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1

Postawić kwantową teorię pola z głowy na nogi. Część 3

Chankowski Piotr

Postępy Fizyki

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2024

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T. 75, z. 1

2--17

PL

Wbrew wrażeniu jakie można odnieść z większości standardowych podręczników, równania falowe Diraca, Kleina-Gordona i inne nie są podstawą kwantowej teorii pola. W niniejszym artykule staram się pokazać, jak powinna być ona poprawnie formułowana i omawiam pewne jej aspekty, które na ogół nie są przedstawiane właściwie. Celem artykułu jest spowodowanie zmiany w nauczaniu kwantowej teorii pola. Tekst został podzielony na trzy części. W niniejszej części 3 (i ostatniej) omawiam przepis LSZ pozwalający wyznaczać elementy macierzy S i inne wielkości fizyczne bez czynienia zwykłych, bardzo restrykcyjnych założeń oraz zalety i słabości formułowania kwantowej teorii pola za pomocą całek po trajektoriach.

EN

Despite the impression that can be gained from most of the standard textbooks, Dirac, Klein-Gordon and other wave equations do not constitute the basis of quantum field theory. In this article I attempt to show how it should be formulated properly and discuss some of its aspects which usually are presented unsatisfactorily. The aim of the text is to cause the change in the way quantum field theory is taught. The text is split into three parts. In this part 3 (the last one) I discuss the LSZ prescription which allows to extract S-matrix elements without making the usual, very restrictive assumptions and advantages and weak sides of formulating quantum field theory with the help of path integrals.

2

Postawić kwantową teorię pola z głowy na nogi. Część I

Chankowski Piotr

Postępy Fizyki

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2023

|

T. 74, z. 3

5--14

PL

Wbrew wrażeniu jakie można odnieść z lektury większości standardowych podręczników, równania falowe Diraca, Kleina-Gordona i inne nie są podstawą relatywistycznej kwantowej teorii pola. W niniejszym artykule staram się pokazać, jak powinna być ona poprawnie formułowana i omawiam pewne jej aspekty, które na ogół nie są przedstawiane właściwie. Moim celem jest spowodowanie zmiany w nauczaniu kwantowej teorii pola. Tekst został podzielony na trzy części. W pierwszej przypominam krótko historyczny rozwój kwantowej teorii pola i omawiam jej sformułowanie jako kwantowej teorii oddziałujących cząstek (relatywistycznych lub nierelatywistycznych).

EN

Despite the impression that can be gained from most of the standard textbooks, Dirac, Klein-Gordon and other wave equations do not constitute the basis of relativistic quantum field theory. In this article I attempt to show how it should be formulated properly and discuss some of its aspects which usually are presented unsatisfactorily. My aim is to cause the change in the way quantum field theory is taught. The text is split into three parts. In the first one I briefly recall the quantum field theory historical development and present its formulation as a quantum theory of interacting particles (relativistic or nonrelativistic).

3

Postawić kwantową teorię pola z głowy na nogi. Część 2

Chankowski Piotr

Postępy Fizyki

|

2023

|

T. 74, z. 4

5--15

PL

Wbrew wrażeniu jakie można odnieść z lektury większości standardowych podręczników, równania falowe Diraca, Kleina-Gordona i inne nie są podstawą relatywistycznej kwantowej teorii pola. W niniejszym artykule staram się pokazać, jak powinna być ona poprawnie formułowana i omawiam pewne jej aspekty, które na ogół nie są przedstawiane właściwie. Moim celem jest spowodowanie zmiany w nauczaniu kwantowej teorii pola. Tekst został podzielony na trzy części. W niniejszej drugiej części omawiam sformułowanie kwantowej teorii pola jako teorii oddziałujących pól oraz sens fizyczny procedury renormalizacji.

EN

Despite the impression that can be gained from most of the standard textbooks, Dirac, Klein-Gordon and other wave equations do not constitute the basis of relativistic quantum field theory. In this article I attempt to show how it should be formulated properly and discuss some of its aspects which usually are presented unsatisfactorily. My aim is to cause the change in the way quantum field theory is taught. The text is split into three parts. In the second part I discuss the formulation of quantum field theory as a theory of interacting fields as well as the physical sense of the renormalization procedure.

4

Stabilization of the wave equation with a nonlinear delay term in the boundary conditions

Ghecham Wassila, Rebiai Salah-Eddine, Sidiali Fatima Zohra

Journal of Applied Analysis

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2022

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Vol. 28, nr 1

35--55

EN

A wave equation in a bounded and smooth domain of ℝn with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established. The proof of existence of solutions relies on a construction of suitable approximating problems for which the existence of the unique solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behavior of the energy and of the solutions to an appropriate dissipative ordinary differential equation.

