De Donder Research Articles

Conformal symmetry in quantum gravity

We study the problem of how to derive conformal symmetry in the framework of quantum gravity. We start with a generic gravitational theory which is invariant under both the general coordinate transformation (GCT) and Weyl transformation (or equivalently, local scale transformation), and then construct its BRST formalism by fixing the gauge symmetries by the extended de Donder gauge and scalar gauge conditions. These gauge-fixing conditions are invariant under global GL(4) and global scale transformations. The gauge-fixed and BRST invariant quantum action possesses a huge Poincaré-like IOSp(10|10) global symmetry, from which we can construct an extended conformal symmetry in a flat Minkowski background in the sense that the Lorentz symmetry is replaced with the GL(4) symmetry. Moreover, we construct the conventional conformal symmetry out of this extended symmetry. With a flat Minkowski background ⟨gμν⟩=ημν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle g_{\\mu \ u } \\rangle = \\eta _{\\mu \ u }$$\\end{document} and a non-zero scalar field ⟨ϕ⟩≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\phi \\rangle \ e 0$$\\end{document}, the GL(4) and global scale symmetries are spontaneously broken to the Lorentz symmetry, thereby proving that the graviton and the dilaton are respectively the corresponding Nambu–Goldstone bosons, and therefore they must be exactly massless at nonperturbative level. One of remarkable aspects in our findings is that in quantum gravity, a derivation of conformal symmetry does not depend on a classical action, and its generators are built from only the gauge-fixing and the FP ghost actions. Finally, we address a generalized Zumino theorem in quantum gravity.

Canonical lifts in multisymplectic De Donder–Weyl Hamiltonian field theories

We define canonical lifts of vector fields to the multisymplectic multimomentum bundles of De Donder–Weyl Hamiltonian (first-order) field theories and to the appropriate premultisymplectic embedded constraint submanifolds on which singular field theories are studied. These new canonical lifts are used to study the so-called natural Noether symmetries present in both regular and singular Hamiltonian field theories along with their associated conserved quantities obtained from Noether’s theorem. The Klein–Gordon field, the Polyakov bosonic string, and Einstein–Cartan gravity in 3+1 dimensions are analyzed in depth as applications of these concepts; as a peripheral result obtained in the analysis of the bosonic string, we provide a new geometrical interpretation of the well-known Virasoro constraint.

Schwarzschild Black Hole from Perturbation Theory to All Orders.

Applying the quantum field theoretic perturbiner approach to Einstein gravity, we compute the metric of a Schwarzschild black hole order by order in perturbation theory. Using recursion, this calculation can be carried out in de Donder gauge to all orders in Newton's constant. The result is a geometric series which is convergent outside a disk of finite radius, and it agrees within its region of convergence with the known de Donder gauge metric of a Schwarzschild black hole. It thus provides a first all-order perturbative computation in Einstein gravity with a matter source, and this series converges to the known nonperturbative expression in the expected range of convergence.

Massive propagating modes of torsion

The dynamics of the torsion field is analyzed in the framework of the Covariant Canonical Gauge Theory of Gravity (CCGG), a De Donder–Weyl Hamiltonian formulation of gauge gravity. The action is quadratic in both, the torsion and the Riemann–Cartan tensor. Since the latter adds the derivative of torsion to the equations of motion, torsion is no longer identical to spin density, as in the Einstein–Cartan theory, but an additional propagating degree of freedom. As torsion turns out to be totally anti-symmetric, it can be parametrised via a single axial vector. It is shown in this paper that, in the weak torsion limit, the axial vector obeys a wave equation with an effective mass term which is partially dependent on the scalar curvature. The source of torsion is thereby given by the fermion axial current which is the net fermionic spin density of the system. Possible measurable effects and approaches to experimental analysis are addressed. For example, neutron star mergers could act as a dipoles or quadrupoles for torsional radiation, and an analysis of radiation of pulsars could lead to a detection of torsion wave background radiation.

Connecting De Donder’s equation with the differential changes of thermodynamic potentials: understanding thermodynamic potentials

Connecting De Donder’s equation with the differential changes of thermodynamic potentials: understanding thermodynamic potentials

Caring neighbourhood and loneliness: Importance of applying a place-based perspective.

