Feynman Path Integral Using Operator Integration in Banach Space

Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.

Innovation Transformation and Industry-Education Integration:A Study on the Integration Path of Guangxi Zhuang’s Intangible Cultural Heritage and Master of Arts Education in China

This study focuses on the master of arts education in higher education institutions in Guangxi Zhuang Autonomous Region of China,explores the path of integrating Guangxi Zhuang’s intangible cultural heritage with the teaching of master of arts,and puts forward the teaching mode of“thinking guidance-autonomous judgement-program construction.”A theoretical model of innovative transformation of intangible cultural heritage is also summarized.Through the development of this study,it is expected to further enrich the practical teaching mechanism of master of arts education in Chinese universities and form a master of arts teaching model with strong local cultural characteristics.At the same time,the teaching reform based on the integration of Guangxi Zhuang’s intangible cultural heritage and master of arts education also has strong practical significance for promoting the inheritance and innovation of Chinese intangible cultural heritage,promoting the development of cultural and creative industries,and serving the economic and social development of Guangxi.

Research on the Path of Integrating Party Construction Aesthetics into Ideological and Political Education in Colleges and Universities

Party construction aesthetics is a valuable aesthetic resource formed in the practical exploration of rural revitalization,and it has logical compatibility with ideological and political education in terms of concepts,goals,tasks,etc.At present,comprehensive deepening reform has entered a deep-water zone,and some erroneous value orientations such as utilitarianism,emptiness,and liberalization are common in colleges and universities,and ideological and political education is caught in a new round of difficulties.As the main position of ideological and political education,colleges and universities should further use party construction aesthetics to strengthen and improve ideological and political education,adhere to the carrier of aesthetic education,carry out diversified integrated education,broaden the ideological and political practical teaching system,build a long-term mechanism for collaborative education of ideological and political education and party construction aesthetics,and fully integrate party construction aesthetics into ideological and political education in colleges and universities.

Path Integrals and Alternative Effective Dynamics in Loop Quantum Cosmology

The alternative dynamics of loop quantum cosmology is examined by the path integral formulation.We consider the spatially flat FRW models with a massless scalar field,where the alternative quantizations inherit more features from full loop quantum gravity.The path integrals can be formulated in both timeless and deparameterized frameworks.It turns out that the effective Hamiltonians derived from the two different viewpoints are equivalent to each other.Moreover,the first-order modified Friedmann equations are derived and predict quantum bounces for contracting universe,which coincide with those obtained in canonical theory.

Path integral solutions for n-dimensional stochastic differential equations underα-stable Lévy excitation

In this paper,the path integral solutions for a general n-dimensional stochastic differential equa-tions(SDEs)withα-stable Lévy noise are derived and verified.Firstly,the governing equations for the solutions of n-dimensional SDEs under the excitation ofα-stable Lévy noise are obtained through the characteristic function of stochastic processes.Then,the short-time transition probability density func-tion of the path integral solution is derived based on the Chapman-Kolmogorov-Smoluchowski(CKS)equation and the characteristic function,and its correctness is demonstrated by proving that it satis-fies the governing equation of the solution of the SDE,which is also called the Fokker-Planck-Kolmogorov equation.Besides,illustrative examples are numerically considered for highlighting the feasibility of the proposed path integral method,and the pertinent Monte Carlo solution is also calculated to show its correctness and effectiveness.

Path Integral Quantization of Non-Natural Lagrangian

Path integral technique is discussed using Hamilton Jacobi method. The Hamilton Jacobi function of non-natural Lagrangian is obtained using separation of variables method. This function makes an important role in path integral quantization. The path integral is obtained as integration over the canonical phase space coordinates, which contains the generalized coordinate q and the generalized momentum p. One illustrative example is considered to explain the application of our formalism.

Research on the Integration Path of Taihang Spirit and Music Education Major in Colleges and Universities

A key strategy for raising students’humanistic aptitude is through music education.Culture is the soul of music and the source of innovation in music education.Under the current social education concept,it is advocated to carry forward excellent traditional and red culture in teaching,hence this paper further explores the path of integrating red culture into music education.The incorporation of the Taihang spirit,the most representative cultural resource in Shanxi,into music education in colleges and universities will promote the concept of Lide Shuren,further enhance students’training,and enable students to fulfill the dual education requirements.