5

An accurate and efficient local one-dimensional method for the 3D acoustic wave equation

Wu Mengling, Jiang Yunzhi, Ge Yongbin

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

528--552

EN

We establish an accurate and efficient scheme with four-order accuracy for solving three-dimensional (3D) acoustic wave equation. First, the local one-dimensional method is used to transfer the 3D wave equation into three one-dimensional wave equations. Then, a new scheme is obtained by the Padé formulas for computation of spatial second derivatives and the correction of the truncation error remainder for discretization of temporal second derivative. It is compact and can be solved directly by the Thomas algorithm. Subsequently, the Fourier analysis method and the Lax equivalence theorem are employed to prove the stability and convergence of the present scheme, which shows that it is conditionally stable and convergent, and the stability condition is superior to that of most existing numerical methods of equivalent order of accuracy in the literature. It allows us to reduce computational cost with relatively large time step lengths. Finally, numerical examples have demonstrated high accuracy, stability, and efficiency of our method.

6

The stability of poro elastic wave equations in saturated porous media

Xiong Fansheng, Sun Weitao, Liu Jiawei

Acta Geophysica

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2021

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Vol. 69, no. 1

65--75

EN

Poro-elastic wave equations are one of the fundamental problems in seismic wave exploration and applied mathematics. In the past few decades, elastic wave theory and numerical method of porous media have developed rapidly. However, the math ematical stability of such wave equations have not been fully studied, which may lead to numerical divergence in the wave propagation simulation in complex porous media. In this paper, we focus on the stability of the wave equation derived from Tuncay’s model and volume averaging method. By analyzing the stability of the frst-order hyperbolic relaxation system, the mathematical stability of the wave equation is proved for the frst time. Compared with existing poro-elastic wave equations (such as Biot’s theory), the advantage of newly derived equations is that it is not necessary to assume uniform distribution of pores. Such wave equations can spontaneously incorporate complex microscale pore/fracture structures into large-scale media, which is critical for unconventional oil and gas exploration. The process of proof and numerical examples shows that the wave equations are mathematically stable. These results can be applied to numerical simulation of wave feld in reservoirs with pore/fracture networks, which is of great signifcance for unconventional oil and gas exploration.

7

Wellposedness and long time behavior for a general class of Moore-Gibson-Thompson equations

Nicaise Serge, Bounadja Hizia

Control and Cybernetics

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2020

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Vol. 49, No. 3

245--276

EN

We consider the well-posedness and the long time behavior of third order in time linear evolution equations, general and abstract version of the Moore-Gibson-Thompson system. We find sufficient but strong conditions that guarantee the exponential decay of the system and present some illustrative examples. Then, by comparing the behavior of the resolvent of the Moore-Gibson-Thompson system with the one of the resolvent of the wave equation with a frictional interior damping, we furnish weaker conditions that guarantee exponential, polynomial or even logarithmic decay of the solution of the Moore-Gibson-Thompson system in a bounded domain of Rn, n ≥ 1.

8

Exact controllability of a string to rest with a moving boundary

Gugat Martin

Control and Cybernetics

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2019

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Vol. 48, No. 1

69--87

EN

We consider the problem of steering a ﬁnite string to the zero state in ﬁnite time from a given initial state by controlling the state at one boundary point while the other boundary point moves. As a possible application we have in mind the optimal control of a mining elevator, where the length of the string changes during the transportation process. During the transportation process, oscillations of the elevator-cable can occur that can be damped in this way. We present an exact controllability result for Dirichlet boundary control at the ﬁxed end of the string that states that there exist exact controls for which the oscillations vanish after ﬁnite time. For the result we assume that the movements are Lipschitz continuous with a Lipschitz constant, whose absolute value is smaller than the wave speed. In the result, we present the minimal time, for which exact controllability holds, this time depending on the movement of the boundary point. Our results are based upon travelling wave solutions. We present a representation of the set of successful controls that steer the system to rest after ﬁnite time as the solution set of two point-wise equalities. This allows for a transformation of the optimal control problem to a form where no partial diﬀerential equation appears. This representation enables interesting insights into the structure of the successful controls. For example, exact bang-bang controls can only exist if the initial state is a simple function and the initial velocity is zero.

9

Solution of the complex-valued Helmholtz equation using a dedicated finite element solver

Chaber B., Szmurło R., Starzyński J.

Przegląd Elektrotechniczny

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2017

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R. 93, nr 3

171--174

EN

This paper describes a numerical FEM model for solving the complex-valued, vector Helmholtz wave equation. The model describes phenomena of electromagnetic wave propagation for high frequencies. The presented model can be used in a larger system seeking an efficient design parameters of electromagnetic energy transfer for high power pulse generation device.