Background: Loneliness is recognized as an important societal challenge, with high policy attention the last years. In Flanders an important policy objective is to counteract loneliness by investing in so-called caring neighbourhoods. Although some research indicates that physical and social neighbourhood characteristics (e.g. public parks, public transportation) affect loneliness, little is known in this respect. 
 Aim: The goal of this workshop is to gain more insight into the relation between the neighbourhood and loneliness. Further, much loneliness research is limited to specific age groups and does not take into account the different dimensions of loneliness (emotional, social and existential). This distinction is nevertheless crucial because interventions that aim to counteract loneliness must be tailored around the unique needs of the individual and must consider specific dimensions of loneliness. 
 Participants: The workshop is for everyone interested in how neighbourhood characteristics impact loneliness among people of all ages, and on how to deal with this in research and practice. Both people with and without experience in the subject are welcome.
 Design of the workshop:
 1.Inspirational presentations (40’)
 •Insight into loneliness prevalence. Hannelore Stegen conducted a systematic literature review and meta-analysis on the prevalence of loneliness among community-dwelling older people. We will discuss how different measurement instruments and modes of datacollection lead to different results.
 •Presentation of the FWO-SBO project ‘A Lonely Planet?’. In this part, we discuss which neighbourhood characteristics relate to the risk of loneliness. Leen Heylen, Liesbeth De Donder and Jasper De Witte present the first results of the interuniversity research project A Lonely Planet? Addressing LONELIness from a PLAce-based perspective: research on the influence of Neighbourhood and Environment. Goal is to increase our understanding on the role of neighbourhood characteristics in loneliness.
 •Third, SAAMO Limburg & West-Vlaanderen (i.e. Community Building Organisation) add a practitioner perspective. Together we dive into possibilities on how to deal with tackling and preventing loneliness in caring neighbourhoods. Which neighbourhood level practices can prevent or alleviate loneliness? They present their experiences with several practical projects. 
 •After each presentation, space is reserved for questions and reflections of the audience. 
 2.Interactive work-tables. Through interactive “work-tables” with the participants, we aim to formulate specific recommendations on how to prevent and counteract loneliness from a neighbourhood perspective. (35’)
 Table 1: Which neighbourhood characteristics relate to the risk of loneliness among adults?
 Table 2: How do neighbourhood characteristics relate to the risk of loneliness among adults?
 Table 3: In which manner can loneliness be prevented and alleviated on a neighbourhood level?
 3.Closing: plenary ‘sharing’ session (15’)
 A participant of each table shares the main take-away-home messages from their table. 
 End goal: The goal of this workshop is to create awareness among the participants about the impact that neighbourhood characteristics can have on the level of loneliness among people of all ages, as well as on how to deal with it in practice, research and policy. The co-created content of the workshop will be gathered in a report and shared on the website of “A Lonely Planet”.

Person-centred integrated care? Looking beyond contemporary needs, wishes and goals. A workshop on the importance of working from a life course perspective in caring neighbourhoods.

Background: The importance of a life course perspective is often not taken into account when detecting the needs, wishes and goals of older adults. Research and practice to date has tended to focus on the 'now'. However, adapting a life-course perspective in person-centred care and caring neighbourhoods could enrich and deepen our understanding of current needs, wishes and goals.
 Aim: The aim of this workshop is to give insight into the use of a life-course approach in relation to research and practice on person-centred integrated care and caring neighbourhoods. It aims to enthuse participants in using a life-course approach in their future academic and professional work.
 Participants: This workshop is important for a variety of actors: researchers, practitioners, politicians, older adults. Experience with the subject is not mandatory. The presence of a diverse group enriches our discussions and reflections.
 Methods:
 1.Introduction: The presentation will start with some questions toward the participants. The answers will lead to a presentation of 5 minutes on the use of a life course approach in relation to person-centred integrated care and caring neighbourhoods. (10min)
 2.Inspirational presentations: Four examples will be given of how a life-course approach can be used throughout 4 different themes: housing, civic engagement, care and the neighbourhood. Presentations will be interactive. (40min)
 
 Ariane Vanbellinghen studies housing and wellbeing through the life-course of older adults. Housing is an essential part in caring neighbourhoods, especially for older adults, where there is often a certain resistance to remodel their current home or relocate.
 Bas Dikmans focuses on civic engagement of older adults in disadvantaged neighbourhoods, making use of a life course perspective in his research to see how life events facilitate or pose obstacles to civic engagement in later life.
 
