Terms Of Functional Integrals Research Articles

Maritime Emission Monitoring: Development and Testing of a UAV-Based Real-Time Wind Sensing Mission Planner Module

Maritime emissions contribute significantly to global pollution, necessitating accurate and efficient monitoring methods. Traditional methods for tracking ship emissions often face limitations in real-time data accuracy, with wind measurement being a critical yet challenging aspect. This paper introduces an innovative mission planner module for unmanned aerial vehicles (UAVs) that leverages onboard wind sensing capabilities to enhance maritime emission monitoring. The module’s primary objective is to assist operators in making informed decisions by providing real-time wind data overlays, thus optimizing flight paths and data collection efficiency. Our experimental setup involves the testing of the module in simulated maritime environments, demonstrating its efficacy in varying wind conditions. The real-time wind data overlays provided by the module enable UAV operators to adjust their flight paths dynamically, reducing unnecessary power expenditure and mitigating the risks associated with low-battery scenarios, especially in challenging maritime conditions. This paper presents the implementation of real-time wind data overlays on an open-source state-of-the-art mission planner as a C# plugin that is seamlessly integrated into the user interface. The factors that affect performance, in terms of communication overheads and real-time operation, are identified and discussed. The operation of the module is evaluated in terms of functional integration and real-time visual representation of wind measurements, and the enhanced situational awareness that it can offer to mission controllers is demonstrated. Beyond presenting a novel application of UAV technology in environmental monitoring, we also provide an extensive discussion of how this work will be extended in the context of complete aerial environmental inspection missions and the future directions in research within the field that can potentially lead to the modernization of maritime emission monitoring practices.

High-Performance Megawatt-Scale MVDC Zonal Electrical Distribution System Based on Power Electronics Open System Interfaces

Integrated Power and Energy Systems (IPESs) will play a critical role in supporting future complex electric ship operations. In particular, flexibility is required to exploit shipboard energy sources and storages for multiple ship functions, supporting a highly dynamic electrical power use. To this aim, the Medium Voltage DC (MVDC) electrical power distribution has been conceived to provide requested power flexibility, along with other expected advantages in terms of functional integration and ship design. For complex and high-performance ship power systems, the Zonal Electrical Distribution System (ZEDS) concept has been proposed to maximize MVDC capabilities even under extremely demanding operational conditions. This paper presents a megawatt-scale test facility of a MVDC ZEDS developed and built in Trieste, Italy, based on power electronics open system interfaces. The demonstrator is described in its power and control design architectures and capabilities. In particular, a successful MW-scale test is presented, to show ZEDS capability in supporting large and fast load variations, while maintaining acceptable MVDC voltage tolerances.

Quantum Mechanics and Quantum Field Theory: Algebraic and Geometric Approaches

This is a non-standard exposition of the main notions of quantum mechanics and quantum field theory, including recent results. It is based on the algebraic approach in which the starting point is a star-algebra and on the geometric approach in which the starting point is a convex set of states. Standard formulas for quantum probabilities are derived from decoherence. This derivation allows us to go beyond quantum theory in the geometric approach. Particles are defined as elementary excitations of the ground state (and quasiparticles as elementary excitations of any translation invariant state). The conventional scattering matrix does not work for quasiparticles (or even for particles if the theory does not have particle interpretation). The analysis of scattering in these cases is based on the notion of an inclusive scattering matrix, which is closely related to inclusive cross-sections. It is proven that the conventional scattering matrix can be expressed in terms of Green functions (LSZ formula) and the inclusive scattering matrix can be expressed in terms of generalized Green functions that appear in the Keldysh formalism of non-equilibrium statistical physics. The derivation of the expression of the evolution operator and other physical quantities in terms of functional integrals is based on the notion of the symbol of an operator; these arguments can be applied in the geometric approach as well. In particular, this result can be used to provide a simple derivation of the diagram technique for generalized Green functions. The notion of an inclusive scattering matrix makes sense in the geometric approach, although it seems that a definition of the conventional scattering matrix cannot be provided in this situation. The geometric approach is used to show that quantum mechanics and its generalizations can be considered as classical theories where our devices are able to measure only a part of the observables.

Functional Integrals in Geometric Approach to Quantum Theory

In quantum mechanics, one can express the evolution operator and other quantities in terms of functional integrals. The main goal of this paper is to prove corresponding results in geometric approach to quantum theory. We apply these results to the formalism of L-functionals.

