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The space-time foundation of quantum
physics Ó
Dangson Tran. All rights reserved Original idea 1999. This web version 2002 ABSTRACT: It
is shown that quantum phenomena are natural consequences of the resolution limits
in classical space and time. A discussion on methodology reveals that many mathematical
treatments of quantum phenomena are in fact classical methods. A new mathematical
approach for electromagnetic radiation is presented and leads to the same solution
given by the method of classical waves. Based on this result it is concluded that the
wave nature traditionally given to electromagnetic radiation is a case of mistaken
identity caused by similar mathematical descriptions. It is suggested that electromagnetic radiations
are fundamentally disturbances in space-time; which explains why they need no medium
to propagate. The same mathematical approach leads to the “Time Indifference Principle”,
which is believed to be the fitting solution for quantum puzzles such as the double-slit experiment with single photons.
Differentiating chaotic and quantum processes My goal in this article is to show that resolution limits of
classical space and time are not only necessary, but also natural in the
description of quantum phenomena. As a start, I will conceptually combine space
and time to the horizontal axis and plot the property of a physical entity
versus space-time. I will call such a plot the state function of the entity. If
for every point in space-time I only find a unique value on the state function,
I conclude that the entity behaves classically
and therefore can be successfully described by the method of classical physics.
The state function of a classical entity is shown on the left picture of figure
1.
It was discovered (first by Poincaré) that there are classical systems whose state functions
are extremely non-linear. In the last few decades Lorentz and others has
developed the investigation of this class of systems into one of the most
exciting fields of modern science, namely the Chaotic
theory. The center picture of figure 1 demonstrates the problem investigated by
this theory. The state function is so sensitive to small changes in space-time
that if the initial condition is slightly off, the system may diverge
dramatically, giving the false impression that the state function is
multi-valued (see state functions A and B of center picture, figure 1.) The
Chaotic theory remedies the situation by using smaller increments for
space-time. This action successfully separates the state functions and return
chaotic systems to classical physics, where they now belong to a special class characterized
by high to extreme levels of local non-linearity. I need to mention the Chaotic theory
because I have seen trade books on science written by well respected scientists
advocating the idea that quantum phenomena are fundamentally chaotic phenomena.
At a quick glance this argument seems to be sound because both classes of
phenomena have something to do with the resolution of space and time. However, when
scientists investigated quantum phenomena at reduced length and time scales,
they did not find the determinism expected of chaotic processes. In addition,
quantum systems do not behave erratically like chaotic systems at macroscopic scale. On the contrary, many
measurable (macroscopic) effects of quantum systems can be predicted to accuracy levels
unheard of in the previous history of science.
CONVENTIONAL CLASSICAL
Figure 2: Chaotic processes still
belongs to the classical realm, while quantum processes involve space and
time ranges out of the reach of classical physics. To settle the “chaotic question” once and for all, I need to ask
you to pay close attention to figure 2. Like I have mentioned in an earlier article1,
both space and time are considered to be indefinitely divisible in classical
physics. In experimental works, however, one can only gather data in non-zero
increments of space and time. In figure 2, I have used single digits to denote experimental space-time “points”. While the single experimental point 1 (see top left of
figure 2) may be sufficient in the analysis of a conventional classical
system; it may be necessary to separate it into multiple experimental
points (i.e., points 1, 2, and 3) to account for chaotic behavior of highly non-linear systems (top right of figure 2).
Please note that 1, 2, and 3 are still points in the classical sense. Quantum
phenomena, on the other hand, take place at scales that classical physics
cannot resolve. I have illustrated this point with the bottom picture of figure
2. The single digits (1 through 6) denote points in the classical sense, the double digits (12 through 56) denote space and
time scales that are the focus of quantum physics. I hope this has made it
clear that quantum phenomena can never be reduced to chaotic phenomena, which
are but special cases of classical phenomena. The reason
for resolution limits But why
resolution limits exist in the first place? To answer this question I have to
return to the “measurable” requirement of physics. I only know something exists
if it responds to my attempt to measure it by, say, giving me a signal on my measurement tool.
If the measurement tool shows me a strong and narrow signal at
position X, I am very certain that something exists at X. Strictly speaking,
since the signal must have a width of size Dx, I should say that something exist within X±Dx/2.
