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PACS №: 03.50.De Henning F. Harmuth Retired from The Catholic University of America, Washington, DC 757 Bayou Drive, Destin, Florida 32541-1846 Nonsinusoidal Waves, Modified Maxwell Equations, Dogma of the Continuum Contents 1. Introduction 19 2. Finite Resolution of Space, Time and Amplitude Measurements 22 3. Electromagnetic Signals in Seawater 23 4. Modified Maxwell Equations and Renormalization 25 5. Small Finite Resolutions ∆x and ∆t Relativistic Quantum Physics 29 6. Summary 34 Abstract Electrical communications as well as physics are strongly based on infinitely extended periodic sinusoidal functions. Neither the causality law nor the conservation law of energy have any meaning for waves represented by such functions. Information theory demands that any physical process starts at a finite time and ends at a finite time since we can neither observe negative or positive infinite times. A corresponding statement holds for space intervals. Maxwell’s equation do generally not have solutions that start at a finite time and thus permit to represent the causality law. Hence, they represent generally steady state solutions rather than transient or signal solutions. The problem with Maxwell’s equations is overcome by permitting magnetic dipole currents that are produced by rotating magnetic dipoles. The modified Maxwell equations make it possible to study the propagation of heavily distorted signals in seawater. When we further replace infinite times and distances by arbitrarily large but finite ones, we avoid the problem of the infinite ‘zero-point energy’ in quantum electrodynamics and eliminate the need for renormalization. Finally, if we observe that infinitesimal intervals dx, dt are no more observable than infinite ones, we find that differential calculus should be replaced in relativistic quantum physics with the calculus of finite differences using arbitrarily small but finite intervals ∆x, ∆t. 1. Introduction in standard courses something that is contradicted in the afternoon in a course on information theory. Consider a first result of information theory that was startling at that time: A periodic sinusoidal wave transmits information at the rate zero This statement directed much interest to nonsinusoidal waves. It did not take long to recognize that nonsinusoidal waves are the only ones we can produce or receive. Mathematicians coined the term causal functions for functions that were zero before a certain time. Adding The three topics nonsinusoidal electromagnetic waves, modified Maxwell equations, and dogma of the continuum do not have an obvious connection. But they are actually closely connected through information theory. To recognize this we have to go back about 60 years when information theory got started. Those who studied electrical communications around 1950 were exposed to this new development. Some were strongly influenced by it, since it is a unique experience for a young student to learn in the morning 19 Henning F. Harmuth the requirement of quadratic integrability assures that quadratically integrable causal functions could represent electromagnetic signals with finite energy. We had a basis for electrical communications that satisfied both the causality law and the conservation law of energy. Once we had a solid foundation for the mathematical representation of electromagnetic signals the question of the propagation velocity of signals in lossy media arose. This developed into a problem of great practical interest in connection with radio communication with deeply submerged submarines [1]. The widely used concept of group velocity was based on the transmission of two periodic sinusoidal waves with almost equal frequency. Since the transmission rate of information of one periodic sinusoidal waves was zero, anything based on two sinusoidal waves had become of dubious value. The strong variation of attenuation with frequency implied great distortions of electromagnetic signals in seawater. The question of propagation velocity of signals in seawater was evidently strongly linked with their distortion. The only reliable way to obtain usable results was to go beyond sinusoidal waves and to find solutions of Maxwell’s equations for signals in lossy media. The simplest signal was a rectangular pulse; it could be further simplified by decomposing it into two time-shifted and amplitude-reversed step functions. A standard method to solve a partial differential equation or a set of them with a step function as boundary condition is Fourier’s method of standing waves. It had been known since Fourier (1768–1830), but its results were stated by integrals and were thus of little practical use before the arrival of electronic computers. By 1980 the computing power of academic scientists had reached a level that permitted them to make plots of electric field strengths E(y, t) and magnetic field strengths H(y, t) as functions of distance y and time t due to an electric step function excitation well’s equations we write them in the following form: ∂D + ge , ∂t ∂B , − curl E = ∂t div D = ρe , div B = 0. curl H = (2) (3) (4) (5) Here E and H stand for the electric and magnetic field strength, D and B for the electric and magnetic flux density, ge and ρe for the electric current and charge density. In addition we need three constitutive equations that have in the simplest case the form D = ²E, B = µH, (6) (7) ge = σE (8) with scalar constants for the permittivity ², permeability µ, and conductivity σ. Consider the electric excitation of a planar wave propagating in the direction y as function of t. We refer the reader to the literature for details in order to be able to concentrate on the important results [2,3,pp. 1–26]. Equations (2) and (3) are brought into the following form for the magnitudes E(y, t) and H(y, t) of the electric and magnetic field strength: ∂E ∂H +µ = 0, ∂y ∂t ∂E ∂H +² + σE = 0. ∂y ∂t (9) (10) Elimination of H from Eqs. (9) and (10) yields a second order equation for E: ∂2E ∂E ∂2E − µ² 2 − µσ = 0. 2 ∂y ∂t ∂t (11) at the boundary y = 0. The surprising result was that such a solution could not be obtained since Maxwell’s equations do not generally satisfy the causality law: If E is obtained from this equation one may obtain the magnitude H(y, t) of the associated magnetic field strength from either Eq. (9) or (10): Z 1 ∂E H(y, t) = − dt + Ht (y), (12) µ ∂y ¶ Z µ ∂E H(y, t) = − ² + σE dy + Hy (t). (13) ∂t Every effect has a sufficient cause that occurred a finite time earlier The words “that occurred a finite time earlier” are a contribution of information theory to the causality law. We will discuss presently why the word “finite” is necessary. Even though half a century has passed since this form of the causality law was recognized as necessary, we are not likely to find it in this form in books on natural philosophy or physics. For a simple explanation of the problem of Max- Let a boundary condition E(0, t) be given and an initial condition E(y, 0). One may then solve Eq. (11) for E = E(y, t). Substitution of E into Eqs. (12) and (13) should then yield H = H(y, t) with undetermined functions Ht (y) and Hy (t). These two functions are determined by the requirement that Eqs. (12) and (13) must yield the same function H(y, t). All this assumes, of