[Excerpt  Be and Become, ProCreative, Sydney 2000]

The observation of anomalies is an excellent fillip for suspecting that our current view of reality is incomplete.

The high degree to which we have biased our perceptions in terms of our local physical senses (sight, hearing, touch, smell and taste) has blinded our consideration of some glaring anomalies in our thinking—anomalies (theoretical paradoxes) which have persisted for nearly two and a half thousand years.

Broken Arrows

Around 450 B.C. Greek philosopher Zeno of Elea introduced a number of paradoxes that revealed how motion (of any kind) was theoretically impossible. What he managed through straight-forward reasoning was to show that our theoretical perception of reality didn’t match our experience of it. And that mis-match between theory and practical experience has persisted ever since. In fact, it has become all the more entrenched in the last few hundreds years since the on-set of the industrial revolution.

The first paradox established that the commencement of movement is theoretically impossible.

Consider a runner, ready to run the 100 metres at the Olympics. She’s a top athlete, but, according to Zeno’s paradox, she can not only never run the race, she can’t ever start it. For the athlete to run the 100 metres, she must first get to the halfway (50 metre) mark. But before she gets there, she must get to the quarter-way (25 metre) mark. And before that, she has to get to the “one-eighth way” (12.5 metre) mark and so on, ad infinitum. And this is were the paradox arises, for in taking time (albeit ever smaller portions of it) to traverse an infinite number of increasingly smaller and smaller initial points in the race, the runner can never (as in ever) get started. Each halfway point is a finite number, requiring finite time to traverse and since there are infinite such traversals required to run a race, the runner requires infinite time to run the race.

Basically, the paradox is that before we move anywhere, we have to first get through or beyond infinity.

The second paradox involved Achilles’s attempt to overtake the tortoise in the classic hare and the tortoise race. Even though Achilles is much more fleet of foot than the tortoise, he can never catch up to and overtake the head start initially given to the tortoise. Essentially, the same problem as the Olympic runner is involved—taking infinite smaller and smaller bits of time to traverse infinite smaller bits of space. It would take anyone forever to traverse infinite small bits of time (no matter how small). The head start cannot ever be beaten now matter how fast Achilles runs or how slow the tortoise crawls (so long as the tortoise does not stop completely).

The third paradox is more helpful in understanding how the paradoxes can be resolved, in that it introduces ideas which are crucial to resolving all three paradoxes. The third paradox involves the motion of an arrow through the air. At any point in time, an arrow (or indeed any object) flying through space must be at rest in that space. The arrow, in real terms, must be completely at rest in some specified section of space at some specified time, for otherwise the arrow could not be said to be anywhere specifically in space. The arrow would not be real—it would instead be some ghost-like apparition, not real-enough or solid enough to pierce armor or its intended target. We can better visualize this by imagining that we have a very fast movie camera taking consecutive snapshots of the arrow in flight. In each frame, the arrow will appear stationary, i.e. it will appear entirely real, simply because it is real (it has a fixed positioned in space). As physicist Fred Alan Wolf explained,

To get around this paradox, philosophers and scientists have imagined that reality could be broken down into infinite minutely-small time-frames, each with its own slightly different “snapshot” of reality. This is another way of treating time as being infinitely divisible. When all these frames are run consecutively, just like a movie-film through a projector, we get the sense and experience of motion. When we have infinite such frames, they end up blending together into the perfectly continuous flowing reality that we normally take for granted. However, this line of thinking does not resolve the paradox, it merely introduces another set of paradoxesi.

Wolf explains:

The theorists believed that “in principle” we could span the infinite, even though in practice it would be impossible. The reason for the acceptance of such theories was that as the time-frame becomes infinitely short or small, we could “in principle” span or traverse them in finite time. As the frame became infinitely small, we could thus reasonably ignore specific consideration of it. In a sense, the mathematicians performed theoretical leaps of reason and logic by jumping over the intervening infinitely small spaces and times. In other words, it became convenient to ignore the spiritual (infinite, immeasurable and unknowable). And this is the crux of the dilemma that has persisted for the last 2.3 millennia.

Mathematicians invented the generic term “infinitesimals” to conveniently label quantities which were infinitely small, but which were still some value greater than zero. They were paradoxically real, in that they were numbers of some size greater than zero, but also, they were unreal, in the sense that no-one could measure or define them. They weren’t able to be defined as being some specific size.

These “infinitesimal” quantities, with their realness having evaporated in the unknowable mist of infinity, were nonetheless paradoxically still considered real, or at least real enough to be used to effectively explain reality. Consequently the scientists’ standard tool for mapping reality became the accepted confluence of real and unreal measures. These unreal measures were soon expanded to include imaginary quantities (based on the square root of -1) which were even less real, in that they were even less able to be meaningfully related to the real world.

1. 1. Fred Alan Wolf, Taking the Quantum Leap, Harper & Row New York 1989, page17
2. 2. Wolf, page 21

[Excerpt  Be and Become, ProCreative, Sydney 2000]

Historically, the mathematicians’ use of unreal, immeasurable numbers to explain real phenomena was initially developed by Aristotle to resolve Zeno’s troubling paradoxes.

