submaximal examples.

A hybrid Euclidean–Lorentzian universe

The limited velocity in a geometry of Lorentzian signature seem to prevent the universe to reach thermodynamic equilibrium as suggested by the cosmic microwave background. Thus it was suggested that the universe which was initially minuscule has reached a more considerable radius in a short duration by a process known as cosmic inflation. However, to drive such a process have led to the suggestion of an ad-hoc scalar field the inflaton, which has no purpose in nature other than driving the cosmic inflation field and then stopping it once the universe reached the right size. In a recent paper it was shown that rapid expansion can occur without postulating an inflation by following Hawking’s suggestion and assuming that primordially the metric of the universe had an Euclidean signature, in which case velocity is not limited and thermalization and rapid expansion are derived without the need to assume an ad-hoc field. However, while in the previous work emphasis was put on the dynamics and physical statistics of the particles in a Euclidean space versus Lorentzian space in which both spaces were given. No mathematical model was given regarding the development of current Lorentzian space-time from the early Euclidean space-time, and how fundamental problems such as the space-time singularity and the homogeneity of the CMB can be solved in the hybrid Euclidean–Lorentzian picture. This lacuna is to be rectified in this paper.

Inflationary Cosmology from Anti-de Sitter Wormholes.

We propose a new type of wave function for the Universe computed from the Euclidean path integral, with asymptotically anti-de Sitter (AdS) boundary conditions. In the semiclassical limit, it describes a Euclidean (half-)wormhole geometry, exhibiting a local maximum of the scale factor at the surface of reflection symmetry, giving rise to an expanding Universe upon analytic continuation to Lorentzian signature. We find that these Euclidean wormholes set natural initial conditions for inflation and that the semiclassical Wheeler-DeWitt wave function can favor a long-lasting inflationary epoch, resolving a well-known issue of the no-boundary proposal. Because of the asymptotic AdS conditions in the Euclidean past, they raise the possibility of describing the physics of inflating cosmologies and their perturbations within the context of holography.

SU(1,1) and SU(2) bundles on AdS4 and complex supergravity solutions

We study eleven dimensional supergravity solutions with an AdS7 factor. By writing AdS7 metric as an SU(1,1) bundle over AdS4 we derive new supergravity solutions. The solutions, however, are obtained in Euclidean signature and thus are necessarily complex. We also discuss AdS7 as an SU(2) bundle over AdS4, and observe that the two sets of solutions are related by analytic continuation of the scale factor of the fiber.

New Well-Posed boundary conditions for semi-classical Euclidean gravity

We consider four-dimensional Euclidean gravity in a finite cavity. Dirichlet conditions do not yield a well-posed elliptic system, and Anderson has suggested boundary conditions that do. Here we point out that there exists a one-parameter family of boundary conditions, parameterized by a constant p, where a suitably Weyl rescaled boundary metric is fixed, and all give a well-posed elliptic system. Anderson and Dirichlet boundary conditions can be seen as the limits p → 0 and ∞ of these. Focussing on static Euclidean solutions, we derive a thermodynamic first law. Restricting to a spherical spatial boundary, the infillings are flat space or the Schwarzschild solution, and have similar thermodynamics to the Dirichlet case. We consider smooth Euclidean fluctuations about the flat space saddle; for p > 1/6 the spectrum of the Lichnerowicz operator is stable — its eigenvalues have positive real part. Thus we may regard large p as a regularization of the ill-posed Dirichlet boundary conditions. However for p < 1/6 there are unstable modes, even in the spherically symmetric and static sector. We then turn to Lorentzian signature. For p < 1/6 we may understand this spherical Euclidean instability as being paired with a Lorentzian instability associated with the dynamics of the boundary itself. However, a mystery emerges when we consider perturbations that break spherical symmetry. Here we find a plethora of dynamically unstable modes even for p > 1/6, contrasting starkly with the Euclidean stability we found. Thus we seemingly obtain a system with stable thermodynamics, but unstable dynamics, calling into question the standard assumption of smoothness that we have implemented when discussing the Euclidean theory.

CD8 radiomics signature to assess inter-lesion spatial heterogeneity and cold liver lesions in advanced non-small cell lung cancers treated with durvalumab.