course, that a solution E(y, t) of Eq. (11) and an associate solution H(y, t) of Eqs. (12) and (13) exists. E(0, t) = E0 S(t) = 0 = E0 20 for t < 0 for t ≥ 0 (1) Электромагнитные Явления, Т.7, №1 (18), 2007 г. Nonsinusoidal Waves, Modified Maxwell Equations, Dogma of the Continuum No problem of existence seems to have been encountered if the time variation of E(y, t) was that of a periodic or an everywhere analytic function. In these cases there is no boundary condition like that of Eq. (1), or t < 0 and t ≥ 0 are replaced by meaningless conditions t < −∞ and t ≥ −∞. A problem arose when E had a boundary condition like Eq. (1) that is required for signals. Periodic or everywhere analytic functions can be substituted for signals only if causality is not required. The proof that Eqs. (11)–(13) yield generally no solution for signals was helped by luck. There are at least some cases of signals for which Eq. (11) yields a solution for the magnitude E(y, t) of the electric field strength but Eqs. (12) and (13) cannot be reconciled and yield a contradiction. It is difficult to prove that a differential equation does not have a solution, since a proof that holds only for Fourier’s method of standing waves would be meaningless. It is much easier to prove that Eqs. (12) and (13) for H(y, t) contradict each other for a solution E(y, t) of Eq. (11). We must again refer to the literature for the proof [2,p. 14]. This proof does not only hold for the step function excitation of Eq. (1) but any increase E0 tn with n = 0, 1, 2,. . . for t ≥ 0 is included. When investigating what could be done to overcome the contradiction of Eqs. (12) and (13) it was found on strictly mathematical reasoning that the following modification of Maxwell’s equations would work: ∂D + ge , ∂t ∂B − curl E = + gm , ∂t div D = ρe , curl H = div B = 0 or div B = ρm , D = ²E, (14) (15) (16) (17) (18) B = µH, ge = σE, (19) (20) gm = sH. (21) Here gm is a magnetic current density with dimension V/m2 , ρm is a magnetic charge density with dimension Vs/m3 , and s is a magnetic conductivity with dimension V/Am. The important new term in Eqs. (14)–(21) is the magnetic current density gm . It is not part of our usual electromagnetic theory. The term ρm in Eq. (17) is not important since the theory will work both for ρm 6= 0 or ρm = 0. It took several years before the physical significance of the magnetic current density gm was recognized. The electric current density ge in Eq. (14) has always stood for at least two types of electric current densities. The first is the current density due to moving positive or negative charges, which we call monopole currents. Such a current cannot flow in vacuum or in the dielectric of a capacitor, which is Electromagnetic Phenomena, V.6, №1 (18), 2007 an insulator for monopole currents. Since we observe electric currents flowing through capacitors we have always admitted the existence of electric currents other than monopole currents. The proper names for them are dipole, quadrupole, and higher order multipole currents. These names based on atomistic thinking were not in common use in Maxwell’s days. Let us explain them with the help of Fig. 1. Fig. 1. Current carried by independent positive and negative charges (a). Dipole current due to an induced dipole (b). Dipole current due to orientation polarization of inherent dipoles (c). On the left we see in Fig. 1a a negative and a positive charge carrier between two metal plates with positive and negative voltage. The charge carriers move toward the plate with opposite polarity. An electric monopole current is flowing as long as the charge carriers move. In Fig. 1b we see how an induced dipole can produce a dipole current. A neutral particle, such as a hydrogen atom, is not pulled in any direction by voltages at the two metal plates. However, the positive nucleus moves toward the plate with negative voltage and the negative electron toward the plate with positive voltage. A restoring force, symbolized by a coil spring, will pull nucleus and electron together once the voltage at the plates is switched off. A dipole current is flowing as long as the positive and the negative charge carriers are moving either apart or back together again. This simple model becomes more complicated if we say that the probability density function for the location of the electron looses its spherical symmetry and is deformed into the shape of an American football with the nucleus off-center in the elongated direction. We note that a dipole current can become a monopole current if the field strength between the plates exceeds what is usually referred to as the ionization field strength. One cannot tell at the beginning whether a dipole current will become a monopole current or not, since this depends not only on the magnitude of the field strength but also on its duration. As a result a term in an equation representing a dipole current must be so that it can change to a monopole current. Vice versa, a term representing monopole currents must be so that it can change to a dipole current, since two particles having charges with opposite polarity may get close enough 21 Henning F. Harmuth to become a neutral particle. The term ge in Eq. (12) satisfies this requirement. Most molecules, from H2 O to barium-titanate, are subject to electric orientation polarization in addition to induced polarization of their atoms. Figure 1c shows charges with opposite polarity at the ends of rigid rods. A positive and a negative voltage applied to the metal plates will rotate these inherent dipoles to line up with the electric field strength. Dipole currents 2iv are carried by each rotating dipole. There are also dipole currents 2ih perpendicular to the field strength but they compensate if there are counterrotating dipoles as shown. Only the currents in the direction of the field strength will remain observable macroscopically if there are many dipoles with random orientation. Let us advance from the electric current density ge in Eq. (14) to the magnetic current density gm in Eq. (15). Just like the electric current density it can represent monopole, dipole, quadrupole and higher order multipole magnetic current densities. There are serious theoretical arguments for the existence of magnetic charges and thus of magnetic monopole currents. However, the experimental proof for their existence is not widely accepted. This is of no importance here. If there are magnetic charges we must use div B = ρm in Eq. (17), otherwise div B = 0 applies. The existence of magnetic dipoles is not disputable. Furthermore, two magnetic dipoles can be combined to a quadrupole, four to an octupole, etc. Any electric power generator contains a rotating magnetic multipole. Hence, gm in Eq. (15) is required to represent dipole currents produced according to Fig. 1c by rotating magnetic dipoles. The term gm is no longer based on mathematics but on physics. 2. 2. The interval in which P is known to be is divided into two equal intervals by the mark 0.11 of the ruler. Is P to the right of this mark? No, 0. 