Lorentzian Signature Research Articles

Spinorial higher-spin gauge theory from IKKT model in Euclidean and Minkowski signatures

We explore the semi-classical relation between the fuzzy 4-hyperboloid HN4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {H}_N^4 $$\\end{document} and non-compact quantized twistor space ℙN1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{P}}_N^{1,2} $$\\end{document} at large N. This provides two backgrounds of the IKKT matrix model via two natural stereographic projections, leading to higher-spin gauge theories with Euclidean and Minkowski signature denoted by HS-IKKT. The resulting higher-spin gauge theory can be understood as an uplift of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super Yang-Mills to twistor space. The action of HS-IKKT is written using a spinor formalism in both in Euclidean and Minkowski signature. We then compute the tree-level amplitudes of the massless sector within the Yang-Mills part of the HS-IKKT model in the flat limit in Euclidean signature. All n-point tree-level scattering amplitudes for n ≥ 4 of this sector are found to vanish in the flat limit.

Exact quantisation of U(1)3 quantum gravity via exponentiation of the hypersurface deformation algebroid

Abstract The U(1)3 model for 3+1 Euclidian signature general relativity (GR) is an interacting, generally covariant field theory with two physical polarisations that shares many features of Lorentzian GR. In particular, it displays a non-trivial realisation of the hypersurface deformation algebroid with non-trivial, i.e. phase space dependent structure functions rather than structure constants. In this paper we show that the model admits an exact quantisation. The quantisation rests on the observation that for this model and in the chosen representation of the canonical commutation relations the density unity hypersurface algebra can be exponentiated on non-degenerate states. These are states that represent a non-degenerate quantum metric and from a classical perspective are the relevant states on which the hypersurface algebra is representable. The representation of the algebra is exact, with no ambiguities involved and anomaly free. The quantum constraints can be exactly solved using groupoid averaging and the solutions admit a Hilbert space structure that agrees with the quantisation of a recently found reduced phase space formulation. Using the also recently found covariant action for that model, we start a path integral or spin foam formulation which, due to the Abelian character of the gauge group, is much simpler than for Lorentzian signature GR and provides an ideal testing ground for general spin foam models. The solution of U(1)3 quantum gravity communicated in this paper motivates an entirely new approach to the implementation of the Hamiltonian constraint in quantum gravity.

Is It Possible to Estimate the Parameters of the 2D Evolution of the Space Metric Signature with Energy from the Correlations of the Azimuthal Characteristics of Particles?

The coplanarity of subcores in γ-ray–hadron families of the so-called long-range near-side “ridge” effect discovered by the CMS Collaboration at the LHC can be described in terms of the FANSY2.0 model, which reproduces the coplanar generation of the most energetic particles in hadron interactions at superhigh energies. Coplanar generation can be explained, in particular, by the hypothesis of change of the signature of the metric of the space–time continuum, namely, a fluctuation transformation of the basicthree-dimensional state into two-dimensional one (3D ↔ 2D). A method is proposed for experimental verification of this hypothesis by studying the azimuthal correlations of different particles in hadroninteractions.

Gravity from holomorphic discs and celestial Lw1+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Lw_{1+\\infty }$$\\end{document} symmetries

The study of physical theories in various signatures has been important for uncovering structures not easily visible or definable in Lorentz signature. In split signature, global twistor constructions for conformally self-dual (SD) gravity and Yang–Mills construct solutions from twistor data that can be expressed in terms of free data without gauge freedom. This is developed for asymptotically flat SD gravity to give a fully nonlinear encoding of the asymptotic gravitational data in terms of a real homogeneous generating function h on the real twistor space. The recently discovered Lw1+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Lw_{1+\\infty }$$\\end{document} celestial symmetries, when real, act locally as passive Poisson diffeomorphisms on the real twistor space. The twistor data, h, generates an imaginary such Poisson transformation that then generates the gravitational field by shifting the real slice of the twistor space. The twistor chiral sigma models, whose correlators yield the Einstein gravity tree-level S-matrix, are reformulated as theories of holomorphic discs in twistor space whose boundaries lie on the deformed real slice determined by h. The real Lw1+∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Lw_{1+\\infty }$$\\end{document} symmetries act on the corresponding formula for the S-matrix geometrically with vanishing Noether currents, but imaginary generators yield graviton vertex operators that generate gravitons in the perturbative expansion. A generating function for the all plus 1-loop amplitude, the analogous framework for Yang–Mills, possible interpretations in Lorentz signature and similar open string formulations of twistor and ambitwistor strings in 4d in split signature, are briefly discussed.

