Configuration Space Variables Research Articles

BMS3 mechanics and the black hole interior

The spacetime in the interior of a black hole can be described by an homogeneous line element, for which the Einstein–Hilbert action reduces to a one-dimensional mechanical model. We have shown in Geiller et al (2021 SciPost Phys. 10 022) that this model exhibits a symmetry under the (2 + 1)-dimensional Poincaré group. Here we extend the Poincaré transformations to the infinite-dimensional BMS3 group. Although the black hole model is not invariant under those extended transformations, we can write it as a geometric action for BMS3, where the configuration space variables are elements of the algebra and the equations of motion transform as coadjoint vectors. The BMS3 symmetry breaks down to its Poincaré subgroup, which arises as the stabilizer of the vacuum orbit. This symmetry breaking is analogous to what happens with the Schwarzian action in AdS2 JT gravity, although in the present case there is no direct interpretation in terms of boundary symmetries. This observation, together with the fact that other lower-dimensional gravitational models (such as the BTZ black hole) possess the same broken BMS3 symmetries, provides yet another illustration of the ubiquitous role played by this group.

A Semilinear Parameter-Varying Observer Method for Fabric-Reinforced Soft Robots.

This paper presents an observer architecture that can estimate a set of configuration space variables, their rates of change and contact forces of a fabric-reinforced inflatable soft robot. We discretized the continuum robot into a sequence of discs connected by inextensible threads; this allows great flexibility when describing the robot’s behavior. At first, the system dynamics is described by a linear parameter-varying (LPV) model that includes a set of subsystems, each of which corresponds to a particular range of chamber pressure. A real-world challenge we confront is that the physical robot prototype exhibits a hysteresis loop whose directions depend on whether the chamber is inflating or deflating. In this paper we transform the hysteresis model to a semilinear model to avoid backward-in-time definitions, making it suitable for observer and controller design. The final model describing the soft robot, including the discretized continuum and hysteresis behavior, is called the semilinear parameter-varying (SPV) model. The semilinear parameter-varying observer architecture includes a set of sub-observers corresponding to the subsystems for each chamber pressure range in the SPV model. The proposed observer is evaluated through simulations and experiments. Simulation results show that the observer can estimate the configuration space variables and their rate of change with no steady-state error. In addition, experimental results display fast convergence of generalized contact force estimates and good tracking of the robot’s configuration relative to ground-truth motion capture data.

Analysis of the polarization effects in the gyrokinetic theory of magnetized plasmas.

By adopting the hybrid coordinates, in which the nonlinearity of polarization displacement is included in the configuration space variables compared to the conventional gyrocenter coordinates, the polarization effects are analyzed by using the modern gyrokinetic (GK) theory of magnetized plasmas. Based on the invariant property, the velocity transformation between the gyrocenter and hybrid coordinates is calculated, and the phase-space velocity in terms of the hybrid coordinates is obtained. The linear and nonlinear polarization distribution functions are defined, and the evolutions for the polarization distribution functions are derived. It is well known that the polarization density is important in the GK calculation of particle density. Analogously, it is shown that the polarization current should be considered in the GK calculation of current density. In the case with electrostatic fluctuations, the roles of the polarization current are illustrated in the derivations of the Hasegawa-Mima equation and the dispersion relation for geodesic acoustic mode. In the case with magnetic fluctuations, the procedure for the GK calculation of perpendicular current is clarified, the dispersion relation for compressional Alfvén wave is derived, in which the effect of polarization current is discussed.

Particle dynamics and Lie-algebraic type of non-commutativity of space–time

In this paper, we present the results of our investigation relating particle dynamics and non-commutativity of space–time by using Dirac's constraint analysis. In this study, we re-parameterise the time t=t(τ) along with x=x(τ) and treat both as configuration space variables. Here, τ is a monotonic increasing parameter and the system evolves with this parameter. After constraint analysis, we find the deformed Dirac brackets similar to the κ-deformed space–time and also, get the deformed Hamilton's equations of motion. Moreover, we study the effect of non-commutativity on the generators of Galilean group and Poincare group and find undeformed form of the algebra. Also, we work on the extended space analysis in the Lagrangian formalism. We find the primary as well as the secondary constraints. Strikingly on calculating the Dirac brackets among the phase space variables, we obtain the classical version of κ-Minkowski algebra.

Analytic, nonperturbative, almost exact QED: The two-point functions

Based on the choice of a special gauge, in which a useful form of scaling invariance holds, a new method is suggested for the analytic, nonperturbative calculation of the n-point functions of QED. A modified functional analysis is employed in configuration space, where the dressed electron and photon propagators (in quenched approximation) are each found to be simple products of the relevant free propagator with an appropriate function of configuration space variables containing all powers of the square of the coupling constant.