3. The interval in which P is known to be is divided into two equal intervals by the mark 0.101 of the ruler. Is P to the right of this mark? Yes, 1. Finite Resolution of Space, Time and Amplitude Measurements We have so far applied two principles of information theory to electrical communications and physics. A third principle states: Information is always finite This claim implies that any distance or time measurement must have a finite resolution ∆x, ∆t but also that any observed distance X or time T must be finite. Furthermore, any observable amplitude A must be finite and have a finite resolution ∆A. This principle excludes the infinite as well as the infinitesimal dx, dt, dA from any experimental science. Most scientists will agree that they do not expect to ever observe anything infinite or infinitesimal. But we shall discuss the principle in more detail to obtain quantitative results. 22 How much information do we acquire by measuring the location of a point—or the distance between two points—by means of a ruler? Refer to Fig.2 for an explanation of how the location x of a point P relative to a ruler can be expressed in bit [4,Sec.2.1]. Figure 2a shows a ruler with arbitrary finite length X. The ruler is marked at its beginning (0) and its end (1). We assume that P is between these two points. The marking of the ruler is changed to that of Fig. 2b. There is a mark 0.0 on the left, a mark 0.1=1/2 in the middle, and a mark 1.0 on the right; binary notation is used for the marks. The point P is located in the interval 0.1 < x/X < 1.0. We say it is located at x/X = 0.1 and that we have 1 bit of information about its location. In Fig. 2c the ruler has the marks 0.00, 0.01=1/4, 0.10=2/4, 0.11=3/4. and 1.00=4/4. The point P is in the interval 0.10 < x/X < 0.11 and we have 2 bit of information about its location. Finally, Fig. 2d shows the rulers marked 0.000, 0.001=1/8, 0.010=2/8, . . . . The point P is located in the interval 0.101 < x/X < 0.110 and we have 3 bit of information. The information in bit is measured by the number of binary digits to the right of the binary point required to denote the equally spaced marks on the ruler. We turn to the measurement of information by yes-no decisions as is appropriate for the calculus of propositions. The following questions have to be asked: 1. The interval in which P is known to be is divided into two equal intervals by the mark 0.1 of the ruler. Is P to the right of this mark? Yes, 1. Let the number of intervals of the ruler in Fig. 2 increase from 23 to 24 , 25 , . . . . The information increases to 4, 5, . . . bit. It is immediately apparent that the number of bit can be very large, but it must always be finite. Otherwise we would have to write— or generally to transmit, process or store—an infinitely long string of binary digits 0 and 1. Consider examples where the concept of finite information leads to. We have already mentioned that the usual causality law “Every effect has a sufficient cause” is augmented by “that occurred a finite time earlier”. For another result consider our usual assumption of a space-time continuum. The straightforward proof of a continuum of physical space and time would be the observation of events at two spatial points x and x + dx or two times t and t + dt. What is physically possible are observations at x and x + ∆x or t and t + ∆t, where ∆x and ∆t may Электромагнитные Явления, Т.7, №1 (18), 2007 г. Nonsinusoidal Waves, Modified Maxwell Equations, Dogma of the Continuum 3. Electromagnetic Signals in Seawater A strong motivation for the investigation of the transmission of rectangular pulses in seawater was provided by the attempts to increase the transmission rate of of electromagnetic pulses to and from deeply submerged submarines [1]. Maxwell’s equations (2)– (5) did not yield a convergent solution for a planar wave, but the modified equations (14)–(21) did. Instead of Eqs. (9) and (10) we obtain from Eqs. (14)– (21): Fig. 2. Information about the location of the point P for x < X. (a) 1 bit, (b) 2 bit, (c) 3 bit, (d) 4 bit. be very small but must be finite. Any finite interval ∆x, ∆t can be subdivided into nondenumerable many subintervals dx, dt, which means we are a long way from a mathematical continuum. If we want to use finite differences ∆x, ∆t instead of differentials dx, dt we must use the calculus of finite differences instead of differential calculus. This is a true generalization since no fixed values for ∆x, ∆t are specified at the beginning of the calculation. No interesting new results have been obtained in non-relativistic quantum mechanics, but this is quite different in the relativistic theory. When solving for the eigenfunctions of a difference equation in relativistic quantum physics we typically get wellbehaved functions if the spatial resolution ∆x is large enough, but sequences of random numbers for small values of ∆x. This is how the calculation represents the Compton effect. The theory goes beyond Heisenberg’s uncertainty relation since it puts a lower limit on ∆x rather than on the product ∆x∆p, and it allows for the effect of electromagnetic fields on ∆x. Differential calculus made the physics of space and time a branch of mathematics. In complete opposition, the calculus of finite differences uses mathematics as a tool for physics. Consider elementary particles within the framework of differential calculus. We choose the mathematical method first and must then match the physical situation to the mathematical method. We do so by defining elementary particles to be point-like to avoid giving them any spatial features. When we start with finite differences ∆x, ∆t due to information theory or experimental experience, we state first the physical situation and then choose the calculus of finite differences as a suitable method to describe the physical situation. We must demand that an elementary particle is smaller than an arbitrarily small but finite distance ∆x to avoid observable spatial features. The difference theory offers the better choice. ∂H ∂E +µ + sH = 0, ∂y ∂t ∂E ∂H +² + σE = 0. ∂y ∂t (22) (23) These two equations describe the propagation of a planar EM wave in a medium characterized by the scalars ², µ, σ, and s. Instead of Eq. (11) we obtain from Eqs. (22) and (23): ∂2E ∂E ∂2E − µ² 2 − (µσ + ²s) − sσE = 0. 2 ∂y ∂t ∂t (24) Equations (12) and (13) are replaced by: µ ¶ Z ∂E st/µ 1 −st/µ H(y, t) = e − e dt + Ht (y) , (25) µ ∂y ¶ Z µ ∂E + σE dy + Hy (t) (26) ² H(y, t) = − ∂t Equations (13) and (26) are equal. The difference between Eqs. (11),(12) and (24),(25) is strictly caused by s. It is important to solve Eq. (24) with s and make perhaps the transition s → 0 at the end of the calculation, since the differential equation (24) yields a solution different from that of the differential equation (11). Furthermore, the s in Eq. (25) assures that we get a convergent expression for the associated magnetic field strength. For the step function excitation of Eq. (1) we obtain from Eq. (24) in normalized notation for t and y the following result [2,Sec. 2.1;5,Sects. 1.1,1.2;6,Ch. 5]: E(ζ, θ) = EE (ζ, θ) ½ · Z1 µ 2 −θ = E0 1 − e ch(1 − η 2 )1/2 θ π 0 + sh(1 − η 2 )1/2 θ (1 − 1/2 η2 ) ¶ sin ζη dη + η cos(η 2 − 1)1/2 θ ¶ ¸¾ 1 2 + Z∞ µ 1/2 sin(η − 1) (η 2 − 1) 1/2 θ sin ζη , dη η θ = σt/2², ζ = (µ/²)1/2 σy/2. (27) Electromagnetic Phenomena, V.6, №1 (18), 2007 23 Henning F. Harmuth Fig. 3. Plots of EE (ζ, θ)/E0 according to Eq.