Canonical Formalism Research Articles

On the breakdown of space–time in general relativity

Based on the canonical quantization of d > 2-dimensional general relativity (GR) via the Dirac constraint formalism (also termed as ‘constraint quantization’), we propose the loss of covariance as a fundamental property of the theory. This breakdown occurs for the first-order Einstein–Hilbert action, whereby the loss of diffeomorphism invariance, besides first-class constraints, second-class constraints also exist leading to non-standard ghost fields that render the path integral non-covariant. We also attempt, for the first time, the canonical quantization via calculation of the path integral for the equivalent Hamiltonian formulation of GR, for which only first-class constraints exist. However, loss of covariance still occurs in this action due to loss of diffeomorphism invariance and structures arising from non-covariant constraints in the path integral. In contrast, we find that covariance as a symmetry is restored and quantization with perturbative calculations is possible in the weak limit of the gravitational field of these actions. Hence, we first establish, for the first time, that the breakdown in space–time is a property of GR itself as its limitation, indicating it to be an effective field theory (EFT). We further propose that the breakdown of space–time occurs as a non-perturbative feature of GR in the strong field limit of the theory. Besides GR, we also note that covariance is preserved when constraint quantization is conducted for non-Abelian gauge theories, such as the Yang–Mills theory. These findings are novel from a canonical gravity formalism and EFT approach, and are consistent with GR singularity theorems, yet are general and extend them, as the singularity theorems indicate breakdown at a strong field limit of GR in black holes. Our findings are in contrast to the asymptotic safety program. They support emergent theories of space–time and gravity, though are unique, as they do not require thermodynamics such as the entropic gravity program. From an EFT view, these indicate that new degrees of freedom and(or) principles in the non-perturbative sector of the full theory are a requirement, whereby covariance as a symmetry is broken in the high-energy (strong field) sector of GR.

Einstein and Jordan-frame covariant Hamiltonians for F(R) gravity and their canonical relationships

This paper is devoted to the analysis of the covariant canonical formalism of F(R) gravity in Einstein frame. We also find canonical transformation between covariant canonical formulation of F(R) gravity in Jordan frame and Einstein frames and we also determine corresponding generating function.

Gravitational-wave energy and other fluxes in ghost-free bigravity

One of the key ingredients for making binary waveform predictions in a beyond-general-relativity (GR) theory of gravity is understanding the energy and angular momentum carried by gravitational waves and any other radiated fields. Identifying the appropriate energy functional is unclear in Hassan-Rosen bigravity, a ghost-free theory with one massive and one massless graviton. The difficulty arises from the new degrees of freedom and length scales which are not present in GR, rendering an Isaacson-style averaging calculation ambiguous. In this article we compute the energy carried by gravitational waves in bigravity starting from the action, using the canonical current formalism. The canonical current agrees with other common energy calculations in GR, and is unambiguous (modulo boundary terms), making it a convenient choice for quantifying the energy of gravitational waves in bigravity or any diffeomorphism-invariant theories of gravity. This calculation opens the door for future waveform modeling in bigravity to correctly include backreaction due to emission of gravitational waves.

State and covariance estimation of a semi-batch reactor for bioprocess applications

In this work, extended Kalman filter (EKF) and moving horizon estimation (MHE)-based approaches are introduced and applied to a Galacto-oligosaccharides (GOS) bioprocess system considering a recently developed enzymatic model. Using this model, plant data is simulated by varying the kinetic parameters and applying white noise to the resulting output. In terms of EKF applications, both the canonical and parametric (Dual EKF) formulations have their covariances specified using a modified direct optimization (DO) algorithm to reduce estimation error. This work also outlines the development and application of a novel Parameter-based Moving Horizon Estimation (P-MHE) approach and directly compares it to traditional MHE formulations. The proposed P-MHE method reduces both the estimation error and computational time of traditional MHE approaches. When compared to EKF-based approaches, P-MHE can compete in terms of estimation error while exhibiting excellent robustness characteristics such as guaranteed feasibility. Despite the increased computational time of P-MHE when compared to the EKF formulations, state estimates can be obtained in under 3 seconds once a measurement arrives, allowing this algorithm to be applied to real-time process monitoring and other process systems.

PH- and Facet-Dependent Surface Chemistry of TiO2 in Aqueous Environment from First Principles.

