Functional Integral Representation Research Articles

Integral Representation of Functions on the Circle

Integral Representation of Functions on the Circle

Theoretical and Numerical Study of Self-Organizing Processes in a Closed System Classical Oscillator and Random Environment

A self-organizing joint system classical oscillator–random environment is considered within the framework of a complex probabilistic process that satisfies a Langevin-type stochastic differential equation. Various types of randomness generated by the environment are considered. In the limit of statistical equilibrium (SEq), second-order partial differential equations (PDE) are derived that describe the distribution of classical environmental fields. The mathematical expectation of the oscillator trajectory is constructed in the form of a functional-integral representation, which, in the SEq limit, is compactified into a two-dimensional integral representation with an integrand: the solution of the second-order complex PDE. It is proved that the complex PDE in the general case is reduced to two independent PDEs of the second order with spatially deviating arguments. The geometric and topological features of the two-dimensional subspace on which these equations arise are studied in detail. An algorithm for parallel modeling of the problem has been developed.

The mean-field limit of quantum Bose gases at positive temperature

We prove that the grand canonical Gibbs state of an interacting quantum Bose gas converges to the Gibbs measure of a nonlinear Schrödinger equation in the mean-field limit, where the density of the gas becomes large and the interaction strength is proportional to the inverse density. Our results hold in dimensions d ⩽ 3 d \leqslant 3 . For d > 1 d > 1 the Gibbs measure is supported on distributions of negative regularity and we have to renormalize the interaction. More precisely, we prove the convergence of the relative partition function and of the reduced density matrices in the L r L^r -norm with optimal exponent r r . Moreover, we prove the convergence in the L ∞ L^\infty -norm of Wick-ordered reduced density matrices, which allows us to control correlations of Wick-ordered particle densities as well as the asymptotic distribution of the particle number. Our proof is based on a functional integral representation of the grand canonical Gibbs state, in which convergence to the mean-field limit follows formally from an infinite-dimensional stationary phase argument for ill-defined non-Gaussian measures. We make this argument rigorous by introducing a white-noise-type auxiliary field, through which the functional integral is expressed in terms of propagators of heat equations driven by time-dependent periodic random potentials and can, in turn, be expressed as a gas of interacting Brownian loops and paths. When the gas is confined by an external trapping potential, we control the decay of the reduced density matrices using excursion probabilities of Brownian bridges.

An Extension of Beta Function by Using Wiman’s Function

The main purpose of this paper is to study extension of the extended beta function by Shadab et al. by using 2-parameter Mittag-Leffler function given by Wiman. In particular, we study some functional relations, integral representation, Mellin transform and derivative formulas for this extended beta function.

A generalized class of α-convex functions with respect to n-symmetric points

In this paper, we investigate certain subclass of analytic functions on the open unit disc. This class generalizes the well-known class of [Formula: see text]-convex functions with respect to n-symmetric points. Some interesting properties such as subordination results, containment relations, integral preserving properties, and the integral representation for functions in this class are obtained.

Properties of a subclass of harmonic mappings with restricted analytic part

In this article, a new subclass of harmonic mappings with restricted analytic part is introduced. Coefficient bounds, Fekete-Szego inequality and integral representation for functions in this subclass are established. Further closure of this class under convex combination is proved. Finally we obtain an upper bound for the Bloch's constant for functions in this class.

Existence of a phase with finite localization length in the double scaling limit of N-orbital models

Among the models of disordered conduction and localization, models with N orbitals per site are attractive both for their mathematical tractability and for their physical realization in coupled disordered grains. However Wegner proved that there is no Anderson transition and no localized phase in the N→∞ limit, if the hopping constant K is kept fixed (Wegner, 1979; Khorunzhy and Pastur, 1993). Here we show that the localized phase is preserved in a different limit where N is taken to infinity and the hopping K is simultaneously adjusted to keep NK constant. We support this conclusion with two arguments. The first is numerical computations of the localization length showing that in the N→∞ limit the site-diagonal-disorder model possesses a localized phase if NK is kept constant, but does not possess that phase if K is fixed. The second argument is a detailed analysis of the energy and length scales in a functional integral representation of the gauge invariant model. The analysis shows that in the K fixed limit the functional integral’s spins do not exhibit long distance fluctuations, i.e. such fluctuations are massive and therefore decay exponentially, which signals conduction. In contrast the NK fixed limit preserves the massless character of certain spin fluctuations, allowing them to fluctuate over long distance scales and cause Anderson localization.