X+Dx/2 X X+Dx/2 Quantum signal Classical signal Resolution limit=Dx X
Figure 3: Classical signal is narrow
and can be considered as being confined to a single point (left). Quantum
signal is spread out and cannot be considered as being confined to a point
(right). If Dx is too small compared to the length scale I’m
interested in, I could say “For all practical purposes, something exists at X”.
This is a typical statement made by classical physics. Although such an overly precise
statement is incorrect because it implies that the entity is of zero size (i.e., Δx=0);
it is a good enough description of reality without undesirable consequences.
(See left picture of figure 3). But if Dx is significant compared to the length scale that I'm interested in,
ignoring it is no longer an option. I’m tempted to say “There is something
existing at X, but its existence is spilled over to its neighborhood, and the
total range is Dx.” But by making such a statement, I have shown a clear bias toward
the exact center of the signal, which doesn’t differ very much from its
immediate neighboring points. I’m therefore forced to say “There are different
levels of existence of the quantum thing within the confinement of Dx.” (See right
picture of figure 3). At this point,
classical physics leads me to a curious problem. Since the quantum thing is in
motion with velocity v, if I call the time corresponding to the leading edge
and trailing edge of the quantum thing tL and tT
respectively, I should get: tL = tT + Dx/v (2) The logic of
classical physics would force me to accept that the leading edge of the quantum
thing is formed first, and the trailing edge is formed
at time Dx/v later. Unfortunately,
this is not consistent with the right picture of figure 3, which shows the
quantum thing existing as an instantaneous whole. This leads me to the
realization that in order to treat quantum phenomena correctly by the method of
classical physics, I must consider Dx as if it is a unit block of space. Likewise I must
consider Dx/v as if it is a unit
block of time. As far as classical physics is concerned, Dx and
Dt
become the smallest units of space and time. In other words, classical
space has to be counted in multiples of Dx, and classical time multiples of Dt. Thus, I have
arrived at the conclusion that Dx is the resolution of classical space, and Dt = Dx/v is the resolution limit of classical time without having to take
the resolution limit postulate as a founding axiom or an ad hoc condition of quantum physics as I have done in an earlier
article3. Logic for
the apparent violation of classical physics within the quantum realm The
existence of classical space-time resolutions has astounding implications because when I
examine a resolution unit of space, I have to assume that within it classical time does
not change! Since all classical laws require the existence of time sequence, within any given quantum entity,
which may spread over light
years, all classical laws may be violated! With this realization, suddenly many
quantum puzzles become completely understandable. Let me cite a few of them: 1. A quantum
entity may appear to achieve superluminal speed (i.e., moving faster than
light, a situation prohibited by Einsteinian physics.) 2. A quantum
entity may penetrate a “prohibited barrier”. This is known as “quantum
tunneling”. 3. Energy
could be created or destroyed within certain time duration. Etc. I would like
to end this section with an important note regarding the law of “cause and
effect”, which has always been taken for granted in classical physics. Since
this law is based on classical space and (the order of) classical time, it is meaningless in the quantum
realm, regardless whether the locations under consideration are right next to
each other or billion of light years apart. I understand that this statement
seems to have widened the conflict between classical and quantum physics,
making it more difficult to reconcile their differences. However, I promise you
that eventually everything will fall in its proper place, because “cause-and-effect”
is only an apparent law. I plan to present an argument for this and other related effects in my future
book “The Science of Space-Time and Existence,” but you can get a glimpse
of my argument in the next section. Classical
analysis of quantum phenomena If all
classical laws may be violated in the quantum realm as I have claimed, then why physicists have
been able to arrive at many quantum results by methods such as wave equations,
S-matrices, sum-over-histories; which assume the validity of familiar classical
laws such as the law of conservation of energy and momentum? Also, since each
wavelength of an electromagnetic wave is a unit of classical space and
corresponds to a unit of classical time, isn’t it curious that the fine details
of electromagnetic waves have been successfully recorded as well-defined
functions of classical space and time? To answer these
puzzling questions and many others, it is necessary to take a closer look at
the nature of space and time. Looking back at figure 2, I see classical space
as a series of dots of zero size separated by spatial segments of Dx’s. Likewise classical time is like a series of dots of
zero size separated by time durations of Dt’s. It is obvious to me that nothing can happen within
these zero-sized dots. I deduce therefore that all physical events must take
root within the various units of Dx’s and Dt’s. This leads me to the conclusion that all physical
phenomena, including classical phenomena, are ultimately quantum phenomena! I anticipate
strong objections from many proponents of classical physics regarding the point
that I’ve just made. Their objections are understandable because, after all,
even quantum mechanics has to incorporate Minkowski-Einstein spacetime, which
is a classical model, in its calculations. This was one reason why I did not
want to introduce this article at this moment in time. But now since I have
started, I may as well complete my statement. I will claim that the
Minkowski-Einstein spacetime model –as it is today- is still incomplete; and in
its complete picture you will see that it, too, is rooted in the quantum realm.