PATH DEPENDENCE OF J-INTEGRAL IN WELDED JOINT WITH AN OVERMATCHING WELD

By elastic-plastic finite deformation finite element analysis (FEA), the conservation of J-integral is investigated in detail for a welded joint with an overmatchingweld in plane stress case. It is indicated that J-integral is path dependent under various conditions at least in the cases studied in this paper. Meanwhile, the above conclusions are verified by the hybrid method results in which combined Moire interferometry with FEA.

Unified treatment of the bound states of the Schioberg and the Eckart potentials using Feynman path integral approach

We obtain analytical expressions for the energy eigenvalues of both the Schioberg and Eckart potentials using an approximation of the centrifugal term. In order to determine the l-states solutions, we use the Feynman path integral approach to quantum mechanics. We show that by performing nonlinear space-time transformations in the radial path integral, we can derive a transformation formula that relates the original path integral to the Green function of a new quantum solvable system. The explicit expression of bound state energy is obtained and the associated eigenfunctions are given in terms of hypergeometric functions. We show that the Eckart potential can be derived from the Schioberg potential. The obtained results are compared to those produced by other methods and are found to be consistent.

PATH INTEGRAL SOLUTION OF NONLINEAR DYNAMIC BEHAVIOR OF STRUCTURE UNDER WIND EXCITATION

A numerical scheme for the nonlinear behavior of structure under wind excitation is investigated. With the white noise filter of turbulent-wind fluctuations, the nonlinear motion equation of structures subjected to wind load was modeled as the Ito＇ s stochastic differential equation. The state vector associated with such a model is a diffusion process. A continuous linearization strategy in the time-domain was adopted. Based on the solution series of its stochastic linearization equations, the formal probabilistic density of the structure response was developed by the path integral technique. It is shown by the numerical example of a guyed mast that compared with the frequency-domain method and the time-domain nonlinear analysis, the proposed approach is highlighted by high accuracy and effectiveness. The influence of the structure non-linearity on the dynamic reliability assessment is also analyzed in the example.

Optimization of press bend forming path of aircraft integral panel

In order to design the press bend forming path of aircraft integral panels,a novel optimization method was proposed, which integrates FEM equivalent model based on previous study,the artificial neural network response surface,and the genetic algorithm.First,a multi-step press bend forming FEM equivalent model was established,with which the FEM experiments designed with Taguchi method were performed.Then,the BP neural network response surface was developed with the sample data from the FEM experiments.Furthermore,genetic algorithm was applied with the neural network response surface as the objective function. Finally,verification was carried out on a simple curvature grid-type stiffened panel.The forming error of the panel formed with the optimal path is only 0.098 39 and the calculating efficiency has been improved by 77%.Therefore,this novel optimization method is quite efficient and indispensable for the press bend forming path designing.

Hooke's Atom in an Arbitrary External Electric Field: Analytical Solutions of Two-Electron Problem by Path Integral Approach

By using the path integral approach, we investigate the problem of Hooke＇s atom （two electrons interacting with Coulomb potential in an external harmonic-oscillator potential） in an arbitrary time-dependent electric field. For a certain infinite set of discrete oscillator frequencies, we obtain the analytical solutions. The ground state polarization of the atom is then calculated. The same result is also obtained through linear response theory.

Generalized path-integral solution and coherent states of the harmonic oscillator in D-dimensions

We construct a general form of propagator in arbitrary dimensions and give an exact wavefunction of a time- dependent forced harmonic oscillator in D（D ≥ 1） dimensions. The coherent states, defined as the eigenstates of annihilation operator, of the D-dimensional harmonic oscillator are derived. These coherent states correspond to the minimum uncertainty states and the relation between them is investigated.