The power of Lorentzian wormholes

As shown by Louko and Sorkin in 1995, topology change in Lorentzian signature involves spacetimes with singular points, which they called crotches. We modify their construction to obtain Lorentzian semiclassical wormholes in asymptotically AdS. These solutions are obtained by inserting crotches on known saddles, like the double-cone or multiple copies of the Lorentzian black hole. The crotches implement swap-identifications, and are classically located near an extremal surface. The resulting Lorentzian wormholes have an instanton action equal to their area, which is responsible for topological suppression in any number of dimensions.We conjecture that including such Lorentzian wormhole spacetimes is equivalent to path integrating over all mostly Euclidean smooth spacetimes. We present evidence for this by reproducing semiclassical features of the genus expansion of the spectral form factor, and of a late-time two point function, by summing over the moduli space of Lorentzian wormholes. As a final piece of evidence, we discuss the Lorentzian version of West-Coast replica wormholes.

Foliated asymptotically safe gravity in the fluctuation approach

The gravitational asymptotic safety program envisions a high-energy completion of gravity based on a non-Gaussian renormalization group fixed point. A key step in this program is the transition from Euclidean to Lorentzian signature spacetimes. One way to address this challenge is to formulate the quantum theory based on the Arnowitt-Deser-Misner decomposition of the metric field. This equips the Euclidean spacetime with a preferred direction which may serve as the time-direction in the Lorentzian setting. In this work we use the Wetterich equation in order to compute the renormalization group flow of the graviton two-point function. The resulting beta functions possess a non-Gaussian renormalization group fixed point suitable for rendering the theory asymptotically safe. The phase diagram underlying the flow of the two-point function is governed by the interplay between this non-Gaussian fixed point, the Gaussian fixed point, and an infrared fixed point. The latter ensures that the renormalized squared graviton mass cannot take negative values. These results are in qualitative agreement with fluctuation computations carried out in the covariant setting. We take this as non-trivial evidence that the asymptotic safety mechanism remains intact when considering quantum gravity on spacetimes carrying a foliation structure. Technically, our work constitutes the first fluctuation computation carried out within the ADM-framework. Therefore, we also provide a detailed discussion of the conceptual framework, highlighting the elements which differ from fluctuation computations in the covariant setting.

Complete separation of variables in the geodesic Hamilton-Jacobi equation in four dimensions

We list all metrics of arbitrary signature in four dimensions which admit complete separation of variables in the Hamilton–Jacobi equation for geodesic Hamiltonians. There are only ten classes of separable metrics admitting Killing vector fields, indecomposable quadratic conservation laws, and coisotropic coordinates. Canonical separable metrics parameterized by several (up to twelve) arbitrary functions of single coordinates are written explicitly. The full set of independent conservation laws in involution for each canonical metrics is also found.

Unified dark energy from Chiral-Quintom model with a mixed potential in Friedmann–Lemaître–Robertson–Walker cosmology

For a spatially Friedmann–Lemaître–Robertson–Walker cosmology, we propose a multi-scalar field gravitational model. Specifically, we consider a two-scalar field cosmological model in which the kinetic components of the scalar fields establish a two-dimensional sphere of Lorentzian signature. For our Chiral-Quintom model we choose a mixed potential term Vleft( phi ,psi right) =V_{0}e^{lambda phi }+U_{0}e^{kappa phi }psi ^{frac{1}{sigma } } and we investigate the asymptotic limits of the cosmological parameters. This model for U_{0}=0, provides a generalization of the hyperbolic inflation where the equation of the state parameter can cross the phantom divide line. When U_{0}ne 0 we observe that this cosmological model exhibits asymptotic solutions that encompass accelerated universes, big rip singularities, and dust-like solutions. Hence, this multi-scalar field model it can be regarded as as a dark energy unify model which describes a variety of asymptotic cosmological scenarios.

Large N partition functions of 3d holographic SCFTs

We study the S1 × {Sigma}_{mathfrak{g}} topologically twisted index and the squashed sphere partition function of various 3d mathcal{N} ≥ 2 holographic superconformal field theories arising from M2-branes. Employing numerical techniques in combination with well-motivated conjectures we provide compact closed-form expressions valid to all orders in the perturbative 1/N expansion for these observables. We also discuss the holographic implications of our results for the topologically twisted index for the dual M-theory Euclidean path integral around asymptotically AdS4 solutions of 11d supergravity. In Lorentzian signature this leads to a prediction for the corrections to the Bekenstein-Hawking entropy of a class of static asymptotically AdS4 BPS black holes.

