Diffeomorphism Transformations Research Articles

Stabilisation of four types of underactuated systems: a FAS approach

ABSTRACT In this paper, four general types of time-varying underactuated systems (UASs) are proposed and investigated with the fully-actuated system (FAS) approach. The proposed UASs consist of both the first-order and the second-order ones, and cover a very wide range of UASs. FAS models for all these types of UASs are obtained under very mild assumptions, and corresponding homeomorphism or diffeomorphism transformations and sets of feasibility are also established. Governed by the FAS theory, once a FAS model of a system is obtained, a stabilising controller for the system can then be easily designed. Particularly, in the case that the set of feasibility coincides with the whole initial value space, a global FAS model is produced for which (and also the original system) global exponential stabilisation is immediately achieved. Otherwise, a sub-FAS with a feasible set is produced for which a sub-stabilising controller can be immediately designed. Meanwhile, a subset in the system initial value space, which is called the region of exponential attraction (REA), is characterised, such that all the solutions of converted sub-FAS starting from this domain are driven exponentially to the origin within the feasible set. Furthermore, in the case that the set of feasibility, or the REA, contains the origin as only a boundary point, the system possesses a ‘nonholonomic’ feature. Several examples are proposed to demonstrate the proposed approach and the power of the FAS approach.

Well-Posedness of the Ambient Metric Equations and Stability of Even Dimensional Asymptotically de Sitter Spacetimes

The vanishing of the Fefferman–Graham obstruction tensor was used by Anderson and Chruściel to show stability of the asymptotically de Sitter spaces in even dimensions. However, the existing proofs of hyperbolicity of this equation contain gaps. We show in this paper that it is indeed a well-posed hyperbolic system with unique up to diffeomorphism and conformal transformations smooth development for smooth Cauchy data. Our method applies also to equations defined by various versions of the Graham–Jenne–Mason–Sparling operators. In particular, we use one of these operators to propagate Gover’s condition of being almost Einstein (basically conformal to Einsteinian metric). This allows us to study initial data also for Cauchy surfaces which cross the conformal boundary. As a by-product we show that on globally hyperbolic manifolds one can always choose a conformal factor such that Branson Q-curvature vanishes.

Magnetic charges in supergravity

In this paper we study the dual charges of mathcal{N} = 1 supergravity in asymptotically flat space-time. The action considered is the usual supergravity action with a topological contribution. This is the Nieh-Yan term and the magnetic term of the free Rarita-Schwinger field. Through methods of the covariant phase space formalism we construct the charges conjugate to supersymmetry, diffeomorphism and Lorentz transformations. The additional term in the action will lead to new, non-vanishing contributions to these charges. The magnetic diffeomorphism charges are equivalent to the ones previously found for gravity, while the dual supersymmetric charges are new and do not appear for the free Rarita-Schwinger field. The dual Lorentz charges serve to regularize the previous two. We find that the asymptotic symmetry group for supergravity can only include globally well-defined super-rotations.

The theory of symmetric tensor field: From fractons to gravitons and back

We consider the theory of a symmetric tensor field in 4D, invariant under a subclass of infinitesimal diffeomorphism transformations, where the vector diff parameter is the 4-divergence of a scalar parameter. The resulting gauge symmetry characterizes the “fracton” quasiparticles and identifies a theory which depends on a dimensionless parameter, which cannot be reabsorbed by a redefinition of the tensor field, despite the fact that the theory is free of interactions. This kind of “electromagnetic gauge symmetry” is weaker that the original diffeomorphism invariance, in the sense that the most general action contains, but is not limited to, linearized gravity, and we show how it is possible to switch continuously from linearized gravity to a mixed phase where both gravitons and fractons are present, without changing the degrees of freedom of the theory. The gauge fixing procedure is particularly rich and rather peculiar, and leads to the computation of propagators which in the massive case we ask to be tachyonic-free, thus constraining the domain of the parameter of the theory. Finally, a closer contact to fractons is made by the introduction of a parameter related to the “rate of propagation”. For a particular value of this parameter the theory does not propagate at all, and we guess that, for this reason, the resulting theory should be tightly related to the fracton excitations.