I am working on connecting loose ends and planning to present my findings on this
very important subject in a not too distant future. Needless to say, I welcome
the possibility that someone else will beat me to the finish line. To avoid
possible confusion, I do need to differentiate my view on physics from that of the
theory of SED (Stochastic Electrodynamics.) To put in simple terms, SED holds
the view that quantum phenomena are background noises of classical phenomena. In
other words, according to SED all phenomena are fundamentally classical. You
can see that my view is the reverse of SED. This difference is very important conceptually,
because stochastic processes are not natural consequences of classical physics. By
starting with classical physics, SED has to treat stochastic process in an
ad hoc manner and therefore will most likely encounter internal consistency
at some point in its combination of the two theories. By starting with quantum
physics, it is very natural to incorporate classical physics later as
macroscopic manifestations of quantum phenomena without any risk of internal
inconsistency. Return to
the main topic, my immediate task is to explain why the “lawless” quantum
entities would conform to classical laws in methods
such as wave mechanics. My favorite analogy is a coin. It seems very lawless to
me. Even though I am its boss, when I really want it to turn up tails, it turns
up heads; and when I try to outsmart it by changing my choice to heads when it is
already in the air, it would tease me by turning up tails somehow. However, if
I toss this coin many times, it would obediently give me a very consistent
heads to tail ratio of almost 1 each and every time. More, if I toss many coins
like it, the ratio is also very close to 1. That is lawful! (If you are
a healthy skeptic, you probably would ask me if I am sure that this ratio will
work out every time. Actually there are conditions. For a full account, I
invite you to read one of my earlier articles, namely “Logic for the End of
Probability”, which is also posted at this website.) It is clear
to me, then, that each quantum event is like a coin toss. Thus, when I analyze
quantum events with the assumption that they would act lawfully (i.e. obey the
laws of classical physics), my analysis is applicable only to a great number of
quantum events. I think I’m justified in adding the strong statement that the
same analysis is completely meaningless for a single quantum event. This logic
forces me to conclude that methods such as wave equations, S-matrices, and
sum-over-histories are not
descriptions of individual quantum events as many quantum textbooks have led me
to believe. I will not elaborate too long on this point, as the textbook interpretation
is based on the theory of probability, and I already refuted the applicability of this
theory to physics on empirical grounds in an earlier article1. It is sufficient to state that all of the methods
that I have just mentioned are classical descriptions of collective behavior of
many quantum entities. I know that I’m going against the mainstream here; but
since I must choose between logic and fashion, my choice is logic. An example
of classical solution of quantum problem This time I
can feel the objections from many proponents of quantum mechanics. They would
tell me that quantum physics needs
probability to work. These objections are also
understandable. After all, it was the probability interpretation by Max Born in
1926 that helped quantum physics overcome a stalemate and marched forward to
its eventual dominant position. Stripping probability from quantum physics
sounds more like an incoherent statement from the mental ward of a hospital
than a scientific proposition. It is
therefore necessary for me to show why quantum physics does not need
probability, and in fact will be much better off when probability is completely
removed from it. For this purpose I will use the simple example of an
electromagnetic wave, which is characterized by two parameters: 1. The speed of light c, a classical parameter. 2. The wavelength l, which is the resolution limit of classical space.
