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Superposition
by
Miles Mathis
Where
the master has failed
What
hope for the student
Had he
obeyed him in all?
—Richard
Wagner, Siegfried
First
posted October 17, 2005
In this paper
I will offer a simple mechanical explanation of superposition. I
will provide an easy visualization as well, one that
simultaneously solves the mystery of superposition and the wave
motion of particles.
Heisenberg
and Bohr assured everyone that this was not possible. The
Copenhagen interpretation, which is still the preferred
interpretation of quantum mechanics by contemporary physicists,
states in no uncertain terms that the mysteries of quantum
physics are categorically unsolvable. That is, they are not only
unsolved, they are impossible to solve. All other
interpretations of quantum mechanics have agreed with this
interpretation, regarding the impossibility of a straightforward
visualization or of a simple mechanical solution. Some variations
have denied other aspects of the Copenhagen interpretation,
especially regarding its opinion of the collapse of the wave
function. Bohm, for instance, has attempted a deterministic
explanation of certain parts of QED, including a reinterpretation
of the wave function and of the Uncertainty Principle. But not
even Bohm or Bell believed that anyone could offer a simple
visualization that would explain superposition or the socalled
waveparticle duality.
Einstein came
closest to this belief. He remained convinced that quantum
mechanics would eventually be explained in a more consistent
manner. But, again, it was mainly the probabilistic nature of
quantum dynamics that bothered him, not the fact that it could
not yield to simple visualizations. He did not like God playing
dice, but he did not expect God to draw us a picture with every
new theory.
I did not
approach the problem intending to find a visualization or an easy
mechanical solution. I only wanted to make better sense of it in
my own mind. But in analyzing the problem I found that the
mechanical difficulties were not nearly as formidable as has been
claimed. I found that I could quite easily visualize the
physical motions, and that I could put these visualizations into
pretty simple words and pictures. One basic discovery allowed me
to do this, and that is what this paper is about.
I believe
that the most efficient way to lead the reader through the
problem is to analyze the current explanation of superposition,
as it is presented in a contemporary text. As my text I will use
David Albert’s Quantum
Mechanics and Experience.
I choose this book for the same reason that the status
quo chose to
publish it: it puts the theory in as clear a form as possible,
for laymen and physicists alike. Albert is a philosophy
professor at Columbia, but he has been embraced and tutored by
many mainstream physicists. This book may therefore be taken as
a representative, if not perfect, expression of current theory.
If it were not it surely would not have been published by Harvard
University Press.
Albert begins
by taking two measurable qualities of an electron. He tells us
that the qualities don’t matter, and that we could call
them color and hardness if we wanted to. In a footnote on page 1
he informs the reader that experimentally he is talking about
xspin and yspin, but he does not elaborate beyond that.
Conveniently, this footnote allows me to make my first major
substantive point. From a logical point of view, an electron
cannot have angular momentum on the x and y axis at the same
time—not if both spins are about an axis through the center
(Albert claims that they are). Imagine the Earth spinning about
its axis. Call that axis the xaxis. Now go to the yaxis, which
also goes through the center but is at a 90^{o}
angle from the xaxis. Try to
imagine spinning the Earth around that axis at the same time that
it is spinning around the xaxis. If you can imagine it, then
you have a very vivid imagination, to say the least.*
To see what I
mean, remember the gyroscope and the phenomenon called
precession. A torque applied to the axis of rotation is
deflected, so that circular motion is not allowed about the
yaxis. You can have circular motion in only one of the two
planes at a time. To see why this is so, think of a point on the
surface of the sphere or on the edge of a wheel. Give it spin in
the xyplane. Now follow its course and see the curve it
describes. Once you have done that, think of giving it a spin in
the zyplane at the same time. You have a second curve applied
to the first curve. But these two curves cannot be added to
create a new curve that the body can follow as a whole. If the
body were free to follow both curves from the first dt, then the
first thing it would do is warp very badly. Very soon it would
be twisted beyond recognition. But real bodies are not free to
warp into any shape possible. They already have structure at
many levels, and this structure is rigid to one degree or
another. So if you try to apply a second circular motion to a
real body, you are applying a force that does not just lead to
motion—you are applying a force that is trying to break the
body itself. It is the molecular bonds themselves that are
resisting you. The body does not want to warp. This is why you
can apply a second spin to a liquid in circular motion. The
liquid does not resist the second orthogonal force. But your
second force ends up destroying the “body” of the
circular motion, which in a liquid was just a pattern anyway.
That
said, it is possible to have simultaneous x and y spins, but you
must apply the second spin to a center outside the object. What
I mean is that the electron must spin end over end, rather than
spin about the axis through its center. To
go back to the Earth example, you can see that we can easily
imagine the Earth hurtling end over end throughout space, since
this end over end motion would not affect its axis spin at all.
