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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 en | |||||||||||||||||||||||||||||||||||||||||||||||||