Random vortex dynamics via functional stochastic differential equations

In this paper, we present a novel, closed, three-dimensional random vortex dynamics system, which is equivalent to the Navier–Stokes equations for incompressible viscous fluid flows. The new random vortex dynamics system consists of a stochastic differential equation which is, in contrast with the two-dimensional random vortex dynamics equations, coupled to a finite-dimensional ordinary functional differential equation. This new random vortex system paves the way for devising new numerical schemes (random vortex methods) for solving three-dimensional incompressible fluid flow equations by Monte Carlo simulations. In order to derive the three-dimensional random vortex dynamics equations, we have developed two powerful tools: the first is the duality of the conditional distributions of a couple of Taylor diffusions, which provides a path space version of integration by parts; the second is a forward type Feynman–Kac formula representing solutions to nonlinear parabolic equations in terms of functional integration. These technical tools and the underlying ideas are likely to be useful in treating other nonlinear problems.

Brain functional topology differs by sex in cognitively normal older adults.

IntroductionLate onset Alzheimer’s disease (AD) is the most common form of dementia, in which almost 70% of patients are women.HypothesisWe hypothesized that women show worse global FC metrics compared to men, and further hypothesized a sex-specific positive correlation between FC metrics and cognitive scores in women.MethodsWe studied cognitively healthy individuals from the Alzheimer’s Disease Neuroimaging Initiative cohort, with resting-state functional Magnetic Resonance Imaging. Metrics derived from graph theoretical analysis and functional connectomics were used to assess the global/regional sex differences in terms of functional integration and segregation, considering the amyloid status and the contributions of APOE E4. Linear mixed effect models with covariates (education, handedness, presence of apolipoprotein [APOE] E4 and intra-subject effect) were utilized to evaluate sex differences. The associations of verbal learning and memory abilities with topological network properties were assessed.ResultWomen had a significantly lower magnitude of the global and regional functional network metrics compared to men. Exploratory association analysis showed that higher global clustering coefficient was associated with lower percent forgetting in women and worse cognitive scores in men.ConclusionWomen overall show lower magnitude on measures of resting state functional network topology and connectivity. This factor can play a role in their different vulnerability to AD.Significance statementTwo thirds of AD patients are women but the reasons for these sex difference are not well understood. When this late onset form dementia arises is too late to understand the potential causes of this sex disparities. Studies on cognitively healthy elderly population are a fundamental approach to explore in depth this different vulnerability to the most common form of dementia, currently affecting 6.2 million Americans aged 65 and older are, which means that >1 in 9 people (11.3%) 65 and older are affected by AD. Approaches such as resting-state functional network topology and connectivity may play a key role in understanding and elucidate sex-dependent differences relevant to late-onset dementia syndromes.

Propagators in the correlated worldline theory of quantum gravity

Starting from a formulation of Correlated Worldline (CWL) theory in terms of functional integrals over paths, we define propagators for particles and matter fields in this theory. We show that the most natural formulation of CWL theory involves a rescaling of the generating functional for the theory; correlation functions then simplify, and all loops containing gravitons disappear from perturbative expansions. The spacetime metric obeys the Einstein equation, sourced by all of the interacting CWL paths. The matter paths are correlated by gravitation, thereby violating quantum mechanics for large masses. We derive exact results for the generating functional and the matter propagator, and for linearized weak field theory. For the example of a two-path experiment, we derive the CWL matter propagator, and show how the results compare with conventional quantum theory and with semiclassical gravity. We also exhibit the structure of low-order perturbation theory for the CWL matter propagator.

S-Matrix of Nonlocal Scalar Quantum Field Theory in Basis Functions Representation

Nonlocal quantum theory of a one-component scalar field in D-dimensional Euclidean spacetime is studied in representations of S -matrix theory for both polynomial and nonpolynomial interaction Lagrangians. The theory is formulated on coupling constant g in the form of an infrared smooth function of argument x for space without boundary. Nonlocality is given by the evolution of a Gaussian propagator for the local free theory with ultraviolet form factors depending on ultraviolet length parameter l. By representation of the S -matrix in terms of abstract functional integral over a primary scalar field, the S form of a grand canonical partition function is found. By expression of S -matrix in terms of the partition function, representation for S in terms of basis functions is obtained. Derivations are given for a discrete case where basis functions are Hermite functions, and for a continuous case where basis functions are trigonometric functions. The obtained expressions for the S -matrix are investigated within the framework of variational principle based on Jensen inequality. Through the latter, the majorant of S (more precisely, of − ln S ) is constructed. Equations with separable kernels satisfied by variational function q are found and solved, yielding results for both polynomial theory φ 4 (with suggestions for φ 6 ) and nonpolynomial sine-Gordon theory. A new definition of the S -matrix is proposed to solve additional divergences which arise in application of Jensen inequality for the continuous case. Analytical results are obtained and numerically illustrated, with plots of variational functions q and corresponding majorants for the S -matrices of the theory. For simplicity of numerical calculation, the D = 1 case is considered, and propagator for free theory G is in the form of Gaussian function typically in the Virton–Quark model, although the obtained analytical inferences are not, in principle, limited to these particular choices. Formulation for nonlocal QFT in momentum k space of extra dimensions with subsequent compactification into physical spacetime is discussed, alongside the compactification process.