Strictly speaking, it is a quantum parameter. My goal is
to model the behavior of this wave inside one of its wavelength, i.e., in
scales smaller than the resolution limits of classical space and time; and I
will try to achieve this goal with the method of classical physics. I
understand that this goal seems to be in conflict with my earlier statement,
that classical space and time have no meaning inside a wavelength. However,
since whatever I can measure inside a wavelength must be the classical
manifestation of the quantum entity, it follows that there must exist a
classical description for it; and there is nothing conceptually wrong with my
attempt to seek such a description. In a
nutshell, my task is to determine a mathematical procedure to transform quantum
behavior into classical manifestations. For simplicity –without sacrificing
scientific rigor- I will only consider one dimensional space. The natural
choice for this dimension is the direction of light propagation. I will call
this spatial dimension x, and the time dimension t. Since the
quantum entity must exist somewhere in space and time, if I equate existence=A and
non-existence=0, there must be a point (xe,
te) in space-time where the “level of existence” LE of the quantum
entity equals A. Here I have resorted to the point assumption, which is a clear
indication that I intend to use the method of classical
physics to treat a quantum phenomenon. I must admit
that the point (xe, te) occupied
by a given quantum entity is a complete mystery to me. If I were a believer in
the Probability theory I would use some kind of symmetry argument and say that “the
probability that the quantum entity exist” is the same at every point in the subset of
space-time that the quantum entity is allowed to occupy; but there are several problems
with this assumption. First,
since the quantum entity is “lawless”, what makes me believe that
it will obey this “equal probability” law that I impose on it. Who knows, it
may purposely appear at a given spot just to show who the real boss is. Second,
since it will eventually interact with my detection tool at only a single point in space-time anyway, how can I be sure that
the “equal probability” law holds? Third, just assume for argument’s sake that
my co-worker decides to make the problem more challenging by allowing energy to
flow in and out of the system. If I stick with the “equal probability” law I
would have to add the condition that energy is conserved by a single quantum
entity, which I already know to be incorrect. Let’s assume
that I decide to sweep all of these conceptual problems under the rug and go
ahead with the calculation procedure provided by the Probability theory. Let’s
further assume that my results match experimental results spectacularly well.
Does this pacify my enquiring mind? No, because I’m still burdened with the
problem of interpreting what the results mean, and I am very sure that each person
will have his or her own interpretation. Since science should be built only on
logic, not personal interpretation; by adopting the concept of probability I
would run into all sorts of problems. But there is a very simple way for me to get
out of all of these problems. Instead of considering one quantum entity at a
time, I will consider a large number of them. Now the same
symmetry argument allows me to deduce that LE is A at every location that a
quantum entity could occupy, because if the number of quantum entities is large
enough, as a group they will act according to the laws of classical physics
(within certain tolerance depending on the level of randomness of the system,
as I have pointed out in an earlier article4). This
relieves me of the perplexing lawlessness of individual quantum entities.
I can now proceed with the confidence that my result will be experimentally
verifiable and conceptually consistent with classical physics. (Obviously
my treatment only applies to a large number of quantum entities; but to avoid
being too impersonal please allow me to continue to say that I’m dealing with a
“quantum entity” with the understanding that I’m referring to a fictitious
average quantum entity.) With
probability out of the picture, I’m ready to continue. Since the spatial
image of a wavelength is an instantaneous description of the quantum entity, and since I'm
using the method of classical physics, I'm forced to treat this image as a function of
classical space only, as if classical time does not
exist. In the language of mathematics, I’d say that the image of a
wavelength is time-indifferent (i.e., it has nothing to do with classical time.) This is an extremely important point that I
will return to later. In reality I know that
the quantum entity must exist in both space and time.
It follows that the instantaneous spatial image of the wavelength is the projection of
a two-dimensional existence onto the (single) dimension of space. Next, I notice
that classical space and time are
related by x = ct, which can be replaced by x = t’ if I define t’ºct. This shows that, as far as the existence of the
quantum entity is concerned, space and time are perfectly equivalent. There is
only one way to describe LE=A everywhere in two equivalent dimensions: The
circumference of a circle of radius A. Since the
classical image of the quantum entity is the spatial projection of its
existence, if I call QS its (observable) spatial existence I will
get: QS=Acosθ (3) I will reluctantly
call QS the spatial amplitude of the quantum entity just because the
term “amplitude” has become well-known in quantum physics. If I had a choice,
I’d rather call it the spatial existence function. You will see the reason for
my reluctance later.
Figure 4: What I can observe of the
quantum entity is QS, which is a one-dimensional projection of
its two dimensional existence. It is
tempting here for me to say that the temporal amplitude of the quantum entity
is the projection of A onto the vertical dimension, which seems to be the
natural choice for time. This would give the temporal amplitude QT =Asinθ.