Green Hyperbolic Complexes on Lorentzian Manifolds

We develop a homological generalization of Green hyperbolic operators, called Green hyperbolic complexes, which cover many examples of derived critical loci for gauge-theoretic quadratic action functionals in Lorentzian signature. We define Green hyperbolic complexes through a generalization of retarded and advanced Green’s operators, called retarded and advanced Green’s homotopies, which are shown to be unique up to a contractible space of choices. We prove homological generalizations of the most relevant features of Green hyperbolic operators, namely that (1) the retarded-minus-advanced cochain map is a quasi-isomorphism, (2) a differential pairing (generalizing the usual fiber-wise metric) on a Green hyperbolic complex leads to covariant and fixed-time Poisson structures and (3) the retarded-minus-advanced cochain map is compatible with these Poisson structures up to homotopy.

Subadditive average distances and quantum promptness

A central property of a classical geometry is that the geodesic distance between two events is additive. When considering quantum fluctuations in the metric or a quantum or statistical superposition of different spacetimes, additivity is generically lost at the level of expectation values. In the presence of a superposition of metrics, distances can be made diffeomorphism invariant by considering the frame of a family of free-falling observers or a pressureless fluid, provided we work at sufficiently low energies. We propose to use the average squared distance between two events as a proxy for understanding the effective quantum (or statistical) geometry and the emergent causal relations among such observers. At each point, the average squared distance defines an average metric tensor. However, due to non-additivity, is not the (squared) geodesic distance associated with it. We show that departures from additivity can be conveniently captured by a bi-local quantity . Violations of additivity build up with the mutual separation between x and y and can correspond to C < 0 (subadditive) or C > 0 (superadditive). We show that average Euclidean distances are always subadditive: they satisfy the triangle inequality but generally fail to saturate it. In Lorentzian signature there is no definite result about the sign of C, most physical examples give C < 0 but there exist counterexamples. The causality induced by subadditive Lorentzian distances is unorthodox but not pathological. Superadditivity violates the transitivity of causal relations. On these bases, we argue that subadditive distances are the expected outcome of dynamical evolution, if relatively generic physical initial conditions are considered.

An Algebraic QFT Approach to the Wetterich Equation on Lorentzian Manifolds

We discuss the scaling of the effective action for the interacting scalar quantum field theory on generic spacetimes with Lorentzian signature and in a generic state (including vacuum and thermal states, if they exist). This is done constructing a flow equation, which is very close to the renown Wetterich equation, by means of techniques recently developed in the realm of perturbative Algebraic Quantum Field theory (pAQFT). The key ingredient that allows one to obtain an equation which is meaningful on generic Lorentzian backgrounds is the use of a local regulator, which keeps the theory covariant. As a proof of concept, the developed methods are used to show that non-trivial fixed points arise in quantum field theories in a thermal state and in the case of quantum fields in the Bunch–Davies state on the de Sitter spacetime.

Higher-order curvature operators in causal set quantum gravity

We construct higher-order curvature invariants in causal set quantum gravity. The motivation for this work is twofold: First, to characterize causal sets, discrete operators that encode geometric information on the emergent spacetime manifold, e.g., its curvature invariants, are indispensable. Second, to make contact with the asymptotic-safety approach to quantum gravity in Lorentzian signature and find a second-order phase transition in the phase diagram for causal sets, going beyond the discrete analog of the Einstein–Hilbert action may be critical. Therefore, we generalize the discrete d’Alembertian, which encodes the Ricci scalar, to higher orders. We prove that curvature invariants of the form R^{2} -2 Box R (and similar invariants at higher powers of derivatives) arise in the continuum limit.

On the study of the space metrics signature from correlations of particles in hadron interactions

Long-range near-side “ridge” effect discovered by the CMS Collaboration at the LHC, and the coplanarity of subcores in gamma-ray-hadron families, can be reproduced in the framework of the coplanar generation of the most energetic hadrons in hadron interactions, explained, in particular, by the hypothesis of change of the signature of the metric of the space-time continuum, namely, the transformation of the basic three-dimensional state into two-dimensional one (3D ↔ 2D). A method is proposed for experimental verification of this hypothesis by studying the azimuthal correlations of particles in hadron interactions.