Generalized Fourth-Order Decompositions of Imaginary Time Path Integral:Implications of the Harmonic Oscillator

The imaginary time path integral formalism offers a powerful numerical tool for simulating thermodynamic properties of realistic systems.We show that,when second-order and fourth-order decompositions are employed,they share a remarkable unified analytic form for the partition function of the harmonic oscillator.We are then able to obtain the expression of the thermodynamic property and the leading error terms as well.In order to obtain reasonably optimal values of the free parameters in the generalized symmetric fourth-order decomposition scheme,we eliminate the leading error terms to achieve the accuracy of desired order for the thermodynamic property of the harmonic system.Such a strategy leads to an efficient fourth-order decomposition that produces thirdorder accurate thermodynamic properties for general systems.

Finite Size Effect in Path Integral Monte Carlo Simulations of ~4He Systems

Path integral Monte Carlo （PIMC） simulations are a powerful computational method to study interacting quantum systems at finite temperatures. In this work, PIMC has been applied to study the finite size effect of the simulated systems of ^4He. We determine the energy as a function of temperature at saturated-vapor-pressure （SVP） conditions in the temperature range of T ∈ [1.0 K,4.0 K], and the equation of state （EOS） in the grmmd state For systems consisted of 32, 64 and 128 ^4He atoms, respectively, We find that the energy at SVP is influenced significantly by the size of the simulated system in the temperature range of T ∈ [2.1 K, 3.0 K] and the larger the system is, the better results are obtained in comparison with the experimental values; while the EOS appeared to be unrelated to it.

Path integral Monte Carlo study of(H_2)_n@C_(70)(n = 1, 2, 3)

The path integral Monte Carlo(PIMC) method is employed to study the thermal properties of C70 with one, two,and three H2 molecules confined in the cage, respectively. The interaction energies and vibrationally averaged spatial distributions under different temperatures are calculated to evaluate the stabilities of(H2)n@C70(n = 1, 2, 3). The results show that(H2)2@C70is more stable than H2@C70. The interaction energy slowly changes in a large temperature range,so temperature has little effect on the stability of the system. For H2@C70and(H2)2@C70, the interaction energies keep negative; however, when three H2 molecules are in the cage, the interaction energy rapidly increases to a positive value.This implies that at most two H2 molecules can be trapped by C70. With an increase of temperature, the peak of the spatial distribution gradually shifts away from the center of the cage, but the maximum distance from the center of H2 molecule to the cage center is much smaller than the average radius of C70.

Effect of a force-free end on the mechanical property of a biopolymer--A path integral approach

We study the effect of a force-free end on the mechanical property of a stretched biopolymer.The system can be divided into two parts.The first part consists of the segment counted from the fixed point（i.e.,the origin） to the forced point in the biopolymer,with arclength L_f.The second part consists of the segment counted from the forced point to the force-free end with arclength △L.We apply the path integral technique to find the relationship between these two parts.At finite temperature and without any constraint at the end,we show exactly that if we focus on the quantities related to the first part,then we can ignore the second part completely.Monte Carlo simulation confirms this conclusion.In contrast,the effect for the quantities related to the second part is dependent on what we want to observe.A force-free end has little effect on the relative extension,but it affects seriously the value of the end-to-end distance if △L is comparable to L_f.

Coherent States and a Path Integral for the Relativistic Linear Singular Oscillator

The SU（1,1） coherent states for a relativistic model of the linear singular oscillator are considered. The corresponding partition function is evaluated. The path integral for the transition amplitude between SU（1,1） coherent states is given. Classical equations of the motion in the generalized curved phase space are obtained. It is shown that the use of quasiclassical Bohr Sommerfeld quantization rule yields the exact expression for the energy spectrum.

Establishing path integral in the entangled state representation for Hamiltonians in quantum optics

Based on two mutually conjugate entangled state representations, we establish the path integral formalism for some Hamiltonians of quantum optics in entangled state representations. The Wigner operator in the entangled state representation is presented. Its advantages are explained.

A Note on Calculation of Phase Space Path Integral by Means of Invariants

In this note we use the method of invariant to calculate the phase space path integral appearing in our previous article.