2511 Background: The objective of this study was to assess whether a validated CD8 radiomics signature may help to evaluate patient inter-lesion heterogeneity and to predict the clinical outcome of advanced non-small cell lung cancers (NSCLC) patients treated with durvalumab in Study 1108 phase I/II trial (NCT01693562). Methods: Clinical data and imaging data from patients with naïve and pretreated advanced NSCLC who received durvalumab monotherapy were used. Radiomic features were extracted on contrast-enhanced CT scans and a validated CD8 radiomics signature was applied. A progressive lesion was defined by an increase in lesion longest diameter of 20% at 8 weeks. Dispersion metrics of the radiomics signature were estimated to evaluate the impact of inter-lesion heterogeneity on patient’s response. Results: A total of 188 patients were included in this study, accounting for a total of 1137 lesions (median [IQR] = 4 [3 - 9] lesions per patient) evaluated at baseline using a radiomics approach. A low CD8 radiomics score at baseline was associated with a significantly higher risk of progression at the lesion-level (AUC=0.59, P-value&lt;0.0001), and was especially performant for liver lesions (AUC=0.66, P-value=0.0002). At the patient level, the least infiltrated lesion of a patient according to the radiomics score of CD8 T-cells was positively associated with OS (HR=0.70, P-value=0.029) and PFS (HR=0.68, P-value=0.014), the highest values being associated with the best outcomes. 55 patients had liver lesions, with worse prognosis than patients without liver lesion (HR=2 for OS and PFS, P-value = 0.00012 and 0.00022 respectively). In these patients, the CD8 radiomic score enabled the stratification of patients according to hot and cold liver metastasis (HR=0.6, P-value=0.072 and HR=0.54, P-value=0.038 for OS and PFS respectively). A 4-class stratification of the whole cohort based on the least infiltrated (cold/hot) and liver or non-liver lesion was independently associated with clinical outcomes. These radiomics approaches have shown independent prognostic values when adjusting for PD-L1 status in multivariate analyses (Table). Conclusions: These results confirm the predictive value at a lesion level and the patient level of the biologically inspired CD8 radiomics score for advanced NSCLC patients treated with durvalumab. It has shown interesting results in discriminating outcomes of patients with liver lesions by identifying hot and cold lesions. [Table: see text]

Spinorial description for Lorentzian hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{hs} $$\\end{document}-IKKT

We introduce a novel spinorial description for the higher-spin gauge theory induced by the IKKT matrix model on an FLRW spacetime with Lorentzian signature, called Lorentzian hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{hs} $$\\end{document}-IKKT theory. The new description is based on Weyl spinors transforming under the space-like isometry subgroup SL(2, ℂ) of the structure group SO(2, 4) ≃ SU(2, 2). It allows us to exploit the full power of the spinor formalism in Lorentzian signature, in contrast to a previous formalism based on the compact subgroup SU(2)L × SU(2)R of SU(2, 2). Some cubic vertices of the Yang-Mills sector and the corresponding scattering amplitudes are computed. We observe that the n-point (for n ≥ 4) tree-level amplitudes are typically non-trivial on-shell, but exponentially suppressed in the late-time regime. While Lorentz invariance of the higher-spin amplitudes is not manifest, it is expected to be restored by higher-spin gauge invariance.

Functional Formulation of Quantum Theory of a Scalar Field in a Metric with Lorentzian and Euclidean Signatures.

We study the Schrödinger equation in quantum field theory (QFT) in its functional formulation. In this approach, quantum correlation functions can be expressed as classical expectation values over (complex) stochastic processes. We obtain a stochastic representation of the Schrödinger time evolution on Wentzel-Kramers-Brillouin (WKB) states by means of the Wiener integral. We discuss QFT in a flat expanding metric and in de Sitter space-time. We calculate the evolution kernel in an expanding flat metric in the real-time formulation. We discuss a field interaction in pseudoRiemannian and Riemannian metrics showing that an inversion of the signature leads to some substantial simplifications of the singularity problems in QFT.

