Hamiltonian Formulation Research Articles

Spherical accretion onto higher-dimensional Reissner–Nordström black hole

Abstract We obtain relativistic solutions of spherically symmetric accretion by a dynamical analysis of a generalised Hamiltonian for higher-dimensional Reissner–Nordström (RN) Black Hole (BH). We consider two different fluids namely, an isotropic fluid and a non-linear polytropic fluid to analyse the critical points in a higher-dimensional RN BH. The flow dynamics of the fluids are studied in different spacetime dimensions in the framework of Hamiltonian formalism. The isotropic fluid is found to have both transonic and non-transonic flow behaviour, but in the case of polytropic fluid, the flow behaviour is found to exhibit only non-transonic flow, determined by a critical point that is related to the local sound speed. The critical radius is found to change with the spacetime dimensions. Starting from the usual four dimensions it is noted that as the dimension increases the critical radius decreases, attains a minimum at a specific dimension (D > 4) and thereafter increases again. The mass accretion rate for isotropic fluid is determined using Hamiltonian formalism. The maximum mass accretion rate for RN BH with different equations of state parameters is studied in addition to spacetime dimensions. The flow behaviour and mass accretion rate for a change in BH charge is also studied analytically. It is noted that the maximum mass accretion rate in a higher-dimensional Schwarzschild BH is the lowest, which however, increases with the increase in charge parameter in a higher-dimensional RN BH.

Entanglement in ( 1+1 )-dimensional free scalar field theory: Tiptoeing between continuum and discrete formulations

We review some classic works on ground state entanglement entropy in (1+1)-dimensional free scalar field theory. We point out identifications between the methods for the calculation of entanglement entropy and we show how the formalism developed for the discretized theory can be utilized in order to obtain results in the continuous theory. We specify the entanglement spectrum and we calculate the entanglement entropy for the theory defined on an interval of finite length L. Finally, we derive the modular Hamiltonian directly, without using the modular flow, via the continuum limit of the expressions obtained in the discretized theory. In a specific coordinate system, the modular Hamiltonian assumes the form of a free field Hamiltonian on the Rindler wedge. Published by the American Physical Society 2024

On the Role of Fiducial Structures in Minisuperspace Reduction and Quantum Fluctuations in LQC

Abstract In spatially non-compact homogeneous minisuperpace models, spatial integrals in the Hamiltonian and symplectic form must be regularised by confining them to a finite volume $V_o$, known as the \emph{fiducial cell}. As this restriction is unnecessary in the complete field theory before homogeneous reduction, the physical significance of the fiducial cell has been largely debated, especially in the context of (loop) quantum cosmology. Understanding the role of $V_o$ is in turn essential for assessing the minisuperspace description's validity and its connection to the full theory. In this work we present a systematic procedure for the field theory reduction to spatially homogeneous minisuperspaces within the canonical framework and apply it to a massive scalar field theory and gravity. Our strategy consists in implementing spatial homogeneity via second-class constraints for the discrete field modes over a partitioning of the spatial slice into countably many disjoint cells. The reduced theory's canonical structure is then given by the corresponding Dirac bracket. Importantly, the latter can only be defined on a finite number of cells homogeneously patched together. This identifies a finite region, the fiducial cell, whose physical size acquires then a precise meaning already at the classical level as the scale over which homogenenity is imposed. Additionally, the procedure allows us to track the information lost during homogeneous reduction and how the error depends on $V_o$. We then move to the quantisation of the classically reduced theories, focusing in particular on the relation between the theories for different $V_o$, and study the implications for statistical moments, quantum fluctuations, and semiclassical states. In the case of a quantum scalar field, a subsector of the full quantum field theory where the results from the ``first reduced, then quantised" approach can be reproduced is identified and the conditions for this to be a good approximation are determined.

Higher‐order integrable models for oceanic internal wave–current interactions

AbstractIn this paper, we derive a higher‐order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a “flat‐surface” assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long‐wave, and small‐amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations.