Functional Integral and Stochastic Representations for Ensembles of Identical Bosons on a Lattice

Regularized coherent-state functional integrals are derived for ensembles of identical bosons on a lattice, the regularization being a discretization of Euclidian time. Convergence of the time-continuum limit is proven for various discretized actions. The focus is on the integral representation for the partition function and expectation values in the canonical ensemble. The connection to the grand-canonical integral is exhibited and some important differences are discussed. Uniform bounds for covariances are proven, which simplify the analysis of the time-continuum limit and can also be used to analyze the thermodynamic limit. The relation to a stochastic representation by an ensemble of interacting random walks is made explicit, and its modifications in presence of a condensate are discussed.

Deformation quantization and the tomographic representation of quantum fields

The tomographic representation of quantum fields within the deformation quantization formalism is constructed. By employing the Wigner functional we obtain the symplectic tomogram associated with quantum fields. In addition, the tomographic version of the Wigner map allows us to compute the symbols corresponding to field operators. Finally, the functional integral representation of the tomographic star product is determined. Some possible applications of the formalism to loop quantum cosmology and loop quantum gravity are briefly discussed.

Extended k-Gamma and k-Beta Functions of Matrix Arguments

Various k-special functions such as k-gamma function, k-beta function and k-hypergeometric functions have been introduced and investigated. Recently, the k-gamma function of a matrix argument and k-beta function of matrix arguments have been presented and studied. In this paper, we aim to introduce an extended k-gamma function of a matrix argument and an extended k-beta function of matrix arguments and investigate some of their properties such as functional relations, inequality, integral formula, and integral representations. Also an application of the extended k-beta function of matrix arguments to statistics is considered.

Free-energy functional of instantaneous correlation field in liquids: Field-theoretic derivation of the closures.

This paper presents a unified method for formulating a field-theoretic perturbation theory that encompasses the conventional liquid state theory. First, the free-energy functional of an instantaneous correlation field is obtained from the functional-integral representation of the grand potential. Next, we demonstrate that the instantaneous free-energy functional yields a closure relation between the correlation functions in the mean-field approximation. Notably, the obtained closure relation covers a variety of approximate closures introduced in the liquid state theory.

Functional Central Limit Theorems and P(ϕ)1-Processes for the Relativistic and Non-Relativistic Nelson Models

We construct P(ϕ)1-processes indexed by the full time-line, separately derived from the functional integral representations of the relativistic and non-relativistic Nelson models in quantum field theory. These two cases differ essentially by sample path regularity. Associated with these processes we define a martingale which, under an appropriate scaling, allows to obtain a central limit theorem for additive functionals of these processes. We discuss a number of examples by choosing specific functionals related to particle-field operators.

A Class of Meromorphic Bazilevič-Type Functions Defined by a Differential Operator

We introduce a new subclass of meromorphic Bazilevic-type functions defined with the help of a differential operator. We study some interesting properties of functions from this class, such as the arc length, the growth of coefficients, and the integral representations of functions.

Analytic resolution of time-domain half-space Green's functions for internal loads by a displacement potential-Laplace-Hankel-Cagniard transform method.

A refined yet compact analytical formulation is presented for the time-domain elastodynamic response of a three-dimensional half-space subject to an arbitrary internal or surface force distribution. By integrating Laplace and Hankel transforms into a method of displacement potentials and Cagniard's inversion concept, it is shown that the solution can be derived in a straightforward manner for the generalized classical wave propagation problem. For the canonical case of a buried point load with a step time function, the response is proved to be naturally reducible with the aid of a parametrized Bessel function integral representation to six wave-group integrals on finite contours in the complex plane that stay away from all branch points and the Rayleigh pole except possibly at the starting point of the contours. On the latter occasions, the possible singularities of the integrals can be rigorously extracted by an extended method of asymptotic decomposition, rendering the residual numerical computation a simple exercise. With the new solution format, the arrival time of each wave group is derivable by simple criteria on the contour. Typical results for the time-domain response for an internal point force as well as the degenerate case of a surface point source are included for comparison and illustrations.

Sparse Multiresolution Representations With Adaptive Kernels

Reproducing kernel Hilbert spaces (RKHSs) are key components of non-parametric toolsused in signal processing, statistics, and machine learning. In this work, we aim to address three issues of the classical RKHS-based techniques. First, they require the RKHS to be known a priori , which is unrealistic in many applications. Furthermore, the choice of RKHS affects the shape and smoothness of the solution, thus impacting its performance. Second, RKHSs are ill-equipped to deal with heterogeneous degrees of smoothness, i.e., with functions that are smooth in some parts of their domain but vary rapidly in others. Finally, the computational complexity of evaluating the solution of these methods grows with the number of data points in the training sample, rendering these techniques infeasible for many applications. Although kernel learning, local kernel adaptation, and sparsity have been used to address these issues, many of these approaches are computationally intensive or forgo optimality guarantees. We tackle these problems by leveraging a novel integral representation of functions in RKHSs that allows for arbitrary centers and different kernels at each center. To address the complexity issue, we then write the function estimation problem as a sparse functional program that explicitly minimizes the support of the representation leading to low complexity solutions. Despite their non-convexity and infinite dimensionality, we show these problems can be solved exactly and efficiently by leveraging duality, and illustrate this new approach in simulated and real data.

