Advances in Imaging and Electron Physics (eBook)

Calculus of Finite Differences in Quantum Electrodynamics

Henning Harmuth, Beate Meffert (Autoren)

eBook Download: EPUB

2003 | 1. Auflage
312 Seiten
Elsevier Science (Verlag)
978-0-08-049008-3 (ISBN)

Infos

Among the subjects reviewed in these Advances, the properties and computation of electromagnetic fields have been considered on several occasions. In particular, the early work of H.F. Harmuth on Maxwell's equations, which was highly controversial at the time, formed a supplement to the series

This volume, unlike previous volumes in the series concentrates solely on the research of professors' Harmuth and Meffert.

These studies raise important and fundamental questions concerning some of the basic areas of physics: electromagnetic theory and quantum mechanics. They deserve careful study and reflection for although the authors do not attempt to provide the definitive answer to the questions, their work is undoubtedly a major step towards such an answer. This volume essential reading for those researchers and academics working applied mathematicians or theoretical physics

Unlike previous volumes, this book concentrates solely on the new research of professors Harmuth and Meffert

Raises important and fundamental questions concerning electromagnetism theory and quantum mechanics

Provides the steps in finding answers for the highly debated questions

Inhalt

Front Cover 1
Advances in Imaging and Electron Physics 4
Copyright Page 5
Contents 6
Preface 8
Future Contribution 10
Dedication 14
Foreword 16
List of Frequently Used Symbols 20
Chapter 1. Introduction 22
1.1 Modified Maxwell Equations 22
1.2 Summary of Results in Classical Physics 28
1.3 Basic Relations for Quantum Mechanics 35
1.4 Dipole Currents 40
1.5 Infinitesimal and Finite Differences for Space and Time 50
Chapter 2. Differential Equations for the Pure Radiation Field 54
2.1 Pure Radiation Field 54
2.2 Differential Solution for w (Ç,ø) 67
2.3 Hamilton Function for PlanarWave 78
2.4 Quantization of the Differential Solution 88
2.5 Computer Plots for the Differential Theory 92
Chapter 3. Difference Equations for the Pure Radiation Field 98
3.1 Basic Difference Equations 98
3.2 Time Dependent Solution of Ve (Ç,ø) 105
3.3 Solution for Aev (Ç,ø) 122
3.4 Magnetic Potential Amv (Ç,ø) 132
3.5 Hamilton Function for Finite Differences 135
3.6 Quantization of the Difference Solution 150
3.7 Computer Plots for the Difference Theory 158
Chapter 4. Differential Equation for the Klein-Gordon Field 170
4.1 Klein-Gordon Equation with Magnetic Current Density 170
4.2 Step Function Excitation 179
4.3 Exponential Ramp Function Excitation 186
4.4 Hamilton Function and Quantization 189
4.5 Plots for the Differential Theory 195
Chapter 5. Difference Equation for the Klein-Gordon Field 201
5.1 Klein-Gordon Difference Equation 201
5.2 Time Dependent Solution for Psi (Ç,ø) 211
5.3 Exponential Ramp Function as Boundary Condition 222
5.4 Hamilton Function for Difference Equation 226
5.5 Plots for the Difference Theory 231
Chapter 6. Appendix 238
6.1 Calculations for Section 2.2 238
6.2 Inhomogeneous Difference Wave Equation 258
6.3 Differential Derivation of Aev (Ç,ø) 270
6.4 Calculations for Section 3.3 278
6.5 Calculations for Section 3.4 293
6.6 Calculations for Section 3.5 303
6.7 Calculations for Section 4.4 313
6.8 Calculations for Section 5.4 320
References and Bibliography 328
Index 332

Textauszug

−ρ2dϑdθ2−1/211−ρ2dϑ/dθ2d2ϑdθ2+1pqdϑdθ+1q2sinϑ=0θ=tτ,ρ=Rcτ,q=Rτ2m0mmoB=τpτ,p=2m0mmoβξm=τmpτppq=τmpτ=2Rm0τξm,τp=R2m0mmoB,τmp=2m0Rξm

  (34)

For ρ2(dϑ/dθ)2 → 0 one obtains from Eq.(34) the nonrelativistic limit of Eq.(29).

The initial conditions of Eq.(34) are ϑ(θ) = nϑ0 = ϑn and dϑ(0)/dθ = 0 just as for Eq.(29). The velocities υ(θ) and υy(θ) of Eqs.(30) and (31) become:

θc=−Rcτdϑθ=−ρdϑdθ

  (35)

yθc=−ρdϑdθsinϑ

  (36)

Plots of υy(θ)/c are shown in Fig.1.4-11 for p = 1/4, q = 1 or τ = τp, ρ = 4, and various values of ϑ(0) = ϑn = nπ/8. The relativistic limitation υ/c < 1 is not conspicuous but a comparison with Fig. 1.4-9 shows how the peaks of the plots for n = 5, 6, 7 have been flattened.

Figure 1.4-11 The normalized velocity υy(θ)/c according to Eq.(36) for p = 1/4, q = 1, ρ = 4, and ϑ(0) = ϑn = nπ/8 with n = 1, 2, 3, 4, 5, 6, 7, 7.8.