Boundary terms and on-shell action in Ricci-based gravity theories: The Hamiltonian formulation

Boundary terms and on-shell action in Ricci-based gravity theories: The Hamiltonian formulation

Anisotropic odd elasticity with Hamiltonian curl forces

Abstract A host of elastic systems consisting of active components exhibit path-dependent elastic behaviors not found in classical elasticity, which is known as odd elasticity. Odd elasticity is characterized by antisymmetric (odd) elastic modulus tensor. Here, from the perspective of geometry, we construct the Hamiltonian formalism to show the origin of the antisymmetry of the elastic modulus that is intrinsically anisotropic. Furthermore, both non-conservative stress and the associated nonlinear constitutive relation naturally arise. This work also opens the promising possibility of exploring the physics of odd elasticity in dynamical regime by Hamiltonian formalism.

Dimensionless fluctuations balance applied to statistics and quantum physics

This work presents a new method called Dimensionless Fluctuation Balance (DFB), which makes it possible to obtain distributions as solutions of Partial Differential Equations (PDEs). In the first case study, DFB was applied to obtain the Boltzmann PDE, whose solution is a distribution for Boltzmann gas. Following, the Planck photon gas in the Radiation Law, Fermi–Dirac, and Bose–Einstein distributions were also verified as solutions to the Boltzmann PDE. The first case study demonstrates the importance of the Boltzmann PDE and the DFB method, both introduced in this paper. In the second case study, DFB is applied to thermal and entropy energies, naturally resulting in a PDE of Boltzmann’s entropy law. Finally, in the third case study, quantum effects were considered. So, when applying DFB with Heisenberg uncertainty relations, a Schrödinger case PDE for free particles and its solution were obtained. This allows for the determination of operators linked to Hamiltonian formalism, which is one way to obtain the Schrödinger equation. These results suggest a wide range of applications for this methodology, including Statistical Physics, Schrödinger’s Quantum Mechanics, Thin Films, New Materials Modeling, and Theoretical Physics.

A derivation of the Schrödinger equation from Feynman’s path-integral formulation of quantum mechanics

The equation of motion in the standard formulation of non-relativistic quantum mechanics, the Schrödinger equation, is based on the Hamiltonian. In contrast, in Feynman’s path-integral formulation of quantum mechanics, the equation of motion is the propagation equation, which is based on the Lagrangian. That these two different equations of motion are equivalent was shown by Feynman, who provided a derivation of the Schrödinger equation from the propagation equation. Surprisingly, however, while in classical mechanics there exists a simple relationship between the Hamiltonian and the Lagrangian, there is nothing in Feynman’s derivation that gives any indication of this relationship. Here I show that the equations of motion in the Hamiltonian and Lagrangian formulations of quantum mechanics are, in fact, simply related to each other.

Synthesis, general physicochemical behaviour and magnetic studies of the macrocyclic bis-pyridine derivatives [TTDPzM(py)2]·2H2O (M = MnII, CoII, NiII)