Scientists, in the intervening 2,300 years, have further refied the use of such unknowable numbers. Of particular note were Newton and Leibniz who developed (independently of each other) differential calculus in the latter half of the 17thcentury. Differential calculus greatly refined the use of these unknowable infinitesimals to such an extent that it became the mathematical foundation of the industrial revolution.

As physicist and science writer Richard Morris noted:

As a result, calculus is not only used to resounding effect by the physical sciences, but also by the biological and social sciences.

Differential and integral calculusii has become an indispensable tool in our modern technological society. In fact, physicist Richard Morris claimed that

The tremendous success of calculus (and the use of infinitesimals

in general) exacerbated the contrast between the practical real world and the theoretical use of unreal measures. So it is not surprising that with the advent of calculus came those who were at odds with the idea of using unreal measures in order to explain real phenomena.

Richard Morris, in his book “Achilles in the Quantum Universe,” cites one such example.
In 1734, British philosopher Bishop George Berkeley published a book in which he argued that

1. 1. Richard Morris, Achilles in the quantum universe: the definitive history of infinity, Souvenir Press, London 1998, page 63
2. 2. “Calculus,” The Funk & Wagnalls New Encyclopedia, (Electronic Edition), Funk & Wagnalls Corporation, 1994.
3. 3. Morris, page 63.
4. 4. Morris, page 65 (citing Bishop George Berkeley, The Analyst Or a Discourse Addressed to an Infidel Mathematician....)

[Excerpt  Be and Become, ProCreative, Sydney 2000]

In our normal, everyday world we take for granted the ease with which we can observe the strict correspondence between cause and effect. For example, when we shoot a projectile, such as Zeno’s arrow mentioned earlier, we know that it will basically travel in a straight line, save for the curve of trajectory due to gravity or cross wind.

Overall, when we experience the world of macro-sized things—i.e. the normal physical world, things behave as Newton predicted. As covered in the previous chapter in the section “Prisoners of Light” up to the speed of light, things remain pretty much ordinary.

There appears an upper limit to physical reality in terms of how fast things can go. But what about a lower limit? What happens as we examine smaller and smaller segments of both space and time? What do we find as we come closer and closer to the evasive now-moment? Might we start to see the discontinuity of the space-time continuum, as might be expected from consideration of Zeno’s paradox? The study of things microscopic is where quantum research is largely focused. And it is the observation of things at the microscopic level which has shaken, perplexed and indeed shocked scientists the world over for nearly 80 years.

As can be expected from consideration of the flight of Zeno’s arrow, when particles of microscopic size are fired at a target and we attempt to observe their flight, the particles don’t appear to travel a continuous, predictable trajectory. What initially startled the physicists was that the moment-by-moment detailed trajectory of a particle is in fundamental and unavoidable terms, discontinuous and unpredictable.

To appreciate what happens, let’s first consider what happens in our normal everyday world when we shoot an arrow, or throw a ball. As the ball leaves our hand, or the arrow leaves the bow, we can easily observe the trajectory of the ball or arrow. We do so by focussing our eyes, for example, on the light which is reflected off the travelling ball or arrow. The absence of the reflected light makes observations rather difficult, as you will no doubt appreciate if you’ve ever wondered around in the pitch dark without the aid of a flashlight. But in the world of electrons and atoms, when we attempt to “see” the trajectory of the particle, by bouncing light off the particle, we observe bizarre behavior indeed. Behavior which is so bizarre in fact, that one of the pioneers of quantum physics research, Niels Bohr, as mentioned earlier, once made the rather frank admission that he considered the observed behavior and the implications of that behavior as “shocking.”

It seems that we are generally insulated from observing such shocking behavior because we are, in relative terms, big slow dullards, cocooned within the limited perceptions of our local senses (tuned as they are to the macro-sized world of bears, balls and battle-ships). You might say that the world appears as it does due to our unblinking faith in it. If perhaps we were to blink our eyes fast enough, we might be shocked by what we saw (or didn’t see).

What then specifically are these so-called “shocking” behaviors?

[Excerpt  Be and Become, ProCreative, Sydney 2000]

As mentioned in the previous section, when physicists attempt to follow or observe the detailed trajectory of say an electron, things are not so deterministic or certain as in our normal everyday world of balls or arrows. When we throw a ball, for example we can predict quite precisely with mathematics (calculus) the trajectory of the ball. We can do so because the macro-sized world we inhabit appears to be continuous and predictable.

Newton’s laws of physics, for example, are continuously and predictably applicable to the real world of things (putting aside Zeno and his troublesome paradoxes). However, in the world of the quantum, things are not so continuous, predictable or certain.

Perhaps the most significant observation that first began to upset Newton’s mechanistic and predictable model of the world was the observation, around the turn of this century, of the photo-electric effect. The photo-electric effect could not be explained by any of science’s existing schools of factualisms—factualisms which all required continuity and predictability as a necessary condition. In essence, the photo-electric effect occurs when electrons are ejected from a metallic surface when it is irradiated with light (or any electromagnetic energy, such as x-rays and radio waves). The odd thing about the photoelectric effect is that different colored light ejects electrons with different velocity (energy). Brighter light simply ejects more electrons, not faster electrons. This is counter-intuitive to what we might expect—a brighter, more intense light might be expected to burn off an electron more vigorously, perhaps in a similar manner in which normal sunlight focused through a magnifying glass can vigorously burn wood and paper.