 
 
 Mikis Dormaels is a life story writer with the senior care team, part of the Center for Mental Health in Brussels. In 2010, "narrative care" as a treatment method was developed and offered to older people in a vulnerable position.
 Liesbeth De Donder will open the discussion on how to include neighbourhood life stories, as an addition to individual stories, when developing caring neighbourhoods.
 
 3.Round tables: exercises on using a life-course approach (30 min)
 4 workshops will be organised. Participants are free to choose which workshop to attend: 1) housing through the life-course, 2) civic engagement through the life-course, 3) narrative care: life stories in mental care and 4) life stories as a neighbourhood mapping tool. 
 We will interact with the audience trough specific exercises on using a specific method, what the benefits and challenges of the method are, how they can integrate it in their own context. Together with the group input, feedback, and own experiences, among others, knowledge will be co-created.
 Closing: Plenary reflection and summary (10 min)
 In a plenary session, all the main findings that were collected through the workshop will be shared and a representative of each workshop will summarise the key-messages to take home.

On Clifford multivector-valued actions, generalized Dirac equation and quantization of branes

We explore the construction of a generalized Dirac equation via the introduction of the notion of Clifford multivector-valued actions, which was inspired by the work of Kanatchikov [Kanatchikov IV. De Donder-Weyl theory and a hypercomplex extension of quantum mechanics to field theory. Rept Math Phys. 1999;43:157–170; Kanatchikov IV. De Donder-Weyl Hamiltonian formulation and precanonical quantization of Vielbein gravity. J Phys Conf Ser. 2013;442:012041] on the De Donder-Weyl formulation of field theory. Crucial in this construction is the evaluation of the exponentials of multivectors associated with Clifford (hypercomplex) analysis. Exact matrix solutions (instead of spinors) of the generalized Dirac equation in D = 2 , 3 space–time dimensions are found. This formalism can be extended to curved space–time backgrounds. We conclude by proposing a wave-functional equation governing the quantum dynamics of branes living in C-spaces (Clifford spaces), and which is based on the De Donder–Weyl Hamiltonian formulation of field theory.

Celebrating the birth of De Donder’s chemical affinity (1922–2022): from the uncompensated heat to his Ave Maria

Celebrating the birth of De Donder’s chemical affinity (1922–2022): from the uncompensated heat to his Ave Maria

Modelling of surface reactions and diffusion mediated by bulk diffusion.

We develop a continuum framework applicable to solid-state hydrogen storage, cell biology and other scenarios where the diffusion of a single constituent within a bulk region is coupled via adsorption/desorption to reactions and diffusion on the boundary of the region. We formulate content balances for all relevant constituents and develop thermodynamically consistent constitutive equations. The latter encompass two classes of kinetics for adsorption/desorption and chemical reactions-fast and Marcelin-De Donder, and the second class includes mass action kinetics as a special case. We apply the framework to derive a system consisting of the standard diffusion equation in bulk and FitzHugh-Nagumo type surface reaction-diffusion system of equations on the boundary. We also study the linear stability of a homogeneous steady state in a spherical region and establish sufficient conditions for the occurrence of instabilities driven by surface diffusion. These findings are verified through numerical simulations which reveal that instabilities driven by diffusion lead to the emergence of steady-state spatial patterns from random initial conditions and that bulk diffusion can suppress spatial patterns, in which case temporal oscillations can ensue. We include an extension of our framework that accounts for mechanochemical coupling when the bulk region is occupied by a deformable solid. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'.

Multipole expansion of gravitational waves: memory effects and Bondi aspects

In our previous work, we proposed an algorithm to transform the metric of an isolated matter source in the multipolar post-Minkowskian approximation in harmonic (de Donder) gauge to the Newman-Unti gauge. We then applied this algorithm at linear order and for specific quadratic interactions known as quadratic tail terms. In the present work, we extend this analysis to quadratic interactions associated with the coupling of two mass quadrupole moments, including both instantaneous and hereditary terms. Our main result is the derivation of the metric in Newman-Unti and Bondi gauges with complete quadrupole-quadrupole interactions. We rederive the displacement memory effect and provide expressions for all Bondi aspects and dressed Bondi aspects relevant to the study of leading and subleading memory effects. Then we obtain the Newman-Penrose charges, the BMS charges as well as the second and third order celestial charges defined from the known second order and novel third order dressed Bondi aspects for mass monopole-quadrupole and quadrupole-quadrupole interactions.