In our extensive work on phthalocyanine-like macrocycles carrying peripherally attached strongly electron-withdrawing 1,2,5-thia/selenodiazol rings we recently reported on the FeII species of formula [TTDPzFe(py)2].2H2O (TTDPz = tetrakis(thiadiazole)porphyrazinato dianion). The present work deals with the synthesis and extended physicochemical studies of the series of related macrocycles [TTDPzM(py)2].2H2O (M = MnII, CoII, NiII). The diffractograms of three solid samples of the NiII complex [TTDPzNi(py)2]·2H2O, obtained from different synthetic procedures, indicate that the species is reproducibly obtained. All three samples are isomorphous with each other as shown by their X-ray powder diffractograms. Based on the comparison of the IR spectra of [TTDPzNi(py)2].2H2O and the related desolvated [TTDPzNi], the IR absorption peaks belonging to the pyridine molecules could be identified. The systematic presence of the two water molecules in all three species suggests that forms of contact do exist in the form of hydrogen bonds between them and the S and N atoms present peripherally in the thiadiazole groups of the macrocyclic units. As regard to the elimination of water, the thermograms (in inert atmosphere in the range 25–600 °C) indicate in all cases that loss of water occurs below 100 °C but loss of pyridine occurs over a broad range 100–300 °C for the MnII and CoII species whereas for the NiII complex it takes place in the narrow range 200–220 °C. In relation to the parallel series of metallophthalocyanines, i.e. [PcM] (M = MnII, FeII, CoII), it is noted that the NiII analog has not been reported in the literature and in a direct experiment it has been proved that [PcNi] heated in pyridine is obtained unchanged, the different behavior for the NiII TTDPz derivative due to the electron withdrawing effect of the external thiadiazole substituents, which makes favorable axial pyridine coordination in the complex. Completely insoluble in water, the triad of TTDPz derivatives are extremely low-soluble in non-donor or low donor solvents. This precluded any attempt to isolate single crystals for single-crystal X-ray work. Their UV–visible spectra show essential similarities in whatever organic solvent is used with clearly positioned Soret (300–400 nm) and Q-band absorptions (630–650 nm). Variable temperature magnetic susceptibility and magnetization studies reveal that the [TTDPzM(py)2].2H2O complexes all show paramagnetism with ground state spins of 3/2 (MnII), ½ (CoII) and 1 (NiII), supported by fitting the data to spin Hamiltonian formalism, vide infra. Comparisons are made with the magnetism of [PcM] analogues with key differences noted for M = NiII.

Hamiltonian formulation of X-point collapse in an extended magnetohydrodynamics framework

The study of X-point collapse in magnetic reconnection has witnessed extensive research in the context of space and laboratory plasmas. In this paper, a recently derived mathematical formulation of X-point collapse applicable in the regime of extended magnetohydrodynamics is shown to possess a noncanonical Hamiltonian structure composed of five dynamical variables inherited from its parent model. The Hamiltonian and the noncanonical Poisson brackets are both derived, and the latter is shown to obey the requisite properties of antisymmetry and the Jacobi identity (an explicit proof of the latter is provided). In addition, the governing equations for the Casimir invariants are presented, and one such solution is furnished. The above features can be harnessed and expanded in future work, such as developing structure-preserving integrators for this dynamical system.

Constraints of the teleparallel equivalent of general relativity in a gauge

We consider a specific Hamiltonian formulation of the Teleparallel Equivalent of General Relativity, where the canonical variables are expressed by means of differential forms. We show that some “position” variables of this formulation can be always gauge-transformed to zero. In this gauge the constraints of the theory become simpler, and the other “position” variables acquire a nice geometric interpretation that allows for an alternative, clearer form of the constraints. Based on these results we derive some exact solutions to the constraints.

Hamiltonian Formulation for Continuous Systems with Second-Order Derivatives: A Study of Podolsky Generalized Electrodynamics

This paper presents an analysis of the Hamiltonian formulation for continuous systems with second-order derivatives derived from Dirac’s theory. This approach offers a unique perspective on the equations of motion compared to the traditional Euler–Lagrange formulation. Focusing on Podolsky’s generalized electrodynamics, the Hamiltonian and corresponding equations of motion are derived. The findings demonstrate that both Hamiltonian and Euler–Lagrange formulations yield equivalent results. This study highlights the Hamiltonian approach as a valuable alternative for understanding the dynamics of second-order systems, validated through a specific application within generalized electrodynamics. The novelty of the research lies in developing advanced theoretical models through Hamiltonian formalism for continuous systems with second-order derivatives. The research employs an alternative method to the Euler–Lagrange formulas by applying Dirac’s theory to study the generalized Podolsky electrodynamics, contributing to a better understanding of complex continuous systems.