A gyroscope resists a 90^{o}
force, but only because we
have fixed the center of the gyroscope relative to the force. A
gyroscope will not spin in two ways about its center. But if we
put the gyroscope in a spherical container, then we can rotate
the gyroscope around a point on the surface of the sphere. We
can do this even if the gyroscope is firmly attached to the
container. Take a spinning bicycle tire and extend the axle out
so that the diameter of the axle is equal to the diameter of the
wheel. Attach the ends of this axle firmly to a great sphere
with the same diameter, so that the wheel is inside the sphere.
You can now rotate that sphere about any point on the surface of
the sphere, without the internal motion causing precession. This
is because you are no longer attempting to cause two different
rotations about the same center. You have created a center just
beyond the influence of the first axis.
What is even
more interesting is that the circle of this new revolution now
has a center that is not stationary—it travels. And it
travels in a very interesting way. Let us say you have the Earth
spinning about the xaxis, and you give the center of the Earth a
constant velocity in the ydirection. Next, we add an
endoverend spin in this same ydirection. Now, what sort of
total curve would this end over end spin create, for the center
of the Earth? It
would create a wave.
[To see an
animation of this wave motion, you may take these links. The
first is a windows media file, the second requires Quicktime (and
is much faster to download). wave.wmv
4.5Mb. wave.mov
780kb. Expect to wait 30 seconds for the wmv file. Thanks to
Chris Wheeler for use of these files.]
Let that sink
in for a few seconds. Albert assumes that both angular momentums
are measured about the same center. Beyond that, he assumes that
the measured qualities or quantities don’t matter. He
assumes that angular momentum is conceptually equivalent to
velocity or position or any other parameter. He assumes that
because that is what all physicists have so far assumed. What
matters for QED is how these unanalyzed variables plug into
equations. I have just shown that the actual variables matter
very much. The whole explanation for QED lies in the real
motions of these real bodies, and the explanation is capable of
being stated in simple, direct terms, as I did it above. The two
angular momentums not only influence eachother in specific and
distinct ways; the ways they influence eachother provide the
conceptual and physical groundwork for QED—a groundwork
that has so far been ignored.
But let us
return to Albert’s argument. He gives the electron color
and hardness, to simplify the analysis. The electron then has
four states: black, white, hard, soft. The physicist has equally
simple tools. He has a color box and a hardness box. If he
feeds in an unknown electron, the color box tells the physicist
black
or white.
The hardness box tells him hard
or soft.
Now, if the
physicist feeds white or black electrons into a hardness box,
half trip the hard detector and half the soft. Likewise for hard
or soft electrons fed into a color box. This means, according to
Albert, that “the color of an electron apparently entails
nothing whatever about its hardness” or the reverse.
The problem
encountered by Albert’s physicist is that these two simple
detectors seem to work in strange ways, if you set them up in
combination. If the physicist sets up three boxes like this:
color box, hardness box, color box, the percentages at the end
are mystifying. The hardness box in the middle is set up so that
it captures only one emerging color, which Albert lets be white.
The white electrons travel to the middle hardness box, where half
of them make it through and go to the last box. The surprise is
that of those, only half are white when they come out. Our final
color box finds half of them are black. Wow. Albert and QED
tell us this is a big problem. It cannot be explained logically.
Albert says that his physicist tries everything. He builds his
boxes in a variety of ways, to make them more (or even less)
precise. It doesn’t matter. The same 50/50 split comes out
at the end.
This has been one of the central problems of
quantum physics from the very beginning. It has been a mystery
for at least 80 years. But the outcome is easily explainable
once you have my analysis above in hand, regarding the various
spins. Let’s say you have a sample of electrons and are
going to measure angular momentum in both zx and zy planes. If
we have four possible outcomes, then we assume that each momentum
is either clockwise or counterclockwise, relative to some
observer. Now, put yourself in the position of this observer and
see what happens. At the first moment, you look and you see that
the electron is rotating clockwise about its xaxis, with that
axis pointing straight at you. This means that the rotation is
in the zyplane. In other words, you are looking at a little
clock, since it is moving relative to you just like the second
hand on the face of a clock. That clock face exists in the
zyplane. A moment later the electron has rotated a halfturn,
end over end along the xaxis. This rotation is in the zxplane,
about a traveling yaxis. After this halfturn, you look again at
the clock face. Its motion is the same, but it now appears
counterclockwise to you.
If that was
confusing, you can easily perform the above visualization with a
desk clock, provided of course that it is not digital. Hold the
clock in front of you. Its hands are turning clockwise, and they
represent the spin in the xplane. Now give the entire clock a
spin in the yplane, simply by turning it one half turn end over
end. If you do this you will now be looking at the back of the
clock. The second hand is now moving counterclockwise, relative
to you. It is that simple. That is all I am saying. The second
hand of the clock is spinning around an xaxis that is pointed
right at you. Then you spun the whole clock around a yaxis.