A brain network model for depression: From symptom understanding to disease intervention

Understanding the neural substrates of depression is crucial for diagnosis and treatment. Here, we review recent studies of functional and effective connectivity in depression, in terms of functional integration in the brain. Findings from these studies, including our own, point to the involvement of at least four networks in patients with depression. Elevated connectivity of a ventral limbic affective network appears to be associated with excessive negative mood (dysphoria) in the patients; decreased connectivity of a frontal-striatal reward network has been suggested to account for loss of interest, motivation, and pleasure (anhedonia); enhanced default mode network connectivity seems to be associated with depressive rumination; and diminished connectivity of a dorsal cognitive control network is thought to underlie cognitive deficits especially ineffective top-down control of negative thoughts and emotions in depressed patients. Moreover, the restoration of connectivity of these networks-and corresponding symptom improvement-following antidepressant treatment (including medication, psychotherapy, and brain stimulation techniques) serves as evidence for the crucial role of these networks in the pathophysiology of depression.

Geometric expansion of the log-partition function of the anisotropic Heisenberg model

We study the asymptotic expansion of the log-partition function of the anisotropic Heisenberg model in a bounded domain as this domain is dilated to infinity. Using the Ginibre’s representation of the anisotropic Heisenberg model as a gas of interacting trajectories of a compound Poisson process, we find all the non-decreasing terms of this expansion. They are given explicitly in terms of functional integrals. As the main technical tool, we use the cluster expansion method.

Modules and brain mapping

This review highlights the key role of modularity and the additive factors method in functional neuroimaging. Our focus is on structure–function mappings in the human brain and how these are disclosed by brain mapping. We describe how modularity of processing (and possibly processes) was a key point of reference for establishing functional segregation as a principle of brain organization. Furthermore, modularity plays a crucial role when trying to characterize distributed brain responses in terms of functional integration or coupling among brain areas. We consider additive factors logic and how it helped to shape the design and interpretation of studies at the inception of brain mapping, with a special focus on factorial designs. We look at factorial designs in activation experiments and in the context of lesion–deficit mapping. In both cases, the presence or absence of interactions among various experimental factors has proven essential in understanding the context-sensitive nature of distributed but modular processing and discerning the nature of (potentially degenerate) structure–function relationships in cognitive neuroscience.

DisappearingQoperator

In the Schr\"odinger formulation of non-Hermitian quantum theories a positive-definite metric operator $\ensuremath{\eta}\ensuremath{\equiv}{e}^{\ensuremath{-}Q}$ must be introduced in order to ensure their probabilistic interpretation. This operator also gives an equivalent Hermitian theory, by means of a similarity transformation. If, however, quantum mechanics is formulated in terms of functional integrals, we show that the $Q$ operator makes only a subliminal appearance and is not needed for the calculation of expectation values. Instead, the relation to the Hermitian theory is encoded via the external source $j(t)$. These points are illustrated and amplified for two non-Hermitian quantum theories: the Swanson model, a non-Hermitian transform of the simple harmonic oscillator, and the wrong-sign quartic oscillator, which has been shown to be equivalent to a conventional asymmetric quartic oscillator.

Functional Integrals for the Schrodinger Equation on Compact Riemannian Manifolds

In this paper, we represent the solution of the Cauchy problem for the Schrodinger equation on compact Riemannian manifolds in terms of functional integrals with respect to the Wiener measure corresponding to the Brownian motion in a manifold and with respect to the Smolyanov surface measures constructed from the Wiener measure on trajectories in the underlying space. The representation of the solution is obtained for the case of analytic (on some sets) potential and analytic initial condition under certain assumptions on the geometric characteristics of the manifold. In the proof, we use a method due to Doss and the representations via functional integrals of the solution to the Cauchy problem for the heat equation in a compact Riemannian manifold.