Feasibility of a longitudinal statistical atlas model to study aortic growth in congenital heart disease

Studying anatomical shape progression over time is of utmost importance to refine our understanding of clinically relevant processes. These include vascular remodeling, such as aortic dilation, which is particularly important in some congenital heart defects (CHD). A novel methodological framework for three-dimensional shape analysis has been applied for the first time in a CHD scenario, i.e., bicuspid aortic valve (BAV) disease, the most common CHD. Three-dimensional aortic shapes (n = 94) reconstructed from cardiovascular magnetic resonance imaging (MRI) data as surface meshes represented the input for a longitudinal atlas model, using multiple scans over time (n = 2–4 per patient). This model relies on diffeomorphism transformations in the absence of point-to-point correspondence, and on the right combination of initialization, estimation and registration parameters. We computed the shape trajectory of an average disease progression in our cohort, as well as time-dependent parameters, geometric variations and the average shape of the population. Results cover a spatiotemporal spectrum of visual and numerical information that can be further used to run clinical associations. This proof-of-concept study demonstrates the feasibility of applying advanced statistical shape models to track disease progression and stratify patients with CHD.

Gauge \xd7 gauge = gravity on homogeneous spaces using tensor convolutions

A definition of a convolution of tensor fields on group manifolds is given, which is then generalised to generic homogeneous spaces. This is applied to the product of gauge fields in the context of ‘gravity = gauge × gauge’. In particular, it is shown that the linear Becchi-Rouet-Stora-Tyutin (BRST) gauge transformations of two Yang-Mills gauge fields generate the linear BRST diffeomorphism transformations of the graviton. This facilitates the definition of the ‘gauge × gauge’ convolution product on, for example, the static Einstein universe, and more generally for ultrastatic spacetimes with compact spatial slices.

On actions for (entangling) surfaces and DCFTs

The dynamics of surfaces and interfaces describe many physical systems, including fluid membranes, entanglement entropy and the coupling of defects to quantum field theories. Based on the formulation of submanifold calculus developed by Carter, we introduce a new variational principle for (entangling) surfaces. This principle captures all diffeomorphism constraints on surface/interface actions and their associated spacetime stress tensor. The different couplings to the geometric tensors appearing in the surface action are interpreted in terms of response coefficients within elasticity theory. An example of a surface action with edges at the two-derivative level is studied, including both the parity-even and parity-odd sectors. Its conformally invariant counterpart restricts the type of conformal anomalies that can appear in two-dimensional submanifolds with boundaries. Analogously to hydrodynamics, it is shown that classification methods can be used to constrain the stress tensor of (entangling) surfaces at a given order in derivatives. This analysis reveals a purely geometric parity-odd contribution to the Young modulus of a thin elastic membrane. Extending this novel variational principle to BCFTs and DCFTs in curved spacetimes allows to obtain the Ward identities for diffeomorphism and Weyl transformations. In this context, we provide a formal derivation of the contact terms in the stress tensor and of the displacement operator for a broad class of actions.

A general theory of linear cosmological perturbations: bimetric theories

We implement the method developed in [1] to construct the most general parametrised action for linear cosmological perturbations of bimetric theories of gravity. Specifically, we consider perturbations around a homogeneous and isotropic background, and identify the complete form of the action invariant under diffeomorphism transformations, as well as the number of free parameters characterising this cosmological class of theories. We discuss, in detail, the case without derivative interactions, and compare our results with those found in massive bigravity.