1.5 INFINITESIMAL AND FINITE DIFFERENCES FOR SPACE AND TIME

The discussion of finite or infinite divisibility of space and time has been going on for some 2500 years. Zeno of Elea (c.490 – c. 430 B.C.) advanced the paradoxes of the race between Achilles and the turtle or the arrow which cannot fly that were supposed to show that infinite divisibility of space and time was not possible. A quote from Aristotle (384 – 322 B.C.) shows that infinite divisibility and thus the concept of the continuum was a matter of discussion before he wrote his Physica:

Now a motion is thought to be one of those things which are continuous, and it is in the continuous that the infinite first appears; and for this reason, it often happens that those who define the continuous use the formula of the infinite, that is, they say that the continuous is that which is infinitely divisible [Apostle 1969, Book III (Γ), 1, §2].

Aristotle argued the concept of the continuum for space, time and motion so convincingly1 that it does not seem to have been challenged until Max Planck introduced quantum mechanics. Newton (1971) took this concept apparently so much for granted that he did not even mention it, even though he was very meticulous in listing and elaborating his assumptions. The differential calculus of Leibnitz and Newton made us carry the concept of infinite divisible one step further since we distinguish now between dividing a finite interval ΔX into denumerable or non-denumerable many intervals.

A widely held view of a space-continuum is summed up in a quote by Weyl that emphasizes that this concept came from mathematics:

From the essence of space remains in the hands of the mathematician, using such abstraction, only one truth: that it is a three-dimensional continuum (Weyl 1921; 1968, vol. II, p. 213).

The following two quotes give a good summary of our currently accepted thinking about space and time:

So let us conclude that space has a definite real intrinsic structure in its metric, affinity, and topology. This means it has a shape and size in a way I have tried at length to make clear. It shows just how much space is a particular thing (Nehrlich 1976, p. 211).

It is now generally taken for granted that public time is both infinitely divisible, or “dense” as the mathematician terms it, and continuous; that is, not only can we always consider any interval as made up of smaller ones, but we are entitled to apply even irrational numbers to the measurement of time… Our concepts are not immune to revision; and in the case of time, we are already prepared, in some locations, to speak of it as though it were discrete. But to do so consistently would require a fairly radical revision of the concept. We should have to unthink as far back as Aristotle (Lucas 1973, pp. 29, 32, 33).

We avoid all questions of how spatial and time distances can be divided indefinitely since we never find a hint how such a division can be carried out experimentally or can be observed. Instead we replace the differentials dx, dt by arbitrarily small but finite differences Δx, Δt. They can always be equal or even smaller than the shortest observable distance. The difference between a finite distance of 10 −100 m and an infinitesimal distance dx is not directly observable. The question arises whether such small values of Δx would not have to yield the same results as differentials dx. This question was answered by Hölder (1887) who showed that difference equations and differential equations define different classes of functions. In particular the Gamma function satisfies the algebraic difference equation

X+1=XΓX,X=x/Δx

  (1)

but no algebraic differential equation.

The example of the Gamma function shows that difference equations can define continuous and differentiable functions. The use of finite differences Δx, Δt does not imply that only discrete functions defined at integer multiples of Δx and Δt can be obtained as solution of a partial difference equation.

Although this result may be of limited interest to the practical physicist it contributes to the philosophy of science. Our inability to make observations at x and x + dx or t and t + dt prevents any proof that there is a physical space-time continuum, but it also prevents a proof that there is NO physical space-time continuum. The question whether there is or is not a space-time continuum is no more answerable within the confines of a science based on observation than the question how many angels can dance on the point of a needle. We can use mathematics as a tool in physics, but we cannot use it as a source of concepts that are beyond observation.

Let us take one more step in the direction of philosophy of science and quote from Einstein and Infeld (1938, p. 311):

The psychological subjective feeling of time enables us to order our impressions, to state that one event precedes another. But to connect every instant of time with a number, by the use of a clock, to regard time as an one-dimensional continuum, is already an invention. So also are the concepts of Euclidean and non-Euclidean geometry, and our space understood as a three-dimensional continuum.

When we use differential calculus and then permit this mathematical method to define physical space and space-time we make physics a special branch of mathematics. Since mathematics is a science of the thinkable while physics is a science of the observable, mathematics can never be more than a tool in physics or provide inspiration. The succesful solution of a physical problem by means of differential calculus only implies that the assumption of a mathematical continuum can yield results that correspond with physical observation, it does not imply the existence of a physical space-time continuum.

Let us see how this principle works for finite differences. We know from observation that we can only resolve finite space and time differences x and x + Δx or t and t + Δt. If we look what mathematical tools are available that satisfy this requirement we find the calculus of finite differences. This calculus does not define any physical concept of space or space-time. It works for continuous functions, like the Gamma function, but does not suggest any particular topology of space or space-time. We could go one step further and require that x and t are integer multiples of Δx and Δt: x = nΔx, t = mΔt. If we did this we would introduce a cellular space or space-time into physics; we would repeat the mistake we made with differential calculus and the space-time continuum. There is no physical reason to do so.

The use of the calculus of finite differences reduces the concepts of space and space-time to coordinate systems and moving coordinate systems, which are obviously human constructs in line with the quote of Einstein and Infeld above. This is discussed in some detail in a book by...

Produkt-Details

• Erscheint lt. Verlag: 18.12.2003
• Mitarbeit: Herausgeber (Serie): Peter W. Hawkes
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Mathematik / Informatik ► Informatik
• Themenwelt: Medizin / Pharmazie ► Gesundheitsfachberufe
• Themenwelt: Medizin / Pharmazie ► Medizinische Fachgebiete ► Radiologie / Bildgebende Verfahren
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Elektrodynamik
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-08-049008-5 / 0080490085
• ISBN-13: 978-0-08-049008-3 / 9780080490083