TiO2 is a relevant catalytic material, and its chemistry in aqueous environment is a challenging aspect to address. Also, the morphology of TiO2 particles at the nanoscale is often complex, spanning from faceted to spherical. In this work, we study the pH- and facet-dependent surface chemistry of TiO2/water interfaces by performing ab initio molecular dynamics simulations with the grand canonical formulation of species in solution. We first determined the acid-base equilibrium constants at the interface, which allows us to estimate the pH at the point of zero charge, an important experimental observable. Then, based on simulated equilibrium constants, we predict the amount of H+, OH-, and adsorbed H2O species present on the surfaces as a function of the pH, a relevant aspect for water splitting semi-reactions. We approximated the complex morphology of TiO2 particles by considering the rutile (110) and (011), and anatase (101), (001), and (100) surfaces.

Introduction: Assimilation, integration or transnationalism? An overview of theories of migrant incorporation

Introduction: Assimilation, integration or transnationalism? An overview of theories of migrant incorporation

Canonical Formulation of Embedding Gravity in a Form of General Relativity with Dark Matter

Canonical Formulation of Embedding Gravity in a Form of General Relativity with Dark Matter

Black hole and de Sitter microstructures from a semiclassical perspective

We describe two different, but equivalent semiclassical views of black hole physics in which the equivalence principle and unitarity are both accommodated. In one, unitarity is built-in, while the black hole interior emerges only effectively as a collective phenomenon involving horizon (and possibly other) degrees of freedom. In the other, more widely studied approach, the existence of the interior is manifest, while the unitarity of the underlying dynamics can be captured only indirectly by incorporating certain nonperturbative effects of gravity. These two pictures correspond to a distant description and the description based on entanglement islands/replica wormholes, respectively. We also present a holographic description of de Sitter spacetime based on the former approach, in which the holographic theory is located on the stretched horizon of a static patch. We argue that the existence of these two approaches is rooted in the two formulations of quantum mechanics: the canonical and path integral formalisms.

Perusing Buchbinder–Lyakhovich Canonical Formalism for Higher-Order Theories of Gravity

Ostrogradsky’s, Dirac’s, and Horowitz’s techniques in terms of higher-order theories of gravity produce identical phase-space structures. The problem with these techniques is manifested in the case of Gauss–Bonnet–dilatonic coupled action in the presence of higher-order term, in which case, classical correspondence cannot be established. Here, we explore another technique developed by Buchbinder and their collaborators (BL) and show that it also suffers from the same disease. However, when expressing the action in terms of the three-space curvature, and removing ‘the total derivative terms’, if Horowitz’s formalism or even Dirac’s constraint analysis is pursued, all pathologies disappear. Here, we show that the same is true for BL formalism, which appears to be the simplest of all the techniques to handle.

A note on the canonical formalism for gravity

A note on the canonical formalism for gravity

Принцип максимума Понтрягина и непрямой метод спуска в задаче оптимального импульсного управления нелокальным уравнением переноса

We study a singular problem of optimal control of a nonlocal transport equation in the space of probability measures, in which the structure of the drivng vector field with respect to the control variable is somewhat equivalent to the affine one, while the set of controls is norm-unbounded and constrained in the integral sense only. We show that the problem at hand admits an impulse-trajectory relaxation in terms of discontinuous time reparameterization. This relaxation provides a correct statement of the variational problem in the class of control inputs constrained in both pointwise and integral senses. For the relaxed problem, we derive a new form of the Pontryagin’s maximum principle (PMP) with a separate adjoint system of linear balance laws on the space of signed measures. In contrast to the canonical formulation of the PMP in terms of the Hamiltonian equation on the cotangent bundle of the state space, our form allows one to formulate an indirect descent method for optimal impulsive control analogous to classical gradient descent. We expose a version of this method, namely, an algorithm of the steepest descent with an internal line search of the Lagrange multiplier associated with the integral bound on control. The algorithm is proven to monotonically converge to a PMP-extremal up to a subsequence.

Weakly isolated horizons: 3 1 decomposition and canonical formulations in self-dual variables

The notion of Isolated Horizons has played an important role in gravitational physics, being useful from the characterization of the endpoint of black hole mergers to (quantum) black hole entropy. In particular, the definition of weakly isolated horizons (WIHs) as quasilocal generalizations of event horizons is purely geometrical, and is independent of the variables used in describing the gravitational field. Here we consider a canonical decomposition of general relativity in terms of connection and vierbein variables starting from a first order action. Within this approach, the information about the existence of a (weakly) isolated horizon is obtained through a set of boundary conditions on an internal boundary of the spacetime region under consideration. We employ, for the self-dual action, a generalization of the Dirac algorithm for regions with boundary. While the formalism for treating gauge theories with boundaries is unambiguous, the choice of dynamical variables on the boundary is not. We explore this freedom and consider different canonical formulations for non-rotating black holes as defined by WIHs. We show that both the notion of horizon degrees of freedom and energy associated to the horizon is not unique, even when the descriptions might be self-consistent. This represents a generalization of previous work on isolated horizons both in the exploration of this freedom and in the type of horizons considered. We comment on previous results found in the literature.

Lee model and its resolvent analysis

We revisit the relativistic (2+1)-dimensional Lee model on flat space in light-front coordinates and on a space-time with a spatial section given by a compact manifold, in the usual canonical formalism. The simpler 2+1 dimension is chosen because renormalization is needed only for the mass difference but not required for the coupling constant and the wave function. The model is constructed non-perturbatively based on the resolvent formulation [B. T. Kaynak and O. T. Turgut, The relativistic Lee model on Riemannian manifolds, J. Phys. A: Math. Theor. 42(22) (2009) 225402]. The bound state spectrum is studied through its “principal operator” and bounds for the ground state energy are obtained. We show that the formal expression found indeed defines the resolvent of a self-adjoint operator–the Hamiltonian of the interacting system. Moreover, we prove an essential result that the principal operator corresponds to a self-adjoint holomorphic family of type-A, in the sense of Kato.

La modernité baudelairienne et ses parages

The Baudelairean modernity and its environs The following text takes up the problem of modernity in its canonical formulation proposed in 1863 by Charles Baudelaire in his important essay entitled Le peintre de la vie moderne. The author of the article tries to show the basic framework of this concept with the definition of modernity itself and the category of beauty, remaining at Baudelaire’s conception in natural connection with it. The author highlights the importance of Baudelaire’s earlier views and the influence that the Paris World Exhibition of 1855 had on him.

Comparison of combustion duration and end-gas autoignition in inwardly and outwardly propagating flames induced by different ignition configurations

Recently, multiple spark ignition has received great attention since it helps to increase thermal efficiency and to reduce misfire in engines. Multiple spark ignition also affects the combustion duration and thereby it can be used for knock control. However, previous studies reported opposite trends in terms of how multiple spark ignition affects engine knock. This work aims to assess and interpret the influence of flame propagation direction induced by different ignition configurations on combustion duration and end-gas autoignition/engine knock. Two simplified and idealised ignition configurations are studied theoretically and numerically. One is with infinite number of sparks at side circular wall, which induces an inwardly propagating flame (IPF); and the other is with a single central spark, which induces an outwardly propagating flame (OPF). In the asymptotic theoretical analysis, the canonical 1D formulations for IPF and OPF are reduced to 0D model. Based on the 0D model, OPF and IPF at different initial temperatures are studied and compared. Counterintuitively, it is found that the combustion duration of OPF is shorter than that of IPF when there is no end-gas autoignition. On the other hand, the combustion duration of IPF is shorter than that of OPF when end-gas autoignition occurs. Furthermore, end-gas autoignition is found to be more prone to occur in IPF than OPF. These interesting observations are interpreted through assessing the ignition delay time and different components of the absolute flame propagation speed. The theoretical results are validated by transient simulations considering detailed chemistry and transport which are conducted for IPF and OPF in an iso-octane/air mixture at different initial temperatures and pressures. Both theoretical and numerical results suggest that compared to infinite number of ignition sparks at side wall, the single central ignition has the advantages in shortening the combustion duration and reducing the tendency of end-gas autoignition.

Van der waals equation of state

Van der Waals (vdW) invented his equation of state about 150 year ago. This equation described the liquidgas phase transition and predicted an existence of the critical point. The vdW equation of state remains today the most familiar and one of the most successful model in molecular physics. It is presented in all text books on statistical physics. However, several important questions concerning the vdW model had no answers. Only very recently these questions were discussed in the literature, and several important extensions of the vdW equation of state were developed. These new results presented in this article correspond to the grand canonical ensemble formulation, the effects of quantum statistics, and applications of the vdW model to nuclear matter taking into account Fermi statistics for interacting protons and neutrons. A special attention is also paid to the particle number fluctuations in a vicinity of the vdW critical point.

