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Quantum Mechanics

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from the website http://www.bottomlayer.com/bottom/argument/Argument4.pdf

IV. Continuity and Discontinuity in Observed Behaviors

A. "Quantum leaps," as though there were no time or space between quantum events

In our experience, things move from one end to the other by going through the middle; they get from cold to hot by going through warm; they get from slow to fast by going through medium; and so on. Phenomena move from a lower state to a higher state in a ramp-like fashion -- continuously increasing until they reach the higher state.

Even if the transition is quick, it still goes through all of the intermediate states before reaching the new, higher state.

In quantum mechanics, however, there is no transition at all. Electrons are in a low energy state on one observation, and in a higher energy state on the next; they spin one way at first, and in the opposite direction next. The processes proceed step-wise; but more than step-wise, there is no time or space in which the process exists in any intermediate state.

It is a difficult intellectual challenge to imagine a physical object that can change from one form into another form, or move from one place to another place, without going through any transition between the two states.

Zeno’s paradoxes offer a rigorously logical examination of this concept, with results that have frustrated analysts for millennia. In brief, Zeno appears to have "proved" that motion is not possible, because continuity (smooth transitions) between one state and the next implies an infinite number of transitions to accomplish any change whatsoever.

Zeno’s paradoxes imply that space and time are discontinuous -- discrete points and discrete instants with nothing in between, not even nothing. Yet the mind reels to imagine space and time as disconnected, always seeking to understand what lies between two points or two instants which are said to be separate.

the pre-computer analogy

Before computer animation there was the motion picture. Imagine that you are watching a movie. The motion on the screen appears to be smooth and continuous. Now, the projectionist begins to slow the projection rate. At some point, you begin to notice a certain jerkiness in the picture. As the projection rate slows, the jerkiness increases, and you are able to focus on one frame of the movie, followed by a blanking of the screen, followed by the next frame of the movie.

Eventually, you see that the motion which seemed so smooth and continuous when projected at 30 frames per second or so is really only a series of still shots. There is no motion in any of the pictures, yet by rapidly flashing a series of pictures depicting intermediate positions of an actor or object, the effective illusion is one of motion.

the computer analogy

Computers create images in the same manner. First, they compose a still image and project it; then they compose the next still image and project that one. If the computer is quick enough, you do not notice any transition. Nevertheless, the computer’s "time" is completely discrete, discontinuous, and digital. One step at a time.

Similarly, the computer's "space" is discrete, discontinuous, and digital. If you look closely at a computer monitor, you notice that it consists of millions of tiny dots, nothing more. A beautifully rendered image is made up of these dots.

The theory and architecture of computers lend themselves to a step-by-step approach to any and all problems. It appears that there is no presently conceived computer architecture that would allow anything but such a discrete, digitized time and space, controlled by the computer's internal clock ticking one operation at a time.

Accordingly, it seems that this lack of continuity, so bizarre and puzzling as a feature of our natural world, is an inherent characteristic of a computer simulation.

B. The breakdown at zero, yielding infinities, as though the universe were being run by a computer clock on a coordinate grid

Quantum theory assumes that space and time are continuous. This is simply an assumption, not a necessary part of the theory. However, this assumption has raised some difficulties when performing calculations of quantum mechanical phenomena. Chief among these is the recurring problem of infinities.

In quantum theory, all quantum units which appear for the purpose of measurement are conceived of as dimensionless points. These are assigned a place on the coordinate grid, described by the three numbers of height, depth, and width as we have seen, but they are assigned only these three numbers.

By contrast, if you consider any physical object, it will have some size, which is to say it will have its own height, width, and depth. If you were to exactly place such a physical object, you would have to take into account its own size, and to do so you would have to assign coordinates to each edge of the object.

When physicists consider quantum units as particles, there does not seem to be any easy way to determine their outer edges, if, in fact, they have any outer edges. Accordingly, quantum "particles" are designated as simple points, without size and, therefore, without edges. The three coordinate numbers are then sufficient to locate such a pointlike particle at a single point in space.

The difficulty arises when the highly precise quantum calculations are carried out all the way down to an actual zero distance (which is the size of a dimensionless point -- zero height, zero width, zero depth). At that point [sic], the quantum equations return a result of infinity, which is as meaningless to the physicist as it is to the philosopher.

This result gave physicists fits for some twenty years (which is not really so long when you consider that the same problem had been giving philosophers fits for some twenty-odd centuries). The quantum mechanical solution was made possible when it was discovered that the infinities disappeared if one stopped at some arbitrarily small distance -- say, a billionth-of-a-billionth-of-a-billionth of an inch -- instead of proceeding all the way to an actual zero.

One problem remained, however, and that was that there was no principled way to determine where one should stop. One physicist might stop at a billionth-of-a-billionth-of-a-billionth of an inch, and another might stop at only a thousandth-of-a-billionth- of-a-billionth of an inch. The infinities disappeared either way.