Effective quantum gravity observables and locally covariant QFT

Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to construct models of quantum field theories (QFTs) on a general class of Lorentzian manifolds. Recently, this idea has been applied also to perturbative quantum gravity (QG), treated as an effective theory. The difficulty was to find the right notion of observables that would in an appropriate sense be diffeomorphism invariant. In this paper, I will outline a general framework that allows to quantize theories with local symmetries (this includes infinitesimal diffeomorphism transformations) with the use of the Batalin–Vilkovisky (BV) formalism. This approach has been successfully applied to effective QG in a recent paper by Brunetti, Fredenhagen and myself. In the same paper, we also proved perturbative background independence of the quantized theory, which is going to be discussed in the present work as well.

A conformally invariant theory of gravitation in metric measure space

In this manuscript, a conformally invariant theory of gravitation in the context of metric measure space is studied. The proposed action is invariant under both diffeomorphism and conformal transformations. Using the variational method, a generalization of the Einstein equation is obtained, wherein the conventional tensors are replaced by their conformally invariant counterparts, living in metric measure space. The invariance of the geometrical part of the action under a diffeomorphism leads to a generalized contracted second Bianchi identity. In metric measure space, the covariant derivative is the same as it is in the Riemannian space. Hence, in contrast to the Weyl space, the metricity and integrability are maintained. However, it is worth noting that in metric measure space the divergence of a tensor is not simply the contraction of the covariant derivative operator with the tensor that it acts on. Despite the fact that metric measure space and integrable Weyl space, are constructed based on different assumptions, it is shown that some relations in these spaces, such as the contracted second Bianchi identity, are completely similar.

Exact Linearization Algorithm of Tight Coupling Nonlinear System Based on Differential Manifold

Dealt with the short precision the traditional Taylor series expansion induced and shortage that the system mechanics is hard to match with the exact linearization conditions, a novel exact linearization algorithm of tight coupling nonlinear system based on differential manifold to the missile attitude control system is proposed. A dimension expansion method is proposed, the method solves the problem that the input and output dimensions can not meet the exact linearization conditions; the algorithm application range is widened. Using the principle of the differential manifold, the missile velocity and height information are selected as the measure output, the exact linearization of the missile attitude system is derived based on the diffeomorphism transformations. The simulations are performed on the missile attitude control system. Simulation results show that the effectiveness of the algorithm proposed.

D-branes in generalized geometry and Dirac-Born-Infeld action

The purpose of this paper is to formulate the Dirac-Born-Infeld (DBI) action in a framework of generalized geometry and clarify its symmetry. A D-brane is defined as a Dirac structure where scalar fields and gauge field are treated on an equal footing in a static gauge. We derive generalized Lie derivatives corresponding to the diffeomorphism and B-field gauge transformations and show that the DBI action is invariant under non-linearly realized symmetries for all types of diffeomorphisms and B-field gauge transformations. Consequently, we can interpret not only the scalar field but also the gauge field on the D-brane as the generalized Nambu-Goldstone boson.

Remarks on the “Non-canonicity Puzzle”: Lagrangian Symmetries of the Einstein-Hilbert Action

Given the non-canonical relationship between variables used in the Hamiltonian formulations of the Einstein-Hilbert action (due to Pirani, Schild, Skinner (PSS) and Dirac) and the Arnowitt-Deser-Misner (ADM) action, and the consequent difference in the gauge transformations generated by the first-class constraints of these two formulations, the assumption that the Lagrangians from which they were derived are equivalent leads to an apparent contradiction that has been called “the non-canonicity puzzle”. In this work we shall investigate the group properties of two symmetries derived for the Einstein-Hilbert action: diffeomorphism, which follows from the PSS and Dirac formulations, and the one that arises from the ADM formulation. We demonstrate that unlike the diffeomorphism transformations, the ADM transformations (as well as others, which can be constructed for the Einstein-Hilbert Lagrangian using Noether’s identities) do not form a group. This makes diffeomorphism transformations unique (the term “canonical” symmetry might be suggested). If the two Lagrangians are to be called equivalent, canonical symmetry must be preserved. The interplay between general covariance and the canonicity of the variables used is discussed.