Very elementary, but it shows us that the xspin of the electron
must be variable, if you measure it relative to an observer
external to the electron. If the electron has both xspin and
yspin, then the xspin will be variable, measured by a
stationary device. Only an observer traveling with the electron
would measure its spin as consistently CW or CCW. The same thing
applies in reverse, of course. If you are measuring the other
angular momentum, then you get a periodic variance in the first
one.
You could say
that the spin changes due to relativity, but that would actually
be overcomplicating the situation. We don’t need any
transforms here, and the kind of simple relativity I have just
described was known long before Einstein. It is true that my
analysis used relativity to find a solution, but it is the
simplest, preEinstein sort of relativity. It is just to say
that an observer must pay attention to how the object he is
measuring is changing over time. A measuring device, whether it
is an eyeball or an electron detector, is a constant frame of
reference, and a spinning electron will show variance with regard
to that device at different times, as I have just shown. There
is nothing esoteric about it, although I suppose it is a subtle
thing to have to notice.
Once we apply
this to our measuring devices, whatever they are, we see that
this must affect our outcomes quite positively. Let us go inside
the first box. It was measuring color, so let us assign color to
the clockface rotation. White is CW, black is CCW. The box
finds that some electrons are white and some black. To
differentiate, it must apply some field or force to them over
some dt. Let us imagine, to simplify, that the box feeds the
electrons into a chute, like cattle, and then puts them all
through the same door. This door is like the metal detector at
the airport, except that it takes a picture of the electron as it
rushes through. It has a very fast fstop, an fstop of dt. If
the electron was CW at that dt, then the box ejects it from the
white door. If the electron was CCW at that dt, then the box
ejects it from the black door.
This is, in
fact, very much like the way detectors work. They don’t
take pictures, of course, but some sort of force or field
separates the white and black electrons. The field may not be
limited to a dt, but the first impression of the field is
crucial. The electrons are moving quite fast, and the time
periods are therefore quite small. The field doesn’t have
time to snap a bunch of pictures and start changing its mind.
What this all
means is that whiteness and blackness and softness and hardness
are not constants. Every electron is both black and white and
hard and soft, at different times. But it is all those things
only if you sum over some extended period of time. At each dt,
it is either hard or soft, black or white. It is not both at the
same time. At one measurement, it will be one or the other.
Over a series of measurements, it will be both.
This is the
subtlety that QED has never penetrated. It explains the above
problem like this: If you put electrons like those I have
described through a color box, the color box sees some of them as
black and some as white over the dt measured. But they are
actually not white or black as they come out—they remain
potentially both, depending on the point in the wave you measure.
If you measured the white ones coming out at a different point
in the wave motion, you would find them black, and vice versa.
Now, the color determination is repeatable, since a similar box
will catch the electrons in similar ways. All color boxes tend
to shute and channel electrons in the same way, so that the
exiting group is made coherent. A second color box must then read
them the same way as the first.
What happens
in the second box (the hardness box) solves the mystery. The
second box creates coherence in the second angular momentum.
This assures that other hardness boxes will find the same
hardness. But in creating this coherence, the second box
rerandomizes the first variable. Why does it do this? It does
this because the wavelength of the two angular momentums is
different. If the first wavelength was taken as R, for the
radius of the electron, then we have to take the second
wavelength as 2R, for the diameter. This is simply because the
second wavelength is caused by end over end rotation. If we
cohere the end over end rotation, this must split the measurement
of the axial rotation. If we cohere the axial rotation, this
must split the measurement of the end over end rotation. One is
half the other, so you cannot create coherence in both at the
same time.
I can show
this with simple waves in two dimensions. Study the diagram
below. We have two opposite combinations of ½ and 1
waves. If you synchronize the ½ waves, the 1 waves are
off. If you synchronize the 1 waves, then the ½ waves are
off. You cannot synchronize both. This, in essense, is what is
happening in box two. The hardness waves are being made
coherent, so that the color waves are being thrown out of synch.
The third box then reads them as ½ one and ½ the
other.
You can see that I have simultaneously solved the problem
of superposition and the problem of the wave motion of quantum
particles. I did this simply by noticing that the second angular
momentum must be about a center that is just external to the
object. That is to say, the yspin is end over end.
With the
hindsight this gives me, it seems shocking that this was not seen
earlier. The reason it was not seen is that Heisenberg and Bohr
convinced everyone early on that Quantum Mechanics could not be
explained with straightforward logic and simple visualizations.