DMRG study of the theta-dependent mass spectrum in the 2-flavor Schwinger model

We study the θ-dependent mass spectrum of the massive 2-flavor Schwinger model in the Hamiltonian formalism using the density-matrix renormalization group (DMRG). The masses of the composite particles, the pion and sigma meson, are computed by two independent methods. One is the improved one-point-function scheme, where we measure the local meson operator coupled to the boundary state and extract the mass from its exponential decay. Since the θ term causes a nontrivial operator mixing, we unravel it by diagonalizing the correlation matrix to define the meson operator. The other is the dispersion-relation scheme, a heuristic approach specific to Hamiltonian formalism. We obtain the dispersion relation directly by measuring the energy and momentum of the excited states. The sign problem is circumvented in these methods, and their results agree with each other even for large θ. We reveal that the θ-dependence of the pion mass at m/g = 0.1 is consistent with the prediction by the bosonized model. We also find that the mass of the sigma meson satisfies the semi-classical formula, Mσ/Mπ = 3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\sqrt{3} $$\\end{document}, for almost all region of θ. While the sigma meson is a stable particle thanks to this relation, the eta meson is no longer protected by the G-parity and becomes unstable for θ ≠ 0.

Development and Analysis of novel Integrable Nonlinear Dynamical Systems on Quasi-One-Dimensional Lattices. Parametrically Driven Nonlinear System of Pseudo-Excitations on a Two-Leg Ladder Lattice

Following the main principles of developing the evolutionary nonlinear integrable systems on quasi-one-dimensional lattices, we suggest a novel nonlinear integrable system of parametrically driven pseudo-excitations on a regular two-leg ladder lattice. The initial (prototype) form of the system is derivable in the framework of semi-discrete zero-curvature equation with the spectral and evolution operators specified by the properly organized 3 × 3 square matrices. Although the lowest conserved local densities found via the direct recursive method do not prompt us the algebraic structure of system’s Hamiltonian function, but the heuristically substantiated search for the suitable two-stage transformation of prototype field functions to the physically motivated ones has allowed to disclose the physically meaningful nonlinear integrable system with time-dependent longitudinal and transverse inter-site coupling parameters. The time dependencies of inter-site coupling parameters in the transformed system are consistently defined in terms of the accompanying parametric driver formalized by the set of four homogeneous ordinary linear differential equations with the time-dependent coefficients. The physically meaningful parametrically driven nonlinear system permits its concise Hamiltonian formulation with the two pairs of field functions serving as the two pairs of canonically conjugated field amplitudes. The explicit example of oscillatory parametric drive is described in full mathematical details.

End-to-end complexity for simulating the Schwinger model on quantum computers

The Schwinger model is one of the simplest gauge theories. It is known that a topological term of the model leads to the infamous sign problem in the classical Monte Carlo method. In contrast to this, recently, quantum computing in Hamiltonian formalism has gained attention. In this work, we estimate the resources needed for quantum computers to compute physical quantities that are challenging to compute on classical computers. Specifically, we propose an efficient implementation of block-encoding of the Schwinger model Hamiltonian. Considering the structure of the Hamiltonian, this block-encoding with a normalization factor of O(N3) can be implemented using O(N+log2⁡(N/ε)) T gates. As an end-to-end application, we compute the vacuum persistence amplitude. As a result, we found that for a system size N=128 and an additive error ε=0.01, with an evolution time t and a lattice spacing a satisfying t/2a=10, the vacuum persistence amplitude can be calculated using about 1013 T gates. Our results provide insights into predictions about the performance of quantum computers in the FTQC and early FTQC era, clarifying the challenges in solving meaningful problems within a realistic timeframe.