(27) for ζ = 0, 1,. . . , 10 in the time interval 0 ≤ θ ≤ 60. Plots of EE (ζ, θ) are shown in Fig. 3. The solution for H(ζ, θ) according to Eqs. (25) and (26) is too complicated to write it here. We show instead plots of H(ζ, θ) = HE (ζ, θ) for ζ = 0, 1, 2, 3 in Fig. 4 for the time interval 0 ≤ θ ≤ 25. Fig. 5. Electric field strengths EE,R (ζ, θ)/E0 of a rectangular pulse according to Eq. (28) distorted according to Fig. 3 for ∆Θ = 2 and propagation distances ζ = 1, 3, 5, 7, 9 in the time interval 0 ≤ θ ≤ 30. To obtain a more general concept of propagation velocity we include a detection process. The function ζ = 1 of Fig. 5 is plotted once more in Fig.6a. We want to detect it by cross-correlation with its truncated equal shown in Fig.6b. The cross-correlation of the functions of Figs. 6a and b is shown in Fig. 6c. It has a maximum at the time Θmax . This maximum if of importance in the presence of noise. We use it to define a propagation velocity vS . The following three steps connect Θmax and vS : propagation time of the beginning of a pulse td = y/c, (30) time of the maximum of the cross-correlation function in Fig. 6c Fig. 4. Normalized magnetic field strengths HE (ζ, θ)Z/E0 associated with the electric field strengths EE (ζ, θ) of Fig. 3. The normalized time θ and distance ζ are defined in Eq. (27). Two step functions according to Eq. (1) with a delay of ∆Θ and the delayed function amplitude reversed yield a rectangular pulse: EE,R (0, θ) = E0 [S(θ) − S(θ − ∆Θ)] ∆Θ = σT /2². (31) propagation velocity vS vS = y td + Tmax y/td c = = 1 + Tmax /td 1 + Tmax /tD = c/(1 + Θmax /ζ). (32) (28) (29) Plots of such pulses for ∆Θ = 2 distorted according to Fig. 3 are shown in Fig. 5 for the propagation distances ζ = 1, 3, 5, 7, 9. We recognize that the leading edge of a rectangular pulse always propagates with the velocity c of light, but the trailing edge requires an infinite time to decay to zero. This implies a propagation velocity zero for the trailing edge. The group velocity has obviously no meaning in a distorting medium. 24 Θmax /ζ = Tmax /td , In the presence of thermal noise we may crosscorrelate the truncated sample function of Fig. 6b with the noise and calculate the probability pe of obtaining an output that would reduce the peak amplitude of the cross-correlation function in Fig. 6c to zero or less. This probability pe is the error probability for binary transmission. This error probability depends on the values of ∆Θf or Θmax in Fig. 6. With the help of Eq. (31) we may use the relative velocity vS /c instead of Θmax for various distances ζ. For the pulse duration Электромагнитные Явления, Т.7, №1 (18), 2007 г. Nonsinusoidal Waves, Modified Maxwell Equations, Dogma of the Continuum Fig. 6. Distorted rectangular pulse received at the distance ζ from the transmitter (a). Truncated sample signal stored at the receiver (b). Cross-correlation function between received distorted rectangular pulse and sample signal (c). ∆Θ = 2 of Fig. 5 and the distance ζ = 5 we obtain the error probability pe as function of the signal-tonoise ration PS /PN with vS /c as parameter as shown in Fig. 7. For vS /c = 0.526 we need a signal-to-noise ratio PS /PN = 12.5 dB to achieve an error probability of 10−6 . For lower relative velocities vS /c = 0.472, . . . , 0.198 we need an ever lower signal-to-noise ratio, but there is a diminishing return as we approach 0.198. The propagation velocity is now linked not only to the signal distortions but also to the signal-to-noise ratio. To provide a comparison we show in Fig. 7 plots of the error probability versus the signal-to-noise ratio for the conventional coherent phase and frequency shifting PSK and FSK. These two plots assume sinusoidal pulses propagating in a loss-free medium and additive thermal noise. We have seen so far that the transmission of information by initially rectangular electromagnetic pulses in seawater is rather complicated if detection by cross-correlation according to Fig.6 is used. However, one can give a simple answer to the question “How far Electromagnetic Phenomena, V.6, №1 (18), 2007 Fig. 7. Error probability pe as function of the signal-tonoise ratio PS /PN for the distance ζ = 5 and various values of the relative propagation velocity vS /c = 0.526, . . . , 0.198 for the pulse duration ∆Θ = 2. can we transmit information if we can tolerate a signal energy attenuation α and use pulses of duration ∆T .” For seawater we have the following constants: . . µ = µ0 = 4π × 10−7 Vs/Am, σ = 4 A/Vm . . ² = 80²0 = 7.1 × 10−10 As/Vm For the calculation we have to refer to the literatur [6,Sec.5.5]. The plot of Fig. 8 is obtained. It shows the distance y on the horizontal scale and the attenuation α of the signal energy on the vertical scale. A pulse of duration ∆T = 10 ms will be attenuated α = 20 dB over a distance of 700 m. Teletype pulses are usually 20 ms long. They can readily be transmitted with an attenuation of somewhat more than 10 dB over that distance. We cannot discuss the many other applications of nonsinusoidal waves but must instead refer to a few representative books and papers [7–15]. 4. Modified Maxwell Equations and Renormalization It is well known that the electric and magnetic field strengths E and H of Eqs. (2) and (3) can be expressed 25 Henning F. Harmuth form [3,Sec. 3.1]: ∂φe ∂Amx + = 0, ∂t ∂x ∂φe ∂Amy + = 0, ∂t ∂y ∂φe ∂Amz + = EE , ∂z ¶ µ ∂t ∂Amy ∂Amz − c/Z = HE , ∂y ∂z µ ¶ ∂Amz ∂Amx − c/Z = 0, ∂z ∂x µ ¶ ∂Amx ∂Amy c/Z = 0. − ∂x ∂y Fig. 8. Energy attenuation α(y, ∆T ) as function of the propagation distance y in seawater for originally rectangular pulses of duration ∆T . Note that electromagnetic pulses with a duration of 10 to 100 ms may be transmitted over distances of several kilometer without undue attenuation by absorption. with the help of a vector potential Am and a scalar potential φe : ∂Am − grad φe , ∂t c H = curl Am . Z E= (33) (34) These equations are of fundamental importance for quantum field theory. It turns out that Am is not determined or is represented by a divergent integral if the exciting field strength is a signal that is zero before a finite time t < 0. We have already pointed out that a signal or a “causal function” must be zero before a finite time in order to satisfy the causality law. One will suspect that a lack of convergence of Am is a sufficient cause for at least some of the divergences that plague quantum field theory and that are sidestepped currently by means of renormalization. The modified Maxwell equations do not have a problem with convergence and do not need renormalization. To show the problem of Eqs. (32) and (33) for signals wse use an electric step function excitation according to Eq. (1). Let the electric field strength E of Eq. (32) be linearly polarized and point in the direction z of a right-handed Cartesian coordinate system. The magnetic field strength H of Eq. (33) shall point in the direction x: E = EE (ζ, θ)ex = EE ex , H = HE (ζ, θ)ex = HE ex . (35) (36) Equations (32) and (33) are rewritten in component 26 (37) (38) (39) (40) (41) (42) Since E and H in Eqs. (34) and (35) represent a planar wave propagating in the direction y all derivatives with respect to x or z are zero: ∂φe ∂φe = = 0, ∂x ∂z ∂Amy ∂Amy ∂Amx ∂Amx = = = = 0. ∂z ∂x ∂z ∂x (43) (44) Using the normalized notation θ and ζ of Eq. (27) we obtain: ∂Amx = 0, ∂θ µ ¶ ∂φe ∂Amy + = 0, (Zσ/2) c ∂θ ∂ζ ∂Amx (Zcσ/2) = EE (ζ, θ), ∂θ ∂Amx = HE (ζ, θ), (cσ/2) ∂ζ ∂Amx = 0. (cσ/2) ∂ζ (45) (46) (47) (48) (49) We see from Eqs. (44) and (48) that Amx is independent of ζ and θ, which means it is a constant. Equations (45)–(47) yield: Z ∂Amy φe = −c dζ, (50) ∂θ Z 2 Amz = EE (ζ, θ)dθ, (51) Zcσ Z 2 Amz = HE (ζ, θ)dθ. (52) cσ In Eq. (49) we may choose any value we want for Amy and get a defined value of φe . This freedom to choose reflects the fact that a variety of gauges for Am yield the same field strengths E and H. In Eq. (50) we may substitute Eq. (27) for EE (ζ, θ) and obtain a defined result for Amz according to the plots of Fig. 3. But in Eq. (51) we must substitute a divergent integral for HE (ζ, θ) [3,Eq. (6.2-51)] which leaves Amz undefined. Hence, Eqs. (50) and (51) contradict each Электромагнитные Явления, Т.7, №1 (18), 2007 г. Nonsinusoidal Waves, Modified Maxwell Equations, Dogma of the Continuum other. Equations (32) and (33) must be wrong. More details may be found in the cited literature. It has been known at least since Weisskopf and Wigner [16,17] that the quantization of the electromagnetic field based on Eqs. (32) and (33) leads to divergent integrals but it was generally believed that the problem was caused by the quantization process rather than by Maxwell’s equations. We repeat the calculations of Eqs. (32)–(51) for the modified Maxwell equations of Eqs. (14)–(21). The field strengths E and H are now represented by two more symmetric functions: ∂Am − grad φe , ∂t c ∂Am H = curl Am − − grad φm . Z ∂t E = −Z curl Ae − (53) (54) Here Ae is an additional electric vector potential due to magnetic (dipole) currents and φm an additional magnetic scalar potential [3,Sec. 1.6]. For ρm = 0 in Eq. (17) we get φm = 0. Using again Eqs. (34) and (35) we obtain the following equations instead of Eqs. (36)– (41): ¶ ∂Aey ∂Amx ∂φe ∂Aez − − − −Zc ∂y ∂z ∂t ∂x µ ¶ ∂Aez ∂φe ∂Aex ∂Amy − − −Zc − ∂z ∂x ∂t ∂y µ ¶ ∂Aey ∂Aex ∂Amz ∂φe −Zc − − − ∂x ∂y ∂t ∂z µ ¶ ∂Amy ∂φm ∂Aex c ∂Amz − − − Z ∂y ∂z ∂t ∂x µ ¶ ∂Amz ∂φm c ∂Amx ∂Aey − − − Z ∂z ∂x ∂t ∂y µ ¶ ∂Amx ∂φm c ∂Amy ∂Aez − − − Z ∂x ∂y ∂t ∂z µ = 0, (55) = 0, (56) = EE , (57) = HE , (58) = 0, (59) = 0. (60) The conditions of Eqs. (42) and (43) for a plane wave propagating in the direction y are augmented for φm and Ae : ∂φm ∂φm = = 0, ∂x ∂z ∂Aey ∂Aey ∂Aex ∂Aex = = = = 0. ∂z ∂x ∂z ∂x (61) (62) ∂Aez ∂Amx + = 0, ∂ζ ∂θ ∂Amy ∂φe c + = 0, ∂θ ∂ζ µ ¶ ∂Aex ∂Amz Zcσ Z − = EE (ζ, θ), 2 ∂ζ ∂θ (66) (67) (68) Equations (63) and (66) yield φe and φm as functions of Amy and Aey : Z ∂Amy φe = −c dζ, (69) ∂θ Z ∂Aey φm = −c dζ. (70) ∂θ Equations (62) and (67) yield: Z Z 1 1 ∂Amx ∂Amx Aex = − dζ = − dθ, Z ∂θ Z ∂ζ Z Z ∂Aez ∂Aez Amx = −Z dθ = −Z dζ. ∂ζ ∂θ (71) (72) Differentiation of Eq. (64) with respect to θ and of Eq. (65) with respect to ζ yields the one-dimensional, inhomogeneous wave equation for Amz µ ¶ ∂ 2 Amz ∂ 2 Amz 2 ∂HE ∂EE (73) − = + ∂ζ 2 ∂θ2 Zcσ ∂θ ∂ζ and Aex as function of Amz : Aex 1 = Z Z µ ¶ ∂Amz 2 + EE dζ ∂θ Zcσ ¶ Z µ 2 1 ∂Amz − HE dθ. = Z ∂ζ cσ (74) Instead of Eqs. (72) and (73) we may also derive the following relations from Eqs. (64) and (65): µ ¶ ∂ 2 Aez ∂ 2 Aez 2 ∂HE 1 ∂EE − = + , (75) ∂ζ 2 ∂θ2 Zcσ Z ∂ζ ∂θ ¶ Z µ cσ ∂Aex + EE dθ Amz = Z ∂ζ 2 ¶ Z µ ∂Aex 2 =Z + HE dζ. (76) ∂θ Zcσ Equations (72) and (74) are inhomogeneous wave equations with one spatial variable. Their solution is known [18,vol. II,Cha. VII, § 1, Sec. 174, Eq.95]: Using the normalized variables θ and ζ we get: Z µ ¶ cσ ∂Amz ∂Aex = HE (ζ, θ), −Z 2 ∂ζ ∂θ ∂φm ∂Aey + = 0, c ∂θ ∂ζ ∂Aez ∂Amx +Z = 0. ∂ζ ∂θ (63) (64) (65) Electromagnetic Phenomena, V.6, №1 (18), 2007 1 Amz (ζ, θ) = − Zcσ Zθ · 0 0 ζ+(θ−θ )µ Z ∂EE (ζ 0 , θ0 ) ∂θ0 ζ−(θ−θ 0 ) ¶ ¸ ∂HE (ζ 0 , θ0 ) dζ 0 dθ0 , +Z ∂ζ 0 (77) 27 Henning F. Harmuth Aex (ζ, θ) = Amz (ζ, θ)/Z. (78) Equation (76) can be brought into a form that shows that the integrals exist and that Amz as well as Aex can be plotted, which is the most convincing proof that they are actual rather than formal solutions [3,Sec. 3.1]. This concentrated presentation was intended to show that the usual Eqs. (32) and (33) lead to a contradiction if they are applied to electromagnetic signals, which are the only electromagnetic waves that we can produce or observe. Equations (52) and (53) overcome this shortcoming. The detailed proof may be found in the repeatedly referenced book [3], which is mathematically rather demanding. Easier to read texts should eventually become available but this may take some time. The next step is to show that the use of Eqs. (52) and (53) overcomes the problem of infinite ‘zero-point energy’. We cannot compress the calculations from the many pages required in a book [3] to the few pages available here. Hence, we attempt a verbal explanation of the physics involved. It is usual to derive the Hamilton form of Maxwell’s equations for steady state solutions only. This is acceptable since Maxwell’s equations generally do not have solutions that satisfy the causality law and thus they can be valid for the steady state only. If infinitely extended periodic sinusoidal waves are used to represent a steady state solution one must observe that such solutions represent wave with infinite energy. Hence, they are outside both the causality law and the conservation law of energy. One should not be surprised if this leads to problems with infinite energy. It is possible to derive solutions of Maxwell’s equations that satisfy the conservation law of energy but not the causality law. Such solutions, based on the Gaussian 2 bell function e−θ , have been developed [19,20]. They could be extended to the complete orthogonal system of parabolic cylinder functions, but few people seem to be aware of these non-sinusoidal solutions of Maxwell’s equations. If one derives the Hamiltonian form in the conventional way from Eqs. (2) and (3) one obtains a steady state theory that by definition applies to the whole interval −∞ < t < ∞ or −∞ < y = ct < ∞. The following equation is obtained for the sinusoidal waves that represent any electromagnetic wave in this interval [3, Eq.(4.4–9)]: µ ¶ 1 Ef = Em,kλ = 2πf ~ n + , n = 0, 1, 2, . . . (79) 2 For n = 0 one obtains the infinite zero-point energy of the electromagnetic field in vacuum since there are infinitely many sinusoidal oscillations with various frequencies f . Instead of a discussion a quote is cited from a renowned book [21,§ 3. Photons, second paragraph]: But already in this state each oscillator 28 has the ‘zero-point energy’ 2πf ~/2, which differs from zero. The summation over the infinitely many oscillators yields an infinite result. We meet here one of the ‘divergences’ that the existing theory contains because it is not complete and not physically consistent. Becker has the following to say:The ground state represented by [n = 0] corresponds to vacuum; it still contains zero-point vibrations, however, as in the case of the linear oscillator. Since we are dealing with an infinite number of oscillators the mean square values of the field strengths E2 , H2 , must also be infinitely great. A completely satisfactory treatment of this anomaly does not yet exist [22, vol. II, § 52, footnote p. 311]. The anomaly is directly associated with divergent integrals: This divergence has for long been an insuperable difficulty of quantum theory; it has not yet been completely overcome, but has been ingeniously circumvented through the concept of mass renormalization of the electron (Kramer, 1945). [22, vol. II, § 53, p. 319, small print following Eq.