Torsion driving cosmic expansion

We study a cosmological model based on the canonical Hamiltonian transformation theory. Using a linear-quadratic approach for the free gravitational De Donder–Weyl Hamiltonian H_textrm{Gr}, the model contains terms describing a deformation of an AdS spacetime and a fully anti-symmetric torsion in addition to Einstein’s theory. The resulting extension of the Einstein–Cartan theory depends on two initially unknown constants, Omega _textrm{g} and Omega _textrm{s}. Given an appropriate choice of these parameters resulting from the analysis of asymptotics, numerical calculations were performed with Omega _mathrm {Lambda } = 0. Values from the Planck Collaboration (Aghanim in Cosmol Parameters 641, 6, 2020) were used for all other required cosmological parameters. In this way, it is shown that torsion can explain phenomena commonly attributed to dark energy, and thus can replace Einstein’s cosmological constant.

Reaction extent or advancement of reaction: A definition for complex chemical reactions.

The concept of reaction extent (the progress of a reaction, advancement of the reaction, conversion, etc.) was introduced around 100 years ago. Most of the literature provides a definition for the exceptional case of a single reaction step or gives an implicit definition that cannot be made explicit. There are views that the reaction extent somehow has to tend to 1 when the reaction goes to completion as time tends to infinity. However, there is no agreement on which function should tend to 1. Starting from the standard definition by IUPAC and following the classical works by De Donder, Aris, and Croce, we extend the definition of the reaction extent for an arbitrary number of species and reaction steps. The new general, explicit definition is also valid for non-mass action kinetics. We also studied the mathematical properties (evolution equation, continuity, monotony, differentiability, etc.) of the defined quantity, connecting them to the formalism of modern reaction kinetics. Our approach tries to adhere to the customs of chemists and be mathematically correct simultaneously. To make the exposition easy to understand, we use simple chemical examples and many figures, throughout. We also show how to apply this concept to exotic reactions: reactions with more than one stationary state, oscillatory reactions, and reactions showing chaotic behavior. The main advantage of the new definition of reaction extent is that by knowing the kinetic model of a reacting system one can now calculate not only the time evolution of the concentration of each reacting species but also the number of occurrences of the individual reaction events.

Concepts Relevant for the Kinetic Analysis of Reversible Reaction Systems.

The net rate of a reversible chemical reaction is the difference between unidirectional rates of traversal along forward and reverse reaction paths. In a multistep reaction sequence, the forward and reverse trajectories, in general, are not the microscopic reverse of one another; rather, each unidirectional route is comprised of distinct rate-controlling steps, intermediates, and transition states. Consequently, traditional descriptors of rate (e.g., reaction orders) do not reflect intrinsic kinetic information but instead conflate unidirectional contributions determined by (i) the microscopic occurrence of forward/reverse reactions (i.e., unidirectional kinetics) and (ii) the reversibility of reaction (i.e., nonequilibrium thermodynamics). This review aims to provide a comprehensive resource of analytical and conceptual tools which deconvolute the contributions of reaction kinetics and thermodynamics to disambiguate unidirectional reaction trajectories and precisely identify rate- and reversibility-controlling molecular species and steps in reversible reaction systems. The extrication of mechanistic and kinetic information from bidirectional reactions is accomplished through equation-based formalisms (e.g., De Donder relations) grounded in principles of thermodynamics and interpreted in the context of theories of chemical kinetics developed in the past 25 years. The aggregate of mathematical formalisms detailed herein is general to thermochemical and electrochemical reactions and encapsulates a diverse body of scientific literature encompassing chemical physics, thermodynamics, chemical kinetics, catalysis, and kinetic modeling.

Multisymplectic constraint analysis of scalar field theories, Chern-Simons gravity, and bosonic string theory

The (pre)multisymplectic geometry of the De Donder–Weyl formalism for field theories is further developed for a variety of field theories including a scalar field theory from the canonical Klein-Gordon action, the electric and magnetic Carrollian scalar field theories, bosonic string theory from the Nambu-Goto action, and 2+1 gravity as a Chern-Simons theory. The Lagrangians for the scalar field theories and for 2+1 Chern-Simons gravity are found to be singular in the De Donder–Weyl sense while the Nambu-Goto Lagrangian is found to be regular. Furthermore, the constraint structure of the premultisymplectic phase spaces of singular field theories is explained and applied to these theories. Finally, it is studied how symmetries are developed on the (pre)multisymplectic phase spaces in the presence of constraints.