Quartic locality of higher-spin gravity in de Sitter and Euclidean anti-de Sitter space

We consider higher-spin gravity in (A)dS4, defined as the holographic dual of a free O(N) or Sp(N) vector model. At the quartic level, this theory has been judged non-local at distances greater than the (A)dS radius, due to a mismatch of massless (twist=1) exchange-type terms in its boundary OPE behavior. We review the non-locality argument, and note that it relies on a double-lightcone limit, which requires a Lorentzian boundary. In the Euclidean OPE limit, we demonstrate the absence of massless exchange-type non-localities of any spin, by inspecting a known formula for the bulk exchange diagrams in Euclidean AdS, and constructing upper bounds in which the spin-dependence factorizes from the position-dependence. Our results suggest that higher-spin theory is local (at distances greater than the curvature radius) in spacetimes with Euclidean boundary signature. For Lorentzian bulk, this implies locality in de Sitter space, as opposed to anti-de Sitter.

Equivalences between 2D dilaton gravities, their asymptotic symmetries, and their holographic duals

Dilaton gravities in two dimensions can be formulated as particular Poisson sigma models. Target space diffeomorphisms map different models to each other and establish a one-to-one correspondence between their classical solutions. We obtain a general form of such diffeomorphisms in Lorentzian and Euclidean signatures and use them to extend known holographic results, including the Schwarzian action on the asymptotic boundary, from JT to a large class of dilaton gravity models.

Real time lattice correlation functions from differential equations

We report on an exact calculation of lattice correlation functions on a finite four-dimensional lattice with either Euclidean or Minkowskian signature. The lattice correlation functions are calculated by the method of differential equations. This method can be used for Euclidean and Minkowskian signature alike. The lattice correlation functions have a power series expansion in 1/ sqrt{lambda } , where λ is the coupling. We show that this series is convergent for all non-zero values of λ. At small coupling we quantify the accuracy of perturbative approximations. At the technical level we systematically investigate the interplay between twisted cohomology and the symmetries of the twist function.

Paths to gravitation via the gauging of parametrized field theories

In special-relativistic physics, spacetime is imbued with a fixed, non-dynamical metric tensor. A path to gravitational theory is to promote this tensor to a genuine dynamical field. An alternative description of special-relativistic physics involves no fixed geometry but instead the inclusion of scalar fields $X^{I}(x^{\mu})$ which dynamically may take the form of inertial coordinates in spacetime. This suggests an alternative approach to gravity where the invariance of actions under global Poincar\'{e} transformations of $X^{I}$ is promoted to either a local Poincar\'{e} or local Lorentz symmetry via the introduction of gauge fields. Points of commonality and departure of the resulting gravitational theories as compared to General Relativity are discussed. It is shown that the model based on local Lorentz symmetry is an extension of General Relativity that can introduce a standard of time into the dynamics of the gravitational field and allows for spacetimes described by a Minkowski metric or flat Euclidean signature metric despite the gravitational gauge field possessing non-zero curvature.

Pseudo-Riemann’s quartics in Finsler’s geometry—two-dimensional case

Finsler’s geometry usually describes an extension of Riemmann’s geometry into a direction-dependent geometric structure. Historically, the well-known Riemann’s quartic length element example served as the inspiration for this construction. Surprisingly, the covariant Fresnel equation—a fundamental dispersion relation in solid-state electrodynamics—emerges as the exact same quartic expression. As a result, Riemann’s quartic length expression can be regarded of as a mathematical representation of a well-known physical phenomenon. In this study, we offer numerous Riemann’s quartic examples that show Finsler’s geometry, even in the situation of a positive definite Euclidean signature space, is too restrictive for many applications. The strong axioms of Finsler’s geometry are violated in a substantially greater number of distinctive subsets for the spaces having an indefinite (Minkowski) signature. We suggest a weaker characterization of Finsler’s structure based on explicitly calculated two-dimensional examples. In tangential vector space, this concept permits singular subsets. Only open subsets of a manifold’s tangent bundle are required to satisfy the strong axioms of Finsler’s geometry. We demonstrate the distinctive unique subsets of the Riemann’s quartic in two dimensions and briefly discuss their possible physical origins.

The Finsler Spacetime Condition for (α,β)-Metrics and Their Isometries

For the general class of pseudo-Finsler spaces with (α,β)-metrics, we establish necessary and sufficient conditions such that these admit a Finsler spacetime structure. This means that the fundamental tensor has a Lorentzian signature on a conic subbundle of the tangent bundle and thus the existence of a cone of future-pointing time-like vectors is ensured. The identified (α,β)-Finsler spacetimes are candidates for applications in gravitational physics. Moreover, we completely determine the relation between the isometries of an (α,β)-metric and the isometries of the underlying pseudo-Riemannian metric a; in particular, we list all (α,β)-metrics which admit isometries that are not isometries of a.