(5.3–9)]. Let us turn to the modified Maxwell equations (14) and (15). We have shown that they permit solutions that satisfy the causality law, in particular for the step function excitation of Eq.(1). Part of this excitation is that the time t → −∞ is eliminated and replaced by a finite time that we chose to be t = 0. A negative infinite time is not observable. In Section 2 we listed the principle “Information is always finite” and stated that any distance X or time T must be finite. Our calculation still permits the time t → ∞. We correct this violation now by introducing an arbitrarily large but finite time T . An electromagnetic wave in the intervals 0 ≤ t ≤ T or 0 ≤ y = ct ≤ cT may then be represented by a Fourier series rather than by a Fourier integral. We also assume electric and magnetic field strengths with finite amplitudes that will produce waves with finite energy. We obtain the equation Eκ = Eκn = 2πκ~ T µ n+ 1 2 ¶ , n = 0, 1, 2, . . . (80) instead of Eq. (78) [3,Eq.(4.4–8)]. Equations (78) and (79) look very similar since κ = 1, 2, . . . is the number of sinusoidal periods in the time T and we might write f = κ/T . But κ is the number of periods of sinusoidal pulses of duration T while f is the frequency of an infinitely extended sinusoidal function in the whole interval −∞ < t < ∞. The energy Eκ of one photon increases proportionately to κ according to Eq. (79) but the number of photons with energy Eκ decreases like 1/κ3 , which makes the energy of all photons with energy Eκ decrease like 1/κ2 . The zero-point energy becomes finite. The infinite zero-point energy, the historically first divergency of quantum field theory, and the need for renormalization are eliminated. Электромагнитные Явления, Т.7, №1 (18), 2007 г. Nonsinusoidal Waves, Modified Maxwell Equations, Dogma of the Continuum 5. Small Finite Resolutions ∆x and ∆t Relativistic Quantum Physics If we divide arbitrarily large but finite time and space intervals T and X = cT into infinitesimal intervals dt and dx we obtain non-denumerable information about time and location. This is against the principles of information theory. We must replace dx and dt by arbitrarily small but finite differences ∆x and ∆t. The assumption of infinitesimal intervals dx, dt has strongly influenced our thinking about space and time. Space, time, or space-time are almost universally believed to be a mathematical continuum. We credit or blame Aristotle for this state of affairs, since he argued the concept of the continuum so convincingly but made no distinction between what is thinkable and what is observable [29, 30; book V(E) 3, § 5; book VI(Z) 1, 2; book VI(Z) 2 and 9]. As pointed out in Section 2, the straightforward proof of a continuum of physical space and time would be the observation of events at two spatial points x and x+dx or two times t and t+dt. What is physically possible are observations at x and x+∆x or t and t+∆t, where ∆x and ∆t may be very small but must be finite. Any finite interval ∆x, ∆t can be divided into non-denumerable many subintervals dx, dt, which means we are a long way from a mathematical continuum. Mathematics provides us with the tool of the calculus of finite differences instead of the differential calculus, if we want to use ∆x, ∆t instead of dx, dt. This is a true generalization since no particular values of ∆x, ∆t are specified at the beginning of the calculation. Such distinguished values may be encountered in the course of calculation, e.g. due to the Compton effect. The calculus of finite differences is not anywhere as well developed as the differential calculus. Only about ten mathematical books were published during the 20th century. Four of them are referenced [23–26]. The first question, whether the calculus of finite differences can deliver something the differential calculus cannot, is answered by the theorem of Hölder of 1887. It states that the gamma function can be defined by a simple difference equation Γ(x + 1) = xΓ(x) but by no algebraic differential equation. This implies that differential and difference equations define different classes of functions. As an example what the use of finite differences can do consider elementary particles. If we assume that differential calculus has to be used, we we are led to the conclusion that elementary particles must be point-like to avoid giving them any spatial features. Mathematics is chosen and physics is matched to it. With finite differences we assume that elementary particles must be smaller than the smallest resolvable Electromagnetic Phenomena, V.6, №1 (18), 2007 distance ∆x if they have no other observable spatial features. Looking for a mathematical method that matches the physical situation we find the calculus of finite differences. We do not need to belabor the point that matching mathematics to physics is the way to go. The replacement of dx, dt by finite differences ∆x, ∆t forces us to rethink some deeply ingrained concepts. Such familiar elements as straight or curved lines, surfaces, and spaces do not exist, at least not in the form we have become used to since grade school. For an explanation refer to Fig.9. There are 6 spatial markers 0, ∆x, . . . , 5∆x which we would usually say are located on a ruler. We want to avoid this expression since it makes one think there is something more than the marks. Concepts like “straight ruler” creep in. We could think of the marks as the crossings of two searchlight beams at night. Figure 9a shows a dashed line connecting the marks 0 to 5∆x. This line represents nothing more than a beam or ray of light that would pass along the marks if it were turned on. Fig. 9. One-dimensional, Euclidean, straight, discrete coordinate system with 6 marks 0, ∆x, . . . , 5∆x (a). The same 6 marks along a straight line defined by a light ray curved by gravitation (b). Coordinate system of b simplified for drafting (c). The point P is everywhere closest to the mark 3∆x. Let an airplane be observed closer to the mark 3∆x than to any other mark. It is shown as point P in Fig. 9a on the dashed line between 3∆x and 4∆x but it could be far away from this line as long as it is closer to the mark 3∆x than to any other mark. In a gravitational field the Planck relation hν = m0 c2 assures that the light beam or ray will be bended, e.g., along a circle as shown in Fig. 9b. It is easier to plot the dashed line not along a circle but to replace it by straight lines between the marks as shown in Fig. 9c. The extension of Fig. 9c to two dimensions is shown in Fig. 10. One could imagine that two or three searchlights for each one of the 36 marks could implement the marks in the sky at night. A more practical method would be to use a radar that 29 Henning F. Harmuth records observations at 36 azimuth and elevation angles at certain distances. We are, of course, not interested in actually implementing the coordinate systems discussed here. They serve for thinking and calculating. But if we calculate that something should be observed at a certain location and a certain time we must allow for practical and imagined restrictions on coordinate systems. Fig. 10. Extension of a coordinate system according to Fig.9c to two dimensions with 62 marks. The point P is closest