The only requirement was to stop somewhere short of the actual zero point. It seemed much too arbitrary. Nevertheless, this mathematical quirk eventually gave physicists a method for doing their calculations according to a process called "renormalization," which allowed them to keep their assumption that an actual zero point exists, while balancing one positive infinity with another negative infinity in such a way that all of the infinities cancel each other out, leaving a definite, useful number.

In a strictly philosophical mode, we might suggest that all of this is nothing more than a revisitation of Zeno’s Achilles paradox of dividing space down to infinity. The philosophers couldn’t do it, and neither can the physicists. For the philosopher, the solution of an arbitrarily small unit of distance -- any arbitrarily small unit of distance -- is sufficient for the resolution of the paradox. For the physicist, however, there should appear some reason for choosing one small distance over another.

None of the theoretical models have presented any compelling reason for choosing any particular model as the "quantum of length." Because no such reason appears, the physicist resorts to the "renormalization" process, which is profoundly dissatisfying to both philosopher and physicist.

Richard Feynman, who won a Nobel prize for developing the renormalization process, himself describes the procedure as "dippy" and "hocus-pocus." The need to resort to such a mathematical sleight-of-hand to obtain meaningful results in quantum calculations is frequently cited as the most convincing piece of evidence that quantum theory -- for all its precision and ubiquitous application -- is somehow lacking, somehow missing something.

It may be that one missing element is quantized space – a shortest distance below which there is no space, and below which one need not calculate. The arbitrariness of choosing the distance would be no more of a theoretical problem than the arbitrariness of the other fundamental constants of nature -- the speed of light, the quantum of action, and the gravitational constant. None of these can be derived from theory, but are simply observed to be constant values. Alas, this argument will not be settled until we can make far more accurate measurements than are possible today.

quantum time

If space is quantized, then time almost surely must be quantized also. This relationship is implied by the theory of relativity, which supposes that time and space are so interrelated as to be practically the same thing. Thus, relativity is most commonly understood to imply that space and time cannot be thought of in isolation from each other; rather, we must analyze our world in terms of a single concept -- "space-time."

Although the theory of relativity is largely outside the scope of this essay, the reader can see from Zeno’s paradoxes how space and time are intimately related in the analysis of motion.

For the moment, I will only note that the theory of relativity significantly extends this view, to the point where space and time may be considered two sides of the same coin.

The idea of "quantized" time has the intellectual virtue of consistency within the framework of quantum mechanics. That is, if the energies of electron units are quantized, and the wavelengths of light are quantized, and so many other phenomena are quantized, why not space and time? Isn’t it easier to imagine how the "spin" of an electron unit can change from up to down without going through anything in the middle if we assume a quantized time?

With quantized time, we may imagine that the change in such an either/or property takes place in one unit of time, and that, therefore, there is no "time" at which the spin is anywhere in the middle. Without quantized time, it is far more difficult to eliminate the intervening spin directions.

Nevertheless, the idea that time (as well as space) is "quantized," i.e., that time comes in individual units, is still controversial. The concept has been seriously proposed on many occasions, but most current scientific theories do not depend on the nature of time in this sense.

About all scientists can say is that if time is not continuous, then the changes are taking place too rapidly to measure, and too rapidly to make any detectable difference in any experiment that they have dreamed up.

The theoretical work that has been done on the assumption that time may consist of discontinuous jumps often focuses on the most plausible scale, which is related to the three fundamental constants of nature -- the speed of light, the quantum of action, and the gravitational constant.

This is sometimes called the "Planck scale," involving the "Planck time," after the German physicist Max Planck, who laid much of the foundation of quantum mechanics through his study of minimum units in nature.

On this theoretical basis, the pace of time would be around 10-44 seconds. That is one billionth-of-a-billionth-of-a-billionth-of-a-billionth of a second. And that is much too quick to measure by today’s methods, or by any method that today’s scientists are able to conceive of, or even hope for.

mixing philosophy, science, time, and space

We see that the branch of physics known as relativity has been remarkably successful in its conclusion that space and time are two sides of the same coin, and should properly be thought of as a single entity: space-time.

We see also that the philosophical logic of Zeno’s paradoxes has always strongly implied that both space and time are quantized at some smallest, irreducible level, but that this conclusion has long been resisted because it did not seem to agree with human experience in the "real world."

Further, we see that quantum mechanics has both discovered the ancient
paradoxes anew in its mathematics, and provided some evidence of quantized space and time in its essential experimental results showing that "physical" processes jump from one state to the next without transition.

The most plausible conclusion to be drawn from all of this is that space and time are, indeed, quantized. That is, there is some unit of distance or length which can be called "1," and which admits no fractions; and, similarly, there is some unit of time which can be called "1," and which also admits no fractions.

Although most of the foregoing is mere argument, it is compelling in its totality, and it is elegant in its power to resolve riddles both ancient and modern.

Moreover, if we accept the quantization of space and time as a basic fact of the structure of our universe, then we may go on to consider how both of these properties happen to be intrinsic to the operations of a computer, as discussed above at Point IV(A).