Felix Alexandrovich Berezin (A Brief Scientific Biography)

In his relatively brief life (he died in an accident before reaching the age of 50), F. A. Berezin succeeded in doing a great deal in mathematics and mathematical physics. Not only did he leave a deep trace in several branches of mathematics that existed before him (group representation theory, the spectral theory of operators, quantum mechanics, statistical physics, constructive quantum field theory), but he also initiated several new concepts, methods, and theories: a general approach to the quantization problem, the construction of the second quantization formalism in terms of functional integrals, which later became the so-called “calculus of symbols” (a forerunner of the theory of pseudo-differential operators), and finally (this was his most important and long nurtured achievement) the theory of supersymmetry and supermanifolds, i.e., what mathematicians now usually call supermathematics. Further we shall discuss all these topics in more detail. Here I would only like to stress that perhaps the most valuable and important characteristic of Berezin’s mathematical life was not his concrete achievements, but the overall stubborn direction of his research, whose main backbone was mathematical physics. He was one of the very few people who transformed mathematical physics into what it � Original publication in Amer. Math. Soc. Transl. (2) 175, Contemporary mathematical physics

Bayesian approach to inverse quantum statistics: Reconstruction of potentials in the Feynman path integral representation of quantum theory

The Feynman path integral representation of quantum theory is used in a non–parametric Bayesian approach to determine quantum potentials from measurements on a canonical ensemble. This representation allows to study explicitly the classical and semiclassical limits and provides a unified description in terms of functional integrals: the Feynman path integral for the statistical operator and its derivative with respect to the potential, and the integration over the space of potentials for calculating the predictive density. The latter is treated in maximum a posteriori approximation, and various approximation schemes for the former are developed and discussed. A simple numerical example shows the applicability of the method.

MARKOV QUANTUM FIELDS ON A MANIFOLD

We study scalar quantum field theory on a compact manifold. The free theory is defined in terms of functional integrals. For positive mass it is shown to have the Markov property in the sense of Nelson. This property is used to establish a reflection positivity result when the manifold has a reflection symmetry. In dimension d=2 we use the Markov property to establish a sewing operation for manifolds with boundary circles. Also in d=2 the Markov property is proved for interacting fields.

Functional integrals for QCD at nonzero chemical potential and zero density.

In a Euclidean space functional integral treatment of the free energy of QCD, a chemical potential enters only through the functional determinant of the Dirac operator which for any flavor is /D+m-mu(f)gamma(0) (where mu(f) is the chemical potential for the given flavor). Any nonzero mu alters all of the eigenvalues of the Dirac operator relative to the mu=0 value, leading to a naive expectation that the determinant is altered and which thereby alters the free energy. Phenomenologically, this does not occur at T=0 for sufficiently small mu, in contradiction to this naive expectation. The problem of how to understand this phenomenological behavior in terms of functional integrals is solved for the case of an isospin chemical through the study of the spectrum of the operator gamma(0)(/D+m). The case of the baryon chemical potential is briefly discussed.

Nonlinearσmodel for long-range disorder and quantum chaos

We suggest a new scheme of derivation of a non-linear ballistic $\sigma$-model for a long range disorder and quantum billiards. The derivation is based on writing equations for quasiclassical Green functions for a fixed long range potential and exact representation of their solutions in terms of functional integrals over supermatrices $Q$ with the constraint $Q^2=1$. Averaging over the long range disorder or energy we are able to write a ballistic $\sigma$-model for all distances exceeding the electron wavelength. Neither singling out slow modes nor a saddle-point approximation are used in the derivation. Carrying out a course graining procedure that allows us to get rid off scales in the Lapunov region we come to a reduced $\sigma$-model containing a conventional collision term. For quantum billiards, we demonstrate that, at not very low frequencies, one can reduce the $\sigma$-model to a one-dimensional $\sigma$-model on periodic orbits. Solving the latter model, first approximately and then exactly, we resolve the problem of repetitions.

Blurred derivatives and meshless methods

AbstractIn this work we first introduce and describe the concept of blurred derivatives. It is shown how they can be used both to approximate differential equations and to re‐express them in alternative ways. In particular, formulations in terms of functional integrals can be obtained using blurred derivatives and extended to non‐linear problems. Blurred derivatives are shown to provide higher flexibility for selection of approximation functions than strong and weak formulations. Some computational implementations of one‐dimensional problems are discussed and the relationship between several well‐known numerical methods is analysed. Finally a meshless numerical scheme for the Poisson equation is described in detail. Its performance is compared with linear finite elements and generalized finite differences on unstructured meshes of points. Copyright © 2002 John Wiley & Sons, Ltd.

Heat kernel regularization of the effective action for stochastic reaction-diffusion equations.

The presence of fluctuations and nonlinear interactions can lead to scale dependence in the parameters appearing in stochastic differential equations. Stochastic dynamics can be formulated in terms of functional integrals. In this paper we apply the heat kernel method to study the short distance renormalizability of a stochastic (polynomial) reaction-diffusion equation with real additive noise. We calculate the one-loop effective action and its ultraviolet scale dependent divergences. We show that for white noise a polynomial reaction-diffusion equation is one-loop finite in d=0 and d=1, and is one-loop renormalizable in d=2 and d=3 space dimensions. We obtain the one-loop renormalization group equations and find they run with scale only in d=2.