Real time holography for higher spin theories

Real time holography is studied in the context of the embedding space formalism. In the first part of this paper, we present matching conditions for on-shell integer spin fields when going from Euclidean to Lorentzian signature on AdS background. Using the BTZ black hole as an example, we discuss various ways of lifting the physical solution from the AdS surface to the whole embedding space. The BTZ propagator for higher spin field is expressed elegantly in terms of the embedding coordinates. In the second part of the paper, we develop the proposed duality between higher spin theory and vector models. We obtain a specific map between the field configurations of these two theories in real time, so called Lorentzian AdS/CFT map. We conclude by exploring the matching conditions for higher spin fields satisfying the proposed bulk quadratic action. The physical and ghost modes can be treated independently during the Wick rotation; only physical modes are considered to be external modes.

Introduction to loop quantum gravity. The Holst’s action and the covariant formalism

We review Holst formalism and dynamical equivalence with standard GR (in dimension [Formula: see text]). Holst formalism is written for a spin coframe field [Formula: see text] and a Spin(3,1)-connection [Formula: see text] on spacetime [Formula: see text] and it depends on the Holst parameter [Formula: see text]. We show the model is dynamically equivalent to standard GR, in the sense that up to a pointwise Spin(3,1)-gauge transformation acting on (uppercase Latin) frame indices, solutions of the two models are in one-to-one correspondence. Hence, the two models are classically equivalent. One can also introduce new variables by splitting the spin connection into a pair of a Spin(3)-connection [Formula: see text] and a Spin(3)-valued 1-form [Formula: see text]. The construction of these new variables relies on a particular algebraic structure, called a reductive splitting. A weaker structure than requiring that the gauge group splits as the products of two sub-groups, as it happens in Euclidean signature in the selfdual formulation originally introduced in this context by Ashtekar, and it still allows to deal with the Lorentzian signature without resorting to complexifications. The reductive splitting of [Formula: see text] is not unique and it is parameterized by a real parameter [Formula: see text] which is called the Immirzi parameter. The splitting is here done on spacetime, not on space as it is usually done in the literature, to obtain a Spin(3)-connection [Formula: see text], which is called the Barbero–Immirzi connection on spacetime. One obtains a covariant model depending on the fields [Formula: see text] which is again dynamically equivalent to standard GR. Usually in the literature one sets [Formula: see text] for the sake of simplicity. Here, we keep the Holst and Immirzi parameters distinct to show that eventually only [Formula: see text] will survive in boundary field equations.

Characterizing vegetation and return periods in avalanche paths using lidar and aerial imagery

ABSTRACT Snow avalanches are a hazard and ecological disturbance across mountain landscapes worldwide. Understanding how avalanche frequency affects forests and vegetation improves infrastructure planning, risk management, and avalanche forecasting. We implemented a novel approach using lidar, aerial imagery, and a random forest model to classify imagery-observed vegetation within avalanche paths in southern Glacier National Park, Montana, USA. We calculated spatially explicit avalanche return periods using a physically based spatial interpolation method and characterized the vegetation within those return period zones. The automated vegetation classification model differed slightly between avalanche paths, but the combination of lidar and spectral signature metrics provided the best accuracy (88–92 percent) for predicting vegetation classes within complex avalanche terrain rather than lidar or spectral signature metrics alone. The highest frequency avalanche return periods were broadly characterized by grassland and shrubland, but the influence of topography greatly influences the vegetation classes as well as the return periods. Furthermore, statistically significant differences in lidar-derived vegetation canopy height exist between categorical return periods. The ability to characterize vegetation within various avalanche return periods using remote sensing data provides land use planners and avalanche forecasters a tool for assessing the spatial extent of large-magnitude avalanches in individual avalanche paths.

Conformal vector fields on Lie groups: The trans-Lorentzian signature

A pseudo-Riemannian Lie group is a connected Lie group endowed with a left-invariant pseudo-Riemannian metric of signature (p,q). In this paper, we study pseudo-Riemannian Lie groups (G,〈⋅,⋅〉) with non-Killing left-invariant conformal vector fields. Firstly, we prove that if h is a Cartan subalgebra for a semisimple Levi factor of the Lie algebra g, then dim⁡h≤max⁡{0,min⁡{p,q}−2}. Secondly, we classify trans-Lorentzian Lie groups (i.e., min⁡{p,q}=2) with non-Killing left-invariant conformal vector fields, and prove that [g,g] is at most three-step nilpotent. Thirdly, based on the classification of the trans-Lorentzian Lie groups, we show that the corresponding Ricci operators are nilpotent, and consequently the scalar curvatures vanish. As a byproduct, we prove that four-dimensional trans-Lorentzian Lie groups with non-Killing left-invariant conformal vector fields are necessarily conformally flat, and construct a family of five-dimensional ones which are not conformally flat.