PL

W niniejszym artykule opisujemy zbudowany model numeryczny MES rozwiązujący zespolone, wektorowe równanie falowe Helmholtza. Pozwala on na modelowanie zjawisk propagacji fal elektromagnetycznych wysokich częstotliwości. Zaprezentowany model może zostać wykorzystany w systemie poszukującym optymalnego projektu urządzenia służącego do przesyłu energii w postaci fali elektromagnetycznej, do generacji silnych impulsów elektromagnetycznych.

10

An adaptive control scheme for hyperbolic partial differential equation system (drilling system) with unknown coefficient

Farahani H. S., Telabi H. A., Baghermenhaj M.

Archives of Control Sciences

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2017

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Vol. 27, no. 1

63--76

EN

The adaptive boundary stabilization is investigated for a class of systems described by second-order hyperbolic PDEs with unknown coefficient. The proposed control scheme only utilizes measurement on top boundary and assume anti-damping dynamics on the opposite boundary which is the main feature of our work. To cope with the lack of full state measurements, we introduce Riemann variables which allow us reformulate the second-order in time hyperbolic PDE as a system with linear input-delay dynamics. Then, the infinite-dimensional time-delay tools are employed to design the controller. Simulation results which applied on mathematical model of drilling system are given to demonstrate the effectiveness of the proposed control approach.

11

On the numerical solution of the initial-boundary value problem with neumann condition for the wave equation by the use of the laguerre transform and boundary elements method

Litynskyy S., Muzychuk Y., Muzychuk A.

Acta Mechanica et Automatica

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2016

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Vol. 10, no. 4

285--290

EN

We consider a numerical solution of the initial-boundary value problem for the homogeneous wave equation with the Neumann condition using the retarded double layer potential. For solving an equivalent time-dependent integral equation we combine the Laguerre transform (LT) in the time domain with the boundary elements method. After LT we obtain a sequence of boundary integral equations with the same integral operator and functions in right-hand side which are determined recurrently. An error analysis for the numerical solution in accordance with the parameter of boundary discretization is performed. The proposed approach is demonstrated on the numerical solution of the model problem in unbounded three-dimensional spatial domain.

12

The usage of Trefftz functions and Picard's iteration for solving different problems of a two-dimensional wave equation

Krzyszkowska P., Maciąg A., Walaszczyk M.

Journal of Applied Mathematics and Computational Mechanics

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2014

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Vol. 13, nr 3

129--140

EN

The paper presents a method of solving two-dimensional wave equations which describe vibrations of the membrane with variable thickness and with damping. The differential operator is decomposed into two parts. The first one describes vibrations of the membrane with constant thickness without damping. The second contains the rest of the original operator and is treated as inhomogeneity for the first one. Picard’s iterations are used to calculate a successive approximation of the exact solution. Trefftz functions (wave polynomials) are used to solve the problem in each iteration. The presented examples show the usefulness of the method. The approach described in this paper can be used also for solving nonlinear problems for a wave equation.

13

Generalized solutions to a characteristic Cauchy problem

Allaud E, Devoue V

Journal of Applied Analysis

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2013

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Vol. 19, nr 1

1--29

EN

We give a meaning to the nonlinear characteristic Cauchy problem for the wave equation in base form by replacing it by a family of non-characteristic ones. This leads to a well-formulated problem in an appropriate algebra of generalized functions. We prove existence of a solution and we precise how it depends on the choice made. We also check that in the classical case (non-characteristic) our new solution coincides with the classical one.

14

An interval difference method for solving the wave equation

Szyszka B.

Poznan University of Technology Academic Journals. Electrical Engineering

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2012

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No. 69

161--168

EN

In the paper a difference interval method for solving the wave equation together the initial-boundary value problems is presented. Using an interval method together floating-point interval arithmetic guarantee, that obtained interval solutions contain all numerical errors. Additionally, each exact solution is included into interval solution. In numerical experiments it is guarantee contain all numerical errors in obtained interval solutions. Taken into consideration is the central discretization method with regard to space and time. An initial condition is approximated by the third-order Taylor polynomial with local truncation error of order 0(h4). In the paper new formula, which described discretization of the initial condition, is proposed. Therefore more exact solutions are obtained then in the previous considerations.

15

Wpływ bezwładności taboru, nawierzchni kolejowej i podłoża na ich stateczność i prędkości krytyczne pociągów o dużych prędkościach

Ataman M.