EXTENDED HAMILTON–LAGRANGE FORMALISM AND ITS APPLICATION TO FEYNMAN'S PATH INTEGRAL FOR RELATIVISTIC QUANTUM PHYSICS

We present a consistent and comprehensive treatise on the foundations of the extended Hamilton–Lagrange formalism — where the dynamical system is parametrized along a general system evolution parameter s, and the time t is treated as a dependent variable t(s) on equal footing with all other configuration space variables qi(s). In the action principle, the conventional classical action L1d t is then replaced by the generalized action L1d s, with L and L1denoting the conventional and the extended Lagrangian, respectively. It is shown that a unique correlation of L1and L exists if we refrain from performing simultaneously a transformation of the dynamical variables. With the appropriate correlation of L1and L in place, the extension of the formalism preserves its canonical form.In the extended formalism, the dynamical system is described as a constrained motion within an extended space. We show that the value of the constraint and the parameter s constitutes an additional pair of canonically conjugate variables. In the corresponding quantum system, we thus encounter an additional uncertainty relation.As a consequence of the formal similarity of conventional and extended Hamilton–Lagrange formalisms, Feynman's nonrelativistic path integral approach can be converted on a general level into a form appropriate for relativistic quantum physics. In the emerging parametrized quantum description, the additional uncertainty relation serves as the means to incorporate the constraint and hence to finally eliminate the parametrization.We derive the extended Lagrangian L1of a classical relativistic point particle in an external electromagnetic field and show that the generalized path integral approach yields the Klein–Gordon equation as the corresponding quantum description. We furthermore derive the space–time propagator for a free relativistic particle from its extended Lagrangian L1. These results can be regarded as the proof of principle of the relativistic generalization of Feynman's path integral approach to quantum physics.

Constraints in spherically symmetric classical general relativity. I. Optical scalars, foliations, bounds on the configuration space variables, and the positivity of the quasilocal mass.

We examine the constraints of spherically symmetric general relativity with one asymptotically flat region, exploiting both the traditional metric variables and variables constructed from the optical scalars. With respect to the latter variables, there exist two linear combinations of the Hamiltonian and momentum constraints which are related by time reversal. We introduce a one-parameter family of linear extrinsic time foliations of spacetime. The values of the parameter yielding globally valid gauges correspond to the vanishing of a timelike vector in the superspace of spherically symmetric geometries. We define a quasi-local mass on spheres of fixed proper radius which we prove is positive when the constraints are satisfied. Underpinning the proof are various local bounds on the configuration variables. We prove that a reasonable definition of the gravitational binding energy is always negative. Finally, we provide a tentative characterization of the configuration space of the theory in terms of closed bounded trajectories on the parameter space of the optical scalars.

Null hypersurfaces and new variables

Following Ashtekar's (1987) recently revised version of the standard canonical theory, the construction of a new variables canonical formalism for Einstein's theory of gravity is investigated when the time parameter has level sets which are null hypersurfaces. The configuration space variables are the components of a tetrad and the self-dual components of a connection. Because a null time parameter is used, the Hamiltonian formalism has second-class constraints as well as the first-class constraints which are associated with the invariance of the theory under diffeomorphisms and local gauge transformations. The first-class constraint algebra is discussed and reality conditions which relate the complex formalism to real general relativity are displayed.

Self-dual Maxwell field on a null cone

As an introduction to the use of self-dual variables on a null cone, the Hamiltonian for the Maxwell field is set up on a null cone in Minkowski space. In this treatment, the vector potential (the connection) and the self-dual components of the Maxwell field are treated as independent configuration space variables. Because the initial surface is a null cone, additional primary and secondary constraints arise. These constraints can be grouped into first class and second class. The elimination of the second class constraints together with the reality conditions on the vector potential, reduce the independent phase space variables to two. In its final form the Hamiltonian can be expressed in terms of the product of the self-dual Maxwell field and its complex conjugate.

Triad approach to the Hamiltonian of general relativity.

The Sparling-Thirring forms are constructed using a connection over a frame bundle defined in terms of a spacelike triad and lapse and shift functions. From the forms, one can identify a first-order Lagrangian from which to construct the Hamiltonian using triad vector densities as basic configuration-space variables. If one uses the self-dual part of the forms, the resulting first-order Lagrangian (its imaginary part is a total divergence) leads to momenta which are the dual of the Sen connection which was introduced by Ashtekar in a spinor representation. The resulting formalism is equivalent to that of Ashtekar.

Nonequilibrium statistical mechanical theory of linear interacting processes

The thermodynamic Lagrangian formulation of nonequilibrium statistical mechanics is extended to include linear interacting processes that are weakly coupled with wide separation of time scales. The kernel, which is the probability for a transition from one nonequilibrium state to another, is an exponential function of the time integral of the sum of dissipation functions (the thermodynamic Lagrangian). The equilibrium time correlation function is the expectation value of this kernel. In the presence of interactions, the kernel contains a thermodynamic Lagrangian of interaction. Provided that there is a wide separation of time scales between the primary process, having the largest relaxation time, and the weakly coupled set of irreversible processes that make up the environment, the configuration space variables of the processes comprising the environment can be integrated out by the correlation function as the expectation value of the influence functional which is evaluated by means of the unperturbed thermodynamic action. Since energy can be temporarily stored in the environment, which is equivalent to a set of oscillators that are independently coupled to the primary process, noninstantaneous interactions occur in which the primary process at one time can affect itself at a later time. Noninstantaneous interactions are the cause of memory effects that are observed in the transport properties of the primary process. The operator and path integral formulations are shown to be equivalent.

On a class representations of the bilocal current algebra on a light-like ray

A class of representation of the algebra if bilocal currents on a plane perpendicular to a light-like ray is constructed. The saturation of these representations with states of definite momenta conjugate to the configuration space variables on the light-like plane reveals a Fock-space nature of the representation space, generated by quark and anti-quark fields obeying Fermi statistics, projected on the characteristics light-like ray. The mass spectrum, however, is not determined by the algebra. Representations on a subspace of the constructed Fock space are discussed for special momentum configurations.