Light scalars at the cosmological collider

We study the self-energies of weakly interacting scalar fields in de Sitter space with one field much lighter than the Hubble scale. We argue that self-energies drastically simplify in this light limit. We illustrate this in theories with two scalar fields, one heavy and one light, interacting with one another through either cubic or quartic interactions. To regulate infrared divergences, we compute these self-energies in Euclidean de Sitter space and then carefully analytically continue to Lorentzian signature. In particular, we do this for the most general renormalizable theory of two scalar fields with even interactions to leading order in the coupling and the mass of the light field. These self-energies are determined by de Sitter sunset diagrams, whose analytic structure and UV divergences we derive. Even at very weak couplings, the light field can substantially change how the heavy field propagates over long distances. The light field’s existence may then be inferred from how it modifies the heavy field’s oscillatory contribution to the primordial bispectrum in the squeezed limit, i.e. its cosmological collider signal.

Lorentzian Robin Universe

In this paper, we delve into the gravitational path integral of Gauss-Bonnet gravity in four spacetime dimensions, in the mini-superspace approximation. Our primary focus lies in investigating the transition amplitude between distinct boundary configurations. Of particular interest is the case of Robin boundary conditions, known to lead to a stable Universe in Einstein-Hilbert gravity, alongside Neumann boundary conditions. To ensure a consistent variational problem, we supplement the bulk action with suitable surface terms. This study leads us to compute the necessary surface terms required for Gauss-Bonnet gravity with the Robin boundary condition, which wasn’t known earlier. Thereafter, we perform an exact computation of the transition amplitude. Through ħ → 0 analysis, we discover that the Gauss-Bonnet gravity inherently favors the initial configuration, aligning with the Hartle-Hawking no-boundary proposal. Remarkably, as the Universe expands, it undergoes a transition from the Euclidean (imaginary time) to the Lorentzian signature (real time). To further reinforce our findings, we employ a saddle point analysis utilizing the Picard-Lefschetz methods. The saddle point analysis allows us to find the initial configurations which lead to Hartle-Hawking no-boundary Universe that agrees with the exact computations. Our study concludes that for positive Gauss-Bonnet coupling, initial configurations corresponding to the Hartle-Hawking no-boundary Universe gives dominant contribution in the gravitational path-integral.

Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces

We consider perturbations of Minkowski space as well as more general spacetimes on which the wave operator \square_{g} is known to be essentially self-adjoint. We define complex powers (\square_{g}-i\varepsilon)^{-\alpha} by functional calculus, and show that the trace density exists as a meromorphic function of \alpha . We relate its poles to geometric quantities, in particular to the scalar curvature. The results allow us to formulate a spectral action principle which serves as a simple Lorentzian model for the bosonic part of the Chamseddine–Connes action. Our proof combines microlocal resolvent estimates, including radial propagation estimates, with uniform estimates for the Hadamard parametrix. The arguments work in Lorentzian signature directly and do not rely on transition from the Euclidean setting.

Gravitational observatories

We consider four-dimensional general relativity with vanishing cosmological constant defined on a manifold with a boundary. In Lorentzian signature, the timelike boundary is of the form σ × ℝ, with σ a spatial two-manifold that we take to be either flat or S2. In Euclidean signature we take the boundary to be S2 × S1. We consider conformal boundary conditions, whereby the conformal class of the induced metric and trace K of the extrinsic curvature are fixed at the timelike boundary. The problem of linearised gravity is analysed using the Kodama-Ishibashi formalism. It is shown that for a round metric on S2 with constant K, there are modes that grow exponentially in time. We discuss a method to control the growing modes by varying K. The growing modes are absent for a conformally flat induced metric on the timelike boundary. We provide evidence that the Dirichlet problem for a spherical boundary does not suffer from non-uniqueness issues at the linearised level. We consider the extension of black hole thermodynamics to the case of conformal boundary conditions, and show that the form of the Bekenstein-Hawking entropy is retained.