to the mark 3∆x, 2∆y. Without the dashed lines in Fig. 10 it would be difficult to comprehend this illustration. Comprehension becomes a major issue when we advance from two to three dimensions. Even the dashed lines used in Fig.10 become useless. We overcome the problem by using rods in place of the dashed lines. Figure 11 shows the interval 0 ≤ x ≤ 2∆x, 0 ≤ y ≤ 2∆y of Fig.10 extended to three dimensions with the help of the interval 0 ≤ z ≤ 2∆z. To comprehend the illustration we may start at the bottom and recognize 9 marks with coordinates x = 0, ∆x, 2∆x and y = 0, ∆y, 2∆y. This is the level z = 0. At the level z = ∆z we recognize 9 very similar marks for x = 0, ∆x, 2∆x and y = 0, ∆y, 2∆y. Finally at the level z = 2∆z we discern a further set of 9 marks for x = 0, ∆x, 2∆x and y = 0, ∆y, 2∆y. The crossings of the bars in Fig. 11 permits us to visualize what is in front and what is behind. The ending of the rods in the spheres helps enhance this visualization. The large number of marks becomes a problem for higher dimensions. The n1 = 6 marks in Fig. 9c become n2 = 36 marks in Fig. 10, and they would have become n3 = 216 marks in Fig. 11 if we had not restricted n to 3 and obtained n3 = 27. Advancing to four dimensions, even n = 3 and n4 = 81 is too large. We must make do with n = 2 and n4 = 16. This corresponds to the interval 0 ≤ x ≤ ∆x 30 Fig. 11. Extension of the coordinate system of Fig.9c to three dimensions with 33 marks. The dashed lines of Figs.9 and 10 are replaced by solid double lines to help show what is in front and what is behind. A model of this coordinate system can be implemented in our usual three-dimensional physical space with 33 styrofoam spheres and 12 × 3 + 9 × 2 = 54 rods. in Fig.9c, the “square” 0 ≤ x ≤ ∆x, 0 ≤ y ≤ ∆y in Fig.10, and the “three-dimensional cube” 0 ≤ x ≤ ∆x, 0 ≤ y ≤ ∆y, 0 ≤ z ≤ ∆z in Fig.11. In Fig. 12 we show a “four-dimensional cube” with 0 ≤ x ≤ ∆x, 0 ≤ y ≤ ∆y, 0 ≤ z ≤ ∆z, 0 ≤ w ≤ ∆w. Let us first recognize the “three-dimensional cube” with the 8 corner points or marks 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111. This may take some time. Then we try to see the equal “cube” shifted by ∆w and having the 8 corner points or marks 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111. Congratulations, you have visualized a four-dimensional structure from a single two-dimensional projection. The standard in descriptive geometry is to use two two-dimensional projections for a three-dimensional structure. Only the extreme simplicity of the structure in Fig.12 permits us to get away with just one projection. A short reflection makes us recognize that the fourdimensional structure of Fig. 12 can be implemented with 24 = 16 styrofoam spheres and 24 + 23 = 24 rods of equal length in our “three-dimensional space”. The four-dimensional structure of Fig. 12 may Электромагнитные Явления, Т.7, №1 (18), 2007 г. Nonsinusoidal Waves, Modified Maxwell Equations, Dogma of the Continuum Fig. 13. Coordinate system with three spatial variables x, y, z moving as function of time from t = 0 to t = ∆t and to t = 2∆t. Note that the coordinate systems for t = 0 and t = ∆t correspond to the one of Fig. 12 if ∆w is replaced by ∆t and the curved lines by straight rods. Fig. 12. Extension of the coordinate system of Fig. 9c to four dimensions with 24 marks. The representation of Fig. 11 by rods is used again. One needs 24 styrofoam spheres and 12 × 2 + 8 = 32 rods to implement a model in our usual three-dimensional space. It can be represented unambiguously with one projection onto a two-dimensional plane. look somewhat too theoretical, but it represents the movement of a three-dimensional coordinate system. A look at Fig. 13 and some reflection reveal that the two coordinate systems for t = 0 and t = ∆t are in essence Fig. 12 with the coordinate axis w renamed t. The interpretation of Fig. 12 as a moving threedimensional coordinate system creates a strong motivation to extend Fig. 12 to five dimensions to show that we can produce structures with more than the usual four space-time dimensions in our “threedimensional physical space”. This is shown in Fig. 14. The illustration consists of Fig. 12 on the left with marks denoted from 00000 to 01111, and an equal drawing on the right with marks denoted from 10000 to 11111. The two halves of the picture are connected with 16 rods shown by single, wide, black lines. Beyond this introduction the reader will have to study the picture until it is as lucid as the single projection of a five-dimensional structure on a twodimensional plane can be. The structure of Fig.14 can be implemented in our physical space with 32 styrofoam spheres and 80 rods. Before we can give an explanation why a fivedimensional structure can be implemented in our Electromagnetic Phenomena, V.6, №1 (18), 2007 Fig. 14. Extension of the coordinate system of Fig. 12 to five dimensions with 25 marks. A modification of the presentation of Fig. 12 is used for the connection of the left half with the right half of the illustration. One needs 25 styrofoam spheres and 4×12+2×8+1×16 = 80 rods to implement a model in our physical space. It can be represented unambiguously — if not lucidly — with one projection on a two-dimensional plane. physical space with supposedly three dimensions we must make things worse by showing that twoand three-dimensional coordinate systems can be replaced by one-dimensional ones. Hence, the threedimensional physical space can be perceived as an onedimensional space. Refer to Fig.15a. It shows a usual two-dimensional coordinate system with two points A and B. The coordinate distances x = 5 − 2 = 3 and √ y = 5 − 1 = 4 yield the Pythagorean distance 32 + 42 = 5. The 64 coordinate point of Fig. 15a are shown again in Fig. 15b but the points are now numbered consecutively from 0 to 63. The variable y is 31 Henning F. Harmuth Fig. 15. Two-dimensional Cartesian coordinate system (a), its replacement by an one-dimensional coordinate system (b), and the generalization of the principle to coordinate systems with denumerable many points (c). eliminated and we have only one dimension. Most readers will consider Fig.15a more realistic than Fig. 15b for going from A to B. But there are exceptions that we cannot recognize if we have never seen Fig. 15b or an equivalent. For example, if one approaches Montenegro on the Adriatic Sea from the north one faces a steeply climbing road with 27 switchbacks. Figure 15b would be a simplified but quite applicable map for this road. A child being raised in a home at the location A and having to walk to a school at the location B will consider the distance measurement according to Fig. 15b more realistic than the one according to Fig. 15a. On the other hand, a child growing up on an island in the Aegean Sea will observe islands around it and learn that they can be reached by rowing or sailing in a straight line to them. When it later learns about the perpendicular directions east-west and north-south it may be predisposed to discover the Pythagorean theorem. Figure 15c shows a different way to connect the coordinate points of Fig. 15b. The coordinate x in Fig. 15b can have only a finite number of points or marks. In Fig. 15c both x and y can have a finite or denumerable number of marks. The illustration cannot be extended to non-denumerable