Modeling and Finite-Horizon MPC for a Boiler-Turbine System Using Minimal Realization State-Space Model

This paper aims to address a finite-horizon model predictive control (MPC) for non-linear drum-type boiler-turbine system using a system-identification method. Considering that the strong state coupling of a non-linear mechanism model, the subspace identification method is first utilized to obtain a linear state-space model, and transformed into an input–output model. By taking the inputs and outputs of the input–output model as system states, an augmented non-minimal state-space (NMSS) model of state measurable is constructed. In order to reduce the computation burden, the augmented NMSS model is further transformed into a canonical formulation by adopting a Kalman decomposition. Based on the minimal realization state-space model, the MPC controller is parameterized as a finite-horizon optimization problem. Finally, simulations are performed and evaluated the performance of the proposed method, and the simulation results show that: the linear model approximate the non-linear system accurately; the proposed MPC method can achieve a satisfactory stable control performance; and the computation time 18.388 s for the overall optimization problem also illustrates the real-time performance effectively.

The Effect of Magnetic Field on the Thermodynamic Properties of Three-Dimensional GaAs Quantum Dot

The problem of parabolically single three-dimensional semiconductor GaAs quantum dot in the presence of an external magnetic field and including the spin of the electron is studied using canonical formalism. The energy spectrum is obtained as a function of temperature and magnetic field. It is observed that the system tends to absorb the largest amount of heat at critical values of the magnetic field which depends on the temperature. A schottky-like anomaly in heat capacity is observed at low temperature, while at high temperature it converges rapidly and saturates. It was also found that the entropy is inversely proportional to the magnetic field strength, also entropy showed exciting behavior at low temperature, a shoulder is observed which becomes curvier when the magnetic field decreases. However, at high temperature, entropy increases monotonically with temperature and becomes essentially independent of the magnetic field. The substantial effect of magnetic field on the magnetic properties of quantum dot is observed at low temperature. It is shown that, magnetization increases significantly at low temperature and shows strong dependence on magnetic field, while its magnitude increases as the temperature increases and reduces to zero regardless the value of magnetic field strength. In addition, at low temperature susceptibility peaks and is found to be paramagnetic and at high temperature the diamagnetic state is found to be a favored state of the system and susceptibility becomes completely independent of magnetic field. It is also shown that for a fixed value of temperature, susceptibility increases rapidly with magnetic field and beyond a critical value of magnetic field, the increase becomes much slower, susceptibility exhibits a crossing behavior, and magnetization shows a significant drop as the magnetic field increases. So far, no study has been made in literature on the effect of magnetic field on the thermal and magnetic properties of three-dimensional quantum dot.

A pilot study on canonical gravity with mass dimension one fermions

The mass dimension one (MDO) fermionic field is built on a complete set of dual-helicity eigenspinors of the charge conjugation operator, which obeys the statistic of Fermi-Dirac. These spinors are a potential candidates for the description of dark matter. The redefinition of the dual structure of this object holds a local adjacent theory and completely satisfies the Lorentz invariance. In this work, we investigate important aspects of the interaction of this fermion with gravity in the light of a canonical formulation in ADM formalism. We construct an action via tetrad fields using a manifold on a family of spacelike surfaces Σt that carries the MDO matter field, and additionally we propose a condition for the action to have a term associated with dark energy. We found the Hamiltonian and diffeomorphism constraints at the classical level for gravitational dynamics with the immersion of this material content in space-time, which leads us to the interpretation of the contribution of dark matter energy density, parallel to the lapse function of foliation and its directional flux of energy density in the hypersurface of manifold.

Chiral effects in classical spinning gas

We consider a statistical mechanics of rotating ideal gas consisting of classical non-relativistic spinning particles. The microscopic structure elements of the system are massive point particles with a nonzero proper angular momentum. The norm of proper angular momentum is determined by spin. The direction of proper angular momentum changes continuously. Applying the Gibbs canonical formalism for the rotating system, we construct the one-particle distribution function, generalising the usual Maxwell–Boltzmann distribution, and the partition function of the system. The model demonstrates a set of chiral effects caused by interaction of spin and macroscopic rotation, including the change of entropy, heat capacity, chemical potential and angular momentum.