Algebraic symmetries of the observables on the sky: Variable emitters and observers

In this paper we prove a number of exact relations between optical observables, such as trigonometric parallax, position drift, and the proper motion of a luminous source, in addition to the variations of redshift and the viewing angle. These relations are valid in general relativity for any spacetime, and they are of potential interest for astrometry and precise cosmology. They generalize the well-known Etherington’s reciprocity relation between the angular diameter distance and the luminosity distance. Similar to the Etherington’s relation, they hold independently of the spacetime metric, the positions, and the motions of a light source or an observer. We show that those relations follow from the symplectic property of the bilocal geodesic operator, i.e., the geometric object that describes the light propagation between two distant regions of a spacetime. The set of relations we present is complete in the sense that no other relations between those observables should hold, in general. In the meantime, we develop the mathematical machinery of the bilocal approach to light propagation in general relativity and its corresponding Hamiltonian formalism. Published by the American Physical Society 2024

Cartan-like formulation of electric Carrollian gravity

We present a Cartan-like first-order action principle for electric Carrollian gravity. The action is invariant under the local homogeneous Carroll group, albeit in a different representation than the one obtained by gauging the Carroll algebra. Additionally, we show that this first-order action can be derived from a smooth Carrollian limit of the Einstein-Cartan action. The connection with the Hamiltonian and metric forms of the action for electric Carrollian gravity, as well as with previous works in the literature, is also discussed.

Multidimensional Nonhomogeneous Quasi-Linear Systems and Their Hamiltonian Structure

In this paper, we investigate multidimensional first-order quasi-linear systems and find necessary conditions for them to admit Hamiltonian formulation. The insufficiency of the conditions is related to the Poisson cohomology of the admissible Hamiltonian operators. We present in detail the examples of two-dimensional, two-components systems of hydrodynamic type and of a real reduction of the 3-waves system.

Improving Quantum Optimization Algorithms by Constraint Relaxation

Quantum optimization is a significant area of quantum computing research with anticipated near-term quantum advantages. Current quantum optimization algorithms, most of which are hybrid variational-Hamiltonian-based algorithms, struggle to present quantum devices due to noise and decoherence. Existing techniques attempt to mitigate these issues through employing different Hamiltonian encodings or Hamiltonian clause pruning, but they often rely on optimistic assumptions rather than a deep analysis of the problem structure. We demonstrate how to formulate the problem Hamiltonian for a quantum approximate optimization algorithm that satisfies all the requirements to correctly describe the considered tactical aircraft deconfliction problem, achieving higher probabilities for finding solutions compared to previous works. Our results indicate that constructing Hamiltonians from an unconventional, quantum-specific perspective with a high degree of entanglement results in a linear instead of exponential number of entanglement gates instead and superior performance compared to standard formulations. Specifically, we achieve a higher probability of finding feasible solutions: finding solutions in nine out of nine instances compared to standard Hamiltonian formulations and quadratic programming formulations known from quantum annealers, which only found solutions in seven out of nine instances. These findings suggest that there is substantial potential for further research in quantum Hamiltonian design and that gate-based approaches may offer superior optimization performance over quantum annealers in the future.

Inflation and dark matter from the low entropy hypothesis and modeling mechanism of modified gravity

Abstract The hypothesis of low entropy in the initial state of the Universe usually explains the observed entropy increase is in only one time direction: the thermodynamic arrow of time. The Hamiltonian formalism is commonly used in the context of general relativity. The set of Lagrange multipliers are introduced in the formalism, and they are corresponding to the Hamiltonian constraints which are written in terms of ‘weak equality’—the equality is satisfied if the constraints hold. Follow the low-entropy hypothesis, we postulate a modeling mechanism—a weak equality (of modeling) that holds only on the subspace of the theory space of physical models defined by some modeling constraints. By applying the modeling mechanism, we obtain a specific model of modified gravity under specific modeling conditions. We derive a novel equation of modeling from the mechanism, that describes how different gravitational models emerge. The solution of the modeling equation naturally turns out to be the model of R 2-gravity (with additional terms) if ordinary matter is negligible. We also found that this mechanism leads to two models: large-field inflation and wave-like dark matter (DM). Interestingly, the wave-like DM model is supported by the most recent observations of Einstein rings.