Olshanski spherical pairs of semigroups type

Let [Formula: see text] be the infinite semigroup, inductive limit of the increasing sequence of the semigroups [Formula: see text], where [Formula: see text] is the unitary group of matrices and [Formula: see text] is the semigroup of positive hermitian matrices. The main purpose of this work is twofold. First, we give a complete classification of spherical functions defined on [Formula: see text], by following a general approach introduced by Olshanski and Vershik.10 Second, we prove an integral representation for functions of positive-type analog to the Bochner–Godement theorem, and a Lévy–Khinchin formula for functions of negative type defined on [Formula: see text].

Kinetic Theory of Boson Gas

We construct a quantum kinetic theory of a weakly interacting critical boson gas using the expectation values of products of Heisenberg field operators in the grand canonical ensemble. Using a functional representation for the Wick theorem for time-ordered products, we construct a perturbation theory for the generating functional of these time-dependent Green’s functions at a finite temperature. We note some problems of the functional-integral representation and discuss unusual apparent divergences of the perturbation expansion. We propose a regularization of these divergences using attenuating propagators. Using a linear transformation to variables with well-defined scaling dimensions, we construct an infrared effective field theory. We show that the structure of the regularized model is restored by renormalization. We propose a multiplicatively renormalizable infrared effective model of the quantum dynamics of a boson gas.

Integral Representation of Functions of Bounded Variation

Functions of bounded variations form important transition between absolute continuous and singular functions. With Bainov’s introduction of impulsive differential equations having solutions of bounded variation, this class of functions had eventually entered into the theory of differential equations. However, the determination of existence of solutions is still problematic because the solutions of differential equations is usually at least absolute continuous which is disrupted by the solutions of bounded variations. As it is known, if f:[a,bλ]→Rn is of bounded variation then f is the sum of an absolute continuous function fa and a singular function fs where the total variation of fs generates a singular measure τ and fs is absolute continuous with respect to τ. In this paper we prove that a function of bounded variation f has two representations: one is f which was described with an absolute continuous part with respect to the Lebesgue measure λ, while in the other an integral with respect to τ forms the absolute continuous part and t(τ) defines the singular measure. Both representations are obtained as parameter transformation images of an absolute continuous function on total variation domain [a,bν].

Fermionized Heisenberg model on a non-Bravais lattice with exact local constraint: Application to a honeycomb lattice

We study magnetic order of the Heisenberg model on a non-Bravais lattice based on the Popov - Fedotov trick for exactly treating the local constraint on on-site spin number. The spin operators are represented by auxiliary fermions and an imaginary chemical potential is introduced. The following steps are sketched: i) Parameterizing classical ground state by a magnetic ordering vector and angles between the spins within a unit cell. ii) Going to a local coordinate system. ii) Using functional integral representation with Hubbard-Stratonovich for partition function and calculating determinants of block matrices by Silvester-Powel method. For illustration we obtain some explicit expressions for ferromagnetic Heisenberg model on a honeycomb lattice and compare them with the slave boson results.

Lieb and hole-doped ferrimagnetism, spiral, resonating valence-bond states, and phase separation in large-U AB2 Hubbard chains

The ground state (GS) properties of the quasi-one-dimensional AB2 Hubbard model are investigated taking the effects of charge and spin quantum fluctuations on equal footing. In the strong-coupling regime, a functional integral representation allows us to derive a low-energy Lagrangian suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-doped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublattice magnetizations, and total spin per unit cell in the Lieb ferrimagnetic GS of the effective quantum Heisenberg model, in very good agreement with previous results. In the challenging hole doping regime away from half filling, we derive the corresponding Hamiltonian. Under the assumption that charge and spin quantum correlations are decoupled, the evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatially modulated spin structures and the emergence of phase separation in the presence of resonating-valence-bond states. We also calculate the doping-dependent GS energy and total spin per unit cell, including both Zeeman and orbital contributions, in which case it is shown that the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results in the doped regime are in very good agreement with density matrix renormalization group studies, where our assumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in anti-phase with the modulation of the magnetic structure.