Logistyka

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2012

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nr 3

PL

W referacie podano rozwiązanie równania ruchu modelu toru, podtorza i pociągu z uwzględnieniem ich inercji. Wpływ masy w tych modelach zdecydowanie zmienia siły krytyczne, wybaczające tor, jak również prędkości krytyczne. Praca jest poszerzeniem i rozwinięciem opracowania Newlanda [9], w której uwzględniono tylko masę pociągu.

EN

This paper presents solution of the equation of motion of a track, a subgrade and a train, taking into account their inertia. The inclusion of mass in these models changes the critical force, buckling the track, as well as the critical speeds. This paper is an extension of Newland’s study [9], where the effect of mass of the train is included only.

16

Optimal internal dissipation of a damped wave equation using a topological approach

Münch A

International Journal of Applied Mathematics and Computer Science

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2009

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Vol. 19, no 1

15-37

EN

We consider a linear damped wave equation defined on a two-dimensional domain [...], with a dissipative term localized in a subset [...]. We address the shape design problem which consists in optimizing the shape of [...] in order to minimize the energy of the system at a given time T. By introducing an adjoint problem, we first obtain explicitly the (shape) derivative of the energy at time T with respect to the variation in [...]. Expressed as a boundary integral on [...], this derivative is then used as an advection velocity in a Hamilton-Jacobi equation for shape changes. We use the level-set methodology on a fixed working Eulerian mesh as well as the notion of the topological derivative. We also consider optimization with respect to the value of the damping parameter. The numerical approximation is presented in detail and several numerical experiments are performed which relate the over-damping phenomenon to the well-posedness of the problem.

17

Hamowanie bezpieczeństwa górniczego urządzenia wyciągowego z zastosowaniem liniowo narastającej wartości siły hamowania

Kyureghyan K.

Eksploatacja i Niezawodność

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2009

|

nr 1

39-45

PL

Praca przedstawia analizę dynamiki stanu pracy górniczego urządzenia wyciągowego w warunkach hamowania bezpieczeństwa. Proces hamowania bezpieczeństwa przedstawiono za pomocą modelu mechanicznego z uwzględnieniem liniowo narastającej wartości siły hamowania zastosowanej do koła pędnego. Na podstawie modelu mechanicznego zapisano układ równań falowych dla przemieszczeń i odkształceń dowolnych przekrojów poprzecznych lin nośnych i wyrównawczych. Rozwiązanie analityczne układu równań pozwala na wyznaczenie zależności opisujących naprężenia w ustalonych przekrojach lin, wartości obciążenia liny nośnej w miejscu jej zejścia z koła pędnego oraz minimalny przedział czasu t∈<0;t0), w którym następuje wzrost wartości siły hamowania.

18

Solutions of Two-Dimensional Wave Equation using some Form of the Trefftz Functions

Sokała M.

Computational Methods in Science and Technology

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2008

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Vol. 14, nr 2

111-122

EN

The paper presents a specific technique to generate the Trefftz functions for the two-dimensional wave equation. The obtained functions are used to determine approximate solutions of some tested problems. The accuracy of the method is discussed.

19

Arbitrary high-order finite element schemes and high-order mass lumping

Jund S., Salmon S.

International Journal of Applied Mathematics and Computer Science

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2007

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Vol. 17, no 3

375-393

EN

Computers are becoming sufficiently powerful to permit to numerically solve problems such as the wave equation with high-order methods. In this article we will consider Lagrange finite elements of order k and show how it is possible to automatically generate the mass and stiffness matrices of any order with the help of symbolic computation software. We compare two high-order time discretizations: an explicit one using a Taylor expansion in time (a Cauchy-Kowalewski procedure) and an implicit Runge-Kutta scheme. We also construct in a systematic way a high-order quadrature which is optimal in terms of the number of points, which enables the use of mass lumping, up to P5 elements. We compare computational time and effort for several codes which are of high order in time and space and study their respective properties.

20

Wave polynomials in elasticity problems

Maciąg A.

Engineering Transactions

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2007

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Vol. 55, iss. 2

129-153

EN

The paper demonstrates a new technique of obtaining the approximate solution of the two- and threedimensional elasticity problems. The system of equations of elasticity can be converted to the system of wave equation. In this case, as solving functions (Trefftz functions), the so-called wave polynomials can be used. The presented method is useful for a finite body of a certain shape. Then the obtained solutions are coupled through initial and boundary conditions. Recurrent formulas for the two- and three-dimensional wave polynomials and their derivatives are obtained. The methodology for solution of systems of partial differential equations with common initial and boundary conditions by means of solving functions is presented. The advantage of using the method of solving functions is that the solution exactly satisfies the given equation (or system of equations). Some examples are included.