This is, however, not correct; because in the construction of QS I
have used the side condition that time does not exist. Fortunately, since I know that
the temporal component must behave exactly the same way as the spatial
component, all I have to keep in mind is that figure 4 applies to the spatial
as well as the temporal component of the quantum entity. My last step
is to construct the classical two
dimensional existence of the quantum entity from its one-dimensional
projections. The appropriate mathematical operation is multiplying the two
one-dimensional projections (i.e., the spatial and temporal amplitudes). Since
these two projections are identical, if I call the classical existence of the
quantum entity by the more familiar name intensity I, the correct formula is
simply: I = QS2 (4) For the case of electromagnetic waves, it
is: I
=A2cos2θ (5) It is
important to note that when a two dimension existence is projected to 1 dimension,
there is a significant loss of information. The lost information may be
recovered only if the two one-dimensional projections are independent of each
other. Unfortunately, in the case of electromagnetic waves, time and space are
related by x = ct; meaning that they are acting “in sympathy” instead of being
independent of each other. For this reason the constructed existence (i.e., the
quantum existence felt by the classical world) is a distorted version of the
original existence, as equation (5) indicates. I could have
made the above treatment more “rigorous” with complex notations, mathematical transformations,
an account of A, θ, phase angle, boundary conditions, etc.; but I
purposely kept the amount of mathematics at a minimum, because the main point I try to make is that,
once the phenomena are understood, in most cases quantum phenomena can be treated
by the method of classical physics. Equation (4) is the
famous square law of quantum existence. Because it is a
natural consequence of the equivalence of classical space and time, it holds
true for all quantum systems regardless of their level of complexity. “Electromagnetic
waves” are not waves! I notice
that equations (3) and (5) are the correct mathematical descriptions of
electromagnetic waves, and I was able to derive them without appealing to
anything wave-like at all. This prompted me to ask a bold question “Are
electromagnetic waves really ‘waves’?” Of course I was taught in college that
they are waves; but even the experts are at a loss trying to explain why
electromagnetic waves do not need a medium to propagate. I will now offer a
simple alternative explanation: What we have been referring to as
electromagnetic waves are simply “things” confined in finite amounts of space.
Since these finite amounts of space are the resolution limits of classical
space, I will even say that electromagnetic “waves” are no more than localized disturbances
of space-time. In other words, what we have been referring to as
“electromagnetic waves” are not waves at all! (Since “amplitude” is associated with waves, now you understand why I was reluctant in
using this term.) Simply by
treating space-time disturbances as classical objects in Newtonian mechanics,
I can deduce that electromagnetic “waves” should propagate through vacuum at
constant speed. While the real picture is more complex than this, I can safely
say that the so-called “propagation medium” puzzle
of light is conceptually solved. I want to take this opportunity to pay tribute to an amateur
scientist whose line of thought regarding light and wave-particle duality is
very similar to mine. By accident in May 2001, while writing my first book “The
End of Probability and the New Meaning of Quantum Physics”, I ran into a web
page that challenged the “wave-particle duality” interpretation of quantum mechanics5.
According to the information provided, the author of the page is John Murphy of It was a pleasant surprise to read Mr. Murphy’s web page. He and I were miles apart and
probably will never meet each other, but we very much had the same idea. The
details are different because Mr. Murphy was more interested in the application
side of the problem, exemplified by the Fourier transform approach to
light scattering, while I was pursuing the same subject from the
Central Limit Theorem, which incidentally is usually proven in textbooks by
the method of Fourier transform. Even more interesting was that we both arrived at the same
conclusion that light is not a wave as asserted in
many textbooks. The readers are recommended to visit Mr. John Murphy’s website by searching with the keywords
“wave-particle duality” and “light scattering”. The end of
the wave-particle duality For economy
of words and to avoid the word “wave”, from now on I will use the neutral term
“light” in place of “electromagnetic waves”. Quantum
physics historically started in 1900 with the Plank’s equation which relates
the energy E and the frequency n of light with the now well known Plank constant h: E = hn (6) From (6) it
can be shown that the momentum p and the wavelength of light l are related by: pl = h (7) In 1924 de
Broglie made an ingenious suggestion, that other quantum entities would also
obey (7), this is now known as the de Broglie’s hypothesis: pld = h (8) I have added
the subscript “d” to l to remind myself that (8) is applicable to all quantum entities, not
just light. Since light
was thought to be a wave, it was natural to deduce from (8) that other quantum
entities, which had been considered as particles, also acted as waves. Since
“particle” and “wave” had been considered mutually exclusive entities, the
implication of the de Broglie’s hypothesis was mind boggling. This was how the
“wave-particle duality” frenzy started; and it is still going on strong today. Now that I
have arrived at the conclusion that light is not a wave, it is only appropriate
for me to say that the wave-particle duality is not a correct description of
quantum physics. In other words, the wave-particle duality was a myth, no more
and no less. To put it even more bluntly, there is no wave-particle duality! At the same
time, I realize the historical importance of de Broglie’s hypothesis. I
therefore propose that the “wave-particle duality” is kept in physics
textbooks, but with additional comments to clarify its current status. A
practical action is to call the resolution limit of classical space the “de
Broglie resolution limit” or “de Broglie limit” (instead of the existing name
“de Broglie’s wavelength”). Since the mathematical description of the “de
Broglie limit” is identical to the traditional mathematical description of a
“wave”, this adjustment will not have a single detrimental effect on the
calculation power of quantum physics. At the same time it will help quantum
physics to be physically meaningful for the first time. . The Time Indifference
Principle and the solution to the “single photon” puzzle From now on
I will call the spatial resolution limit “the de Broglie limit” to pay respect
to de Broglie’s contribution to quantum physics.