A NOTE ON THE IMPLEMENTATION OF POINCARÉ SYMMETRY IN NONCOMMUTATIVE FIELD THEORY

We argue that Poincaré symmetry can be implemented in noncommutative field theory (NCFT) if we allow the parameter of noncommutative deformation θμν to change as a two-tensor under the corresponding space–time symmetry. The implementation is consistent with the definition of θμν in terms of space–time coordinates and with the Moyal star product. Inspired by the standard definition of a variational symmetry we found a universal way to correct the implementation of the Poincaré symmetry by a term proportional to the variation of θμν in such a way that the new transformation define a symmetry of the theory. Finally we present as an example the case of NCYM theory and comment about the obstructions to implement generalized space–time symmetries in NCFT-like conformal or diffeomorphism transformations.

GRAVITATIONAL OBSERVABLES, INTRINSIC COORDINATES, AND CANONICAL MAPS

It is well known that in a generally covariant gravitational theory the choice of spacetime scalars as coordinates yields phase-space observables (or "invariants"). However, their relation to the symmetry group of diffeomorphism transformations has remained obscure. In a symmetry-inspired approach we construct invariants out of canonically induced active gauge transformations. These invariants may be interpreted as the full set of dynamical variables evaluated in the intrinsic coordinate system. The functional invariants can explicitly be written as a Taylor expansion in the coordinates of any observer, and the coefficients have a physical and geometrical interpretation. Surprisingly, all invariants can be obtained as limits of a family of canonical transformations. This permits a short (again geometric) proof that all invariants, including the lapse and shift, satisfy Poisson brackets that are equal to the invariants of their corresponding Dirac brackets.

Consistent gravitational anomalies for chiral bosons

Exact consistent gravitational anomalies for chiral bosons in two dimensions are treated both with the Schwinger-DeWitt regularization and independently through a cohomological procedure. The diffeomorphism transformations are described by a single ghost which allows one to climb the cohomological chain in a unique way.

Diffeomorphism Symmetry in the Lagrangian Formulation of Gravity

Starting from a knowledge of certain identities in the Lagrangian description, the diffeomorphism transformations of metric and connection are obtained for both the second order (metric) and the first order (Palatini) formulations of gravity. The transformation laws of the connection and the metric are derived independently in the Palatini formulation in contrast to the metric formulation where the gauge variation of the connection is deduced from the gauge variation of the metric.

Can gravitational dynamics be obtained by diffeomorphism invariance of action?

It has recently been suggested that the gravitational dynamics could be obtained by requiring the action to be invariant under diffeomorphism transformations. We argue that the action constructed in a usual way is automatically diffeomorphism invariant in nature, which thus invalidates this alternative perspective to obtain gravitational dynamics. Especially, we also show what is wrong with the technical derivation of gravitational dynamics in the alternative approach.

Renormalization group and infinite algebraic structure inD-dimensional conformal field theory

We consider scalar field theory in the D-dimensional space with nontrivial metric and local action functional of most general form. It is possible to construct for this model a generalization of renormalization procedure and RG-equations. In the fixed point the diffeomorphism and Weyl transformations generate an infinite algebraic structure of D-Dimensional conformal field theory models. The Wilson expansion and crossing symmetry enable to obtain sum rules for dimensions of composite operators and Wilson coefficients.

Observer Linearization by Output Diffeomorphism and Output-Dependent Time-Scale Transformations

In this paper, we study the problem of observer linearization for single output dynamical systems in the presence of output-dependent time scaling changes. An alternative algorithm for the solution of the observer linearization problem is introduced. The algorithm employs an exterior calculus approach, treating the system dynamics as collection of one-forms, i.e., Pfaffian systems. The main advantage of the algorithm is that it provides a simple procedure for the solution of the observer linearization problem under output diffeomorphisms and output dependent time-scale transformations.