marks, which means to the continuum. A possible extension of Fig. 15c to three dimensions is shown in Fig. 16. We need only one dimension to connect a finite or denumerable number of discrete points or marks in three dimensions. A short reflection shows that our result is not limited to three dimensions. As long as we have only a finite or a denumerable number of points or marks we can connect them sequentially by a line and thus reduce them to an one-dimensional structure. In Fig. 17 we show a two-dimensional coordinate 32 Fig. 16. Replacement of a three-dimensional, discrete Cartesian coordinate system with denumerable points by an one-dimensional coordinate system. Электромагнитные Явления, Т.7, №1 (18), 2007 г. Nonsinusoidal Waves, Modified Maxwell Equations, Dogma of the Continuum system based on Fig. 15c that replaces a threedimensional Cartesian coordinate system. Fig. 17. Replacement of a three-dimensional discrete Cartesian coordinate system with denumerably points by a two-dimensiona coordinate system based on Fig. 15c. According to our discussion we can use coordinate systems with 1, 2, 3, 4, . . . dimensions to describe what we can observe in “our three-dimensional space”. Some will be more lucid than others but this is not of primary importance here. If we can assign dimensions so freely they must be human inventions rather than physical features defined and forced on us by nature. But there is no doubt that we think we live in three spatial dimensions. What makes us think so? Immanuel Kant claimed the concepts of time and space are contained in us a priori. In today’s language one would say we have genes for these concepts. Although no such genes have been identified yet it is not impossible that they will be found eventually. But genes are generally developed by evolutionary selection of what works best for reproduction, self preservation being understood as a major contributor to reproduction. It is not clear how the concepts of an one-dimensional time and a three-dimensional space favor reproduction, particularly since many living things reproduce without the concept of dimension. But equal or at least compatible concepts of time Electromagnetic Phenomena, V.6, №1 (18), 2007 and space existed for Europeans, Japanese, Australian aborigines, Incas, Polynesians, etc. at the time they first met. It has been claimed that our concept of time comes from our observation of change. The most important change we observe is our own aging process due to its unique importance for us [4,Sec. 4.1]. Statements like “When I was a child. . . ”, “When I married. . . ”, “When my first child was born. . . ”, “When I prepaid my funeral expenses. . . ” are examples how we use our aging process as a scale of time. Galilee used the human pulse for what were short-time measurements in his time. The pendulum clock, the quartz clock, and the cesium clock obscured the original human clock from 1600 on. It is easy to see how people world wide developed the same concept of time. For the possible origin of our concept of space consider a child of about one year of age. It attempts to stand up on its feet, and learns that standing up requires an effort while falling down does not. Gravity teaches the child a first distinguished direction: updown. Soon after mastering standing-up the child begins to walk. The design of our legs teaches it a second distinguished direction: front-rear. The location of our arms teaches a third distinguished direction: right-left. By age 2 most children of all populations have mastered the most basic concepts of Euclidean geometry, without having any knowledge of the rest of the world. It seems reasonable to conclude that our customary three spatial dimensions are learned at a very young age by experience. The concept of time based on our aging process is learned from middle-aged parents and teachers by education at a much later time. The concept of the continuum made it almost impossible for us to progress beyond three spatial dimensions from the time of Aristotle on. Bolyai (1832), Lobaschevskii (1840), and Riemann (1854) succeeded in generalizing Euclidean geometry within a continuum theory, but the results are far removed from observation, Let us discuss in a light hearted way how the lack of gravity would have affected our development of a concept of space. Consider an octopus living in the sea. The effect of gravity is probably not noticed. When the octopus ejects a jet of water it moves and experiences a first distinguished direction: front-rear. The eye is in front, the arms in the rear. The design of the octopus helps with this discovery just as the design of our legs helps us. There is no obvious second and third distinguished direction, despite the eight arms. An octopus might consider the one-dimensional coordinate systems of Figs.15c and 16 as most practical. This discussion should show that many concepts of a phyiscal space and time are nothing more than concepts of differential calculus. The transition to 33 Henning F. Harmuth the calculus of finite differences does not substitute some cellular space for a continuous space. As long as we cannot observe ‘space’ but only things ‘in space’ we must concentrate on the observable things, their relative distances, their relative changes, etc. A more detailed and mathematical discussion may be found in two recent books [27,28]. 6. [3] Harmuth, H.F., Barrett, T.W., and Meffert, B. Modified Maxwell Equations in Quantum Electrodynamics. – Singapore: World Scientific. – 2001. [4] Harmuth, H.F. Information Theory Applied to Space-Time Physics. – Moscow: MIR. – 1989. (in Russian, transl. A. Patashinski and W.W. Gubarewa) – Singapore: World Scientific. – 1992. (in English) Summary Maxwell’s equations hold generally for the steady state only. The modified Maxwell equations incorporate magnetic dipole currents produced by rotating magnetic dipoles. They permit signal solutions, that start at a certain finite time and have finite energy. The investigation of distortions and propagation velocity in seawater became possible. The restriction of all time and spatial intervals to arbitrarily large but finite eliminated the infinite ‘zeropoint energy’ from quantum electrodynamics and the need for renormalization caused by it. The further demand that small space and time intervals ∆x, ∆t should be arbitrarily small but finite eliminates the infinitesimal intervals dx, dt and calls for the use of the calculus of finite differences instead of the differential calculus in relativistic quantum physics. One result is that elementary particles no longer have to be like mathematical points but only smaller than the smallest resolvable distance ∆x. The infinite selfenergies due to the infinitesimal distance dx disappear from physics. Acknowledgement I want to acknowledge the help I received from four editors over many years, since editors are as important as authors for the publication of ideas that contradict accepted ones: The late Ladislaus L.Marton (Academic Press), the late Richard B. Schulz (IEEE Transactions on Electromagnetic Compatibility), Peter W. Hawkes (Elsevier/Academic Press), and Myron W. Evans (World Scientific Publishers). Manuscript received January 20, 2006 [5] Harmuth, H.F. and Hussain, M.G.M. 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