The left
picture of figure 5 shows a fictitious quantum distribution function, which is obtained
by squaring the spatial or temporal amplitude. I have drawn an arbitrary shape
on purpose, because in the general case there is no rule that says the
distribution has to look beautiful or symmetrical. Since the
spatial amplitude is time-indifferent, and since classical method implies the
collection of many quantum entities, it follows that if I keep detecting
quantum entities and record the location of each occurrence, I will eventually
get a histogram that has the same shape as the quantum distribution (see figure
5). This is an
astounding result, considering that my detection process may only capture one
quantum entity at a time. The quantum distribution somehow acts like a jigsaw
puzzle. Although I can never guess where the next quantum entity will show up,
in the long term the distribution function will be faithfully mapped out as a solved
puzzle (within certain tolerance, of course.) Since this remarkable effect is
“caused” by the fact that the quantum distribution function is time-indifferent,
I will give it the name “the Time Indifference Principle” (TIP). This principle
was covered in my first book “The End of Probability and the New Meaning of
Quantum Physics”, but with a different approach. Every
serious student of modern physics must have run across the “single photon”
puzzle, which is a repeat of Young’s famous double slit experiment (1803), but
with one photon at a time. Amazingly, the pattern of
alternating dark and bright stripes are faithfully mapped out by the
single photons, as if they are conscious and have a prior agreement with one
another. I don’t see any need to elaborate on mathematical details; but with
the Time Indifference Principle, I will claim that this great puzzle of quantum
physics is now solved.
Figure 6: Modern Young’s double slit experiments
(multiple trials with one single photon at a time) Your feed
back Some
materials in this article will be included in my future book “The Science of
Space Time and Existence”. Your feedback is valuable to me because it will help
me improve the quality of the book. Feel free to comment on anything that comes
to your mind. First written
for the Web on September 21, 2002 First revision October 4, 2002 ÓDangSon Tran. All rights reserved References and Notes 1 “The Resolution Limit Interpretation of the
Heisenberg’s Uncertainty Principle”, DangSon Tran, Web article posted 2The lower
limit is theoretically zero, which brings us back to the point assumption that
I have refuted. But this point is academic, because if the resolution limits
are zero there would be no quantum process.
3 “The
Resolution Limits of Space and Time”, DangSon Tran, Web article
posted 4 “Logic for the End of Probability”, DangSon Tran, Web article posted 9/2/2002, first revision 9/22/2002, second revision 9/29/2002 5 “Quantum
theory and Wave/Particle Duality”, John Murphy, web article, web site
http://www.hotquanta.com Now that you have read the article, it is my turn to ask you to contribute your wisdom: -Did you find this article stupid, useless, a waste of time, or having some other negative effect? If so you can help the world by e-mailing me a wake-up message. Who knows? You may be able to wake me up from my delusion so that I won't keep on with this kind of frivolous activities. That would help me personally and spare the internet world of some annoying cyber-pollution. -Do you have any other feedback for me? Like criticism, encouragement, ideas, opinions, etc. Please let me know by email. -Did you find this article good, useful, valuable, or having some other positive effect? If so please help spread the word through your family, your neighbors and friends, your work place, educational and/or academic institutions, internet user groups, etc., so that this article will be read by more people.
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