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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

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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

Source term model for elasticity system with nonlinear dissipative term in a thin domain

Dilmi Mohamed, Dilmi Mourad, Boulaaras Salah, Benseridi Hamid

Demonstratio Mathematica

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2022

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Vol. 55, nr 1

437--451

EN

This article establishes an asymptotic behavior for the elasticity systems with nonlinear source and dissipative terms in a three-dimensional thin domain, which generalizes some previous works. We consider the limit when the thickness tends to zero, and we prove that the limit solution u∗ is a solution of a two-dimensional boundary value problem with lower Tresca’s free-boundary conditions. Moreover, we obtain the weak Reynolds-type 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

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.

8

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.

9

Approximate controllability of semilinear impulsive strongly damped wave equation

Larez H., Leiva H., Rebaza J., Rios A.

Journal of Applied Analysis

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2015

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Vol. 21, nr 1

45--57

EN

Rothe’s fixed-point theorem is applied to prove the interior approximate controllability of a semilinear impulsive strongly damped wave equation with Dirichlet boundary conditions in the space Z1/2 = D((-Δ)1/2) × L2(Ω), where Ω is a bounded domain in Rn (n ≥ 1). Under some conditions we prove the following statement: For all open nonempty subsets ω of Ω the system is approximately controllable on [0, τ]. Moreover, we exhibit a sequence of controls steering the nonlinear system from an initial state z0 to a neighborhood of the final state z1 at time τ > 0.

10

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.

11

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.

12

Exact boundary controllability of coupled hyperbolic equations

Avdonin S., Choque Rivero A., de Teresa L.

International Journal of Applied Mathematics and Computer Science

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2013

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Vol. 23, no. 4

701--710

EN

We study the exact boundary controllability of two coupled one dimensional wave equations with a control acting only in one equation. The problem is transformed into a moment problem. This framework has been used in control theory of distributed parameter systems since the classical works of A.G. Butkovsky, H.O. Fattorini and D.L. Russell in the late 1960s to the early 1970s. We use recent results on the Riesz basis property of exponential divided differences.

13

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.

14

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.

15

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

Kyureghyan K.

Eksploatacja i Niezawodność

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2009

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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.

16

Fizyczne implikacje matematycznego opisu efektu Dopplera

Rafa J.

Biuletyn Wojskowej Akademii Technicznej

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2008

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Vol. 57, nr 4

85-93

PL

W publikacji przedstawiono rozwiązanie dla równania modelującego efekt Dopplera. Poruszające się źródło generuje propagującą się w otaczającej czasoprzestrzeni falę, która emituje okresowy sygnał. W punkcie czasoprzestrzeni sygnał ten jest odbierany ze zmienioną pulsacją wywołaną sprzężeniem ruchu falowego z ruchem źródła tej fali. Wyznaczono efektywne rozwiązanie tego zagadnienia metodą Cagniarda de Hoopa.

EN

In the paper we presented a solution for the equation modelling the Doppler's effect. The moving source generates the wave which is propagated in the time-space with a periodic signal. Using the Cagniard de-Hoop's method we constructed the solution of the above problem in the closed form.

17

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

|

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.

18

Existence and uniqueness of solutions to wave equations with nonlinear degenerate damping and source terms

Barbu V., Lasiecka I., Rammaha A.

Control and Cybernetics

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2005

|

Vol. 34, no 3

665-687

EN

In this article we focus on the global well-posedness of the differential equation u [...] in Omega x(O, T), where j' denotes the derivative of a C1 convex and real valued function j. The interaction between degenerate damping and a source term constitutes the main challenge of the problem. Problems with non-degenerate damping (k = 0) have been studied in the literature (Georgiev and Todorova, 1994; Levine and Serrin, 1997; Vitillaro, 2003). Thus the degeneracy of monotonicity is the main novelty of this work. Depending on the level of interaction between the source and the damping we characterize the domain of the parameters p, m, k, n (see below) for which one obt ains existence, regularity or finite time blow up of solutions. More specifically, when p [is less than or equal to] m + k global existence of generalized solutions in H1 x L2 is proved. For p > m + k, solutions blow up in a finite time. Higher energy solutions are studied as well. For H2 x H1 initial data we obtain both local and global solutions with the same regularity. Higher energy solutions are also proved to be unique.

19

Wave polynomials for solving different types of two-dimensional wave equations

Maciąg A., Wauer J.

Computer Assisted Mechanics and Engineering Sciences

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2005

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Vol. 12, No. 4

364-378

EN

The paper demonstrates a specific power series expansion technique used to obtain the approximate solution of the two-dimensional wave equation in some unusual cases. The solution for inhomogeneous wave equation, for more complicated shape geometry of the body, discrete boundary conditions and a membrane whose thickness is not constant is shown. As solving functions (Trefftz functions), so-called wave polynomials are used. Recurrent formulas for the particular solution are obtained. Some examples are included.

20

Transverse diffraction of weakly nonlinear waves for a first order quasi-linear system involving source-like terms

Conforto F., Giambo S., Valenti G.

Journal of Technical Physics

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2004

|

Vol. 45, no 1

55-69

EN

Making use of the ray theory and the method of asymptotic expansion of multiple scales, a study of the diffraction of weakly nonlinear waves in a direction transverse to the rays, through media described by dissipative or dispersive hyperbolic systems, is proposed. It is shown that the wave amplitude satisfies a generalized Zabolotskaya-Khokhlov equation in the dissipative case, or a generalized Kadomtsev-Petviashvili equation in the dispersive case. Moreover, plane, cylindrical and spherical waves are also investigated. The present approach is used to study wave diffraction in a heat-conducting fluid.