No one has ever bothered to apply a little commonsense to the
physical situation. They were so sure that it couldn’t be
done, that they didn’t even try to tackle the problem on a
visual or mechanical basis. This predicament soon snowballed,
since as more and more great physicists looked at the problem and
failed to explain it, later physicists became more and more sure
that it couldn’t be solved. They did not want to waste
their time combing something that every genius from Bohr to
Feynman had already combed. That seemed not just foolish, but
sacrilegious. But the fact is that there has probably been no
one since Bohr that tried very hard to make classical sense of
the problem. Physicists who came right after Bohr took his word
for it, and contemporary physicists have reached the point where
most don’t even want a mechanical explanation of QED. The
spooky paradoxes are more fun. They make better copy.
You may now
go to my second
paper on superposition, to see a similar experiment solved
even more quickly and transparently. That experiment is the
famous one of two beam splitters and two mirrors. In that paper I
also offer three more diagrams, which may be helpful to many.
A related
problem is that of entanglement, which I
analyze and solve here.
More recently, I have blown
apart the CHSH Bell
tests, unveiling the terrible mathematical cheat at the heart
of these experiments. This leaves entanglement in tatters.
To
see how my solution destroys quantum nonlocality, you may go to
this recent paper,
which even gives you the new wavefunction equations—including
the new degrees of freedom I discovered above.
I think it is
obvious that the end over end spin in the ydirection can be
applied to other problems, including the
propagation of photons, the twoslit
experiment, and so on. In subsequent papers I will apply my
finding to the electron
and proton and to
a large list of mesons, to show that the same four stacked
spins can explain all quantum makeup and motion. I will also
have a lot more to say about other specific problems within QED
and QCD, and
their solution with straightforward logical analysis.
*Addendum,
Feb 2012: A close reader just asked me for clarification on the
spins here. He pointed out that the Earth has a wobble in its
spin. Isn't
that part of a second spin, since it isn't along the original
axis? If we continued the wobble, we could create a whole spin in
either direction. I
answered: Excellent question, and I will even add it to my
super.html paper, to clear up confusion. Let's look at your
Earth wobble, to get to the bottom of this. The Earth's wobble
isn't caused by two spins about two different axes, as in my
example. It is caused by a motion of the first axis. We let the
Earth spin on z, say, then we move z. Yes, we can actually spin
z, moving the north pole to the south, and I think that is what
you are getting at. We then have spins in two planes, which
seems to prove your point. We could then call the spin of z
either x or y, and it looks like I have been refuted. However, I
have not been refuted, since we are talking about different
things. If you now rename the spin of z as xspin, your xspin
is not the same as the xspin I am outlawing. I have outlawed
some x and y spin, right? Well, I am outlawing the original
xspin, the one that is the same sort of motion as the original
zspin. Which is a spin about an axis. You have found a spin of
the axis, not a spin about
the axis. So my point holds.
That xspin about an xaxis is still outlawed. In fact, your new
xspin is the same as my endoverend xspin, since if we give
the Earth any linear motion, your xspin will appear
endoverend. North and south poles switching ends is
endoverend, is it not?
He then
replied, Yes
that clears that up, but there is still the matter of the point
of spin. You say that the endoverend spin needs to spin about a
point on the end of z. I have reminded you that we can spin z
about its center. What gives? And
I answered: I admit that it could be one or the other. Either
way creates what I would call an endoverend spin. But my way
allows me to create my quantum spin equation, which answers a lot
of questions that have been in the shadows. So the argument for
my way is straight from data. The quanta could spin your way,
but in fact I don't think they do. The spin equation wouldn't
fit data. To be specific, if we let the zaxis spin about its
center rather than about one end, we don't get a doubling of the
spin radius with each added spin. We need that. See elecpro.html
for the spin equation I am talking about. As for the physical
reason quanta choose to stack spins that way, by spinning about a
point on the end of z, I don't see the answer yet. I suspect
that it is some analogue of the centrifugal force, and that the
quick first spin pushes following spins out to an "edge."
Could also have something to do with imperfect roundness of the
initial spin. They have supposedly just shown that the electron
is incredibly round, but nothing is perfectly round, I assume.
Any imperfection might cause later spins to be pushed outward
like this. If anyone has a better theory, he or she can email me
with it. I wouldn't say it is crucial, but it would be nice to
figure it out.
Update,
2013. I figured it out myself the next time I reread this
paper. To understand why the photon's second spin spins around a
point on the original spin surface, we just have to look at the
cause of that second spin. I have shown previously it must be
caused by collision with another photon. The first photon stacks
a second spin on top of the first because it cannot spin any
faster on the first axis. It has reached a spin velocity of c,
and if it encounters a positive spin collision that would
increase its spin energy, it can stack that extra energy on only
by creating another spin. Well, since the point of collision is
on the outer surface, the photon naturally spins about that
point. The second spin must take as its new center that point of
collision.
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