On the cosmological constant in the deformed Einstein–Cartan gauge gravity in De Donder–Weyl Hamiltonian formulation

AbstractA modification of the Einstein–Hilbert theory, the Covariant Canonical Gauge Gravity (CCGG), leads to a cosmological constant that represents the energy of the space–time continuum when deformed from its (A)dS ground state to a flat geometry. CCGG is based on the canonical transformation theory in the De Donder–Weyl (DW) Hamiltonian formulation. That framework modifies the Einstein–Hilbert Lagrangian of the free gravitational field by a quadratic Riemann–Cartan concomitant. The theory predicts a total energy‐momentum of the system of space–time and matter to vanish, in line with the conjecture of a “Zero‐Energy‐Universe” going back to Lorentz (1916) and Levi‐Civita (1917). Consequently, a flat geometry can only exist in presence of matter where the bulk vacuum energy of matter, regardless of its value, is eliminated by the vacuum energy of space–time. The observed cosmological constant Λobs is found to be merely a small correction attributable to deviations from a flat geometry and effects of complex dynamical geometry of space–time, namely torsion and possibly also vacuum fluctuations. That quadratic extension of General Relativity, anticipated already in 1918 by Einstein, thus provides a significant and natural contribution to resolving the “cosmological constant problem”.

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.

Metric reconstruction from celestial multipoles

The most general vacuum solution to Einstein’s field equations with no incoming radiation can be constructed perturbatively from two infinite sets of canonical multipole moments, which are found to be mapped into each other under gravitational electric-magnetic duality at the non-linear level. We demonstrate that in non-radiative regions such spacetimes are completely characterized by a set of conserved celestial charges that consist of the Geroch-Hansen multipole moments, the generalized BMS charges and additional celestial multipoles accounting for subleading memory effects. Transitions among non-radiative regions, induced by radiative processes, are therefore labelled by celestial charges, which are identified in terms of canonical multipole moments of the linearized gravitational field. The dictionary between celestial charges and canonical multipole moments allows to holographically reconstruct the metric in de Donder, Newman-Unti or Bondi gauge outside of sources.

Higher-spin gravity’s “string”: new gauge and proof of holographic duality for the linearized Didenko-Vasiliev solution

We consider type-A higher-spin gravity in AdS4, holographically dual to a free U(N ) vector model on the boundary. We study the linearized version of the Didenko-Vasiliev “BPS black hole”, which we view as this theory’s equivalent of the fundamental string. The Didenko-Vasiliev solution consists of gauge fields of all spins generated by a particle-like source along a bulk geodesic, and is holographically dual to a bilocal boundary operator at the geodesic’s endpoints. Our first main result is a new gauge for this solution, which makes manifest its behavior under the boundary field equation. It can be viewed as an AdS uplift of flat spacetime’s de Donder gauge, but is not de Donder in AdS. To our knowledge, this gauge is novel even in the spin-2 sector, and thus provides a new expression for the linearized gravitational field of a massive point particle in (A)dS4. Our second main result is a proof of the holographic duality between the mutual bulk action of two Didenko-Vasiliev solutions and the CFT correlator of two boundary bilocals. As an intermediate step, we show that in a bilocal→local limit, the Didenko-Vasiliev solution reproduces the standard boundary-bulk propagators of all spins. We work in the “metric-like” language of Fronsdal fields, and use the embedding-space formalism.

The Reissner-Nordström-Tangherlini solution from scattering amplitudes of charged scalars

The metric and the electromagnetic potential generated by a static, spherically symmetric charged massive object in any dimension are given by the Reissner-Nordström-Tangherlini solution. We derive the expansion of this solution up to third post-Minkowskian order by computing the classical contribution of scattering amplitudes describing the emission of either a graviton or a photon from a massive charged scalar field up to two loops. In four and five dimensions these amplitudes develop ultraviolet divergences that are cancelled by higher-derivative counterterms in a way that generalises what was recently shown to happen in the chargeless case. This renormalisation procedure produces logarithmic terms that match exactly those produced in the post-Minkowskian expansion of the classical solution in de Donder gauge.