Download Nonsinusoidal Waves, Modified Maxwell Equations, Dogma of the

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.
Propagation of Electromagnetic Signals. –
Singapore: World Scientific. – 1994.
[6] Harmuth, H.F., Boules, R.N., and Hussain,
M.G.M. Electromagnetic Signals: Reflection,
Focusing, Distortion, and Their Practical
Applications.
–
New
York:
Kluwer
Academic/Plenum Publishers. – 1999.
[7] Astanin, L.J. and Kostylev, A.A. Principles of
Utrawideband Radar Measurements. – Moscow:
Radio i Svyaz. – 1989.
[8] Grinev, A.Y., editor. Problems of Subsurface
Radio Location. – Moscow: Radiotekhnika. 2005.
[9] Immorev, I.J. Basic capabilities and features of
ultra wideband (UWB) radars // Radio Physics
and Radio Astronomy – 2002, – V. 7, №. 4. – P.
339–343.
[10] Kolchigin, N.N., Butrym, A.Y., and Kazansky,
O.V. Expanding slot array for wideband
pulse signals // Achievements of Present Day
Radioelectronics – 2005. – № 5. – P. 60–64.
[11] Krymsky, V.V., Bucharin, V.A., and Zalyapin,
V.I. Nonsinusoidal Electromagnetic Wave
Theory. – Chelyabinsk: ChGTU Press. – 1995.
[12] Masalov, S.A., Pochanin, G.P., and Kholod P.V.
UWB loop receiving and transmitting antenna;
in Problems of Subsurface Radio Location (A.Y.
Grinev ed.). – Moscow: Radiotekhnika. – 2005.
[13] Lukin, K.A., Masalov, S.A., and Pochanin, G.P.
Large current radiator with avalanche transistor
switch // IEEE Transaction on Electromagnetic
Compatibility – 1997. – V. 39, № 2. – P. 156–160.
[1] Merril, J. Some early historical aspects of project
Sanguine // IEEE Trans. Communications. –
1974. – V. COM-22. – P. 359–363.
[14] Pochanin, G.P. Problems and promising lines of
development of UWB ground penetrating radio
location // Second International Workshop on
Ultra Wideband and Ultra Short Impulse Signals
September 19–22, 61–66, Sevastopol, Ukraine.
[2] Harmuth, H.F. Propagation of Nonsinusoidal.
– New York. Electromagnetic Waves. Academic
Press: – 1986.
[15] Borisov,
V.V.
Electromagnetic
Fields
of Transient Currents. – St.Petersburg:
St.Petersburg University Press. – 1996.
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Nonsinusoidal Waves, Modified Maxwell Equations, Dogma of the Continuum
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