The photo-electric effect could in the end only be explained by accepting that light waves were quantized, i.e. discontinuous. Einstein received the Nobel Prize in Physics for ultimately solving the dilemma of the photo-electric effect by proposing that light behaved, in such situations, as discrete particles. Up until that time, light was commonly accepted to be a continuous wave. Such wave-like behavior was well established around the turn of the century.

This new development by Einstein was quite revolutionary—light somehow was both wave and/or particle, depending upon how and when it was observed. Unlike previously when light was thought to be wavelike, with Einstein’s development came the uncertainty of how to conceptualize energy (light). What did it really mean for light to behave as either a wave or a particle?

As if that wasn’t enough to upset the deterministic scientists, in 1923 Louis de Broglie submitted his Ph.D. thesis to his physics professors suggesting that electrons (matter) in an atomic orbit were associated with a wave. Not only did light seem to behave as both wave and particle, but here was a young physicist proposing that matter behaved as both wave and particle. His idea was that the observed behavior of an electron in an atomic orbit could be explained by the use of a wave equation.

"Roughly speaking, the electrons in the atom must fit around the nucleus as some sort of standing wave analogous to the waves on a plucked violin or guitar string. As the fit determines the wavelength of the quantum wave, it necessarily determines its energy state. Consequently, atomic systems are restricted to certain discrete, or quantized, energies."11

As physicist Fred Alan Wolf noted

Scientists subsequently realized that

For his bold thesis, de Broglie was to eventually receive the Nobel Prize in Physics. It is very important to realize that the electron does not wiggle around the nucleus in a wave-like manner, but that its range of possible characteristics (such as position and velocity) prior to it actually being observed (or measured by some device) as a discrete particle, will be given by a wave function.

As Norman Friedman explained:

Now, the wave function is one of the cornerstones of quantum physics, so it behoves us to better appreciate its nature. Let’s begin by considering the analogy of the “Mexican wave.” Perhaps you have seen a “Mexican wave” at a football match, whereby various members of the crowd raise their arms in a timely manner to produce a ripple, or wave of hands which “travels” around the stadium. The people don’t travel, only the “wave” travels around the stadium. Assume for the purposes of this analogy that you are quite distant from (or above) the stadium such that you cannot see individual people, only the seamless crowd and the “Mexican wave” rippling around the stadium. Assume further that in being so distant we need to use some mechanical or electronic apparatus (such as a fixed aperture telescope) in order to see individual spectators.

This analogy for it demonstrates the important characteristics of the quantum physical wave. And that is that the wave is comprised of individual “particles”i e.g. electrons) joining together to form the appearance of a wave. The Mexican wave also suggests that each particle (or “spectator-participant”) which forms the wave is aware of the behavior of each other particle (“spectator-participant”) and that the wave is a cooperative process amongst individual particles (“spectator-participants”).

In other words, the “Mexican wave” shows how the wave is comprised of discontinuous, separate parts (particles, “spectators”, anti-particles) joining together with other parts to produce the wave. This analogy is, I believe, of crucial importance in understanding key aspects of quantum physics.

Let’s get back to the historical developments in quantum physics. Additional experimental evidence was soon to show all matter and energy exhibited this strange duality of behaving as either a wave or as a particle, depending again on when and how it was observed. In quantum physics, this phenomena is called the Wave-Particle Duality. There are no exceptions to this Wave-Particle Duality of matter and energy.

In recent years, experimental evidence has shown that matter and energy behave as both waves and particles at the same time. The significance of this Wave-Particle Duality is perhaps at first difficult to appreciate.

Much of science is still based on deterministic mechanisms in which atoms, electrons and photons are believed to act as discrete separate things, much like colliding billiard balls. As a wave, the particle is not anywhere specifically, but is instead spread out or “smeared” across space. This was one of the first “shocking” developments of quantum physics—namely, that when we are not actually watching (measuring) something, it is “everywhere at once.” That is to say, it is everywhere it can be at the same time (i.e. it is “smeared” out in space). However the particle is not diminished, flattened or in any way thinned out, while it is “smeared” around space. It always remains a complete particle in experiments where the wave-particle nature is being investigated—for example, no one has observed half an electron.

As for being “everywhere at once”, Richard Morris explains:

It is helpful here to consider the foregoing in terms of the “Mexican wave” analogy mentioned previously. When we focus our sight or field of view on only one spectator via the telescope, we will be unaware of the wave which travels around the stadium. All that you will observe through the telescope is the spectator raising his or her arms momentarily. Since we are quite distant from the stadium it is only when we look through our fixed aperture telescope that we see individual people. When we turn away from the telescope and look at the stadium with our naked eyes we see a seamless circle of humanity with (in the event of a Mexican wave) a strange ripple proceeding around the stadium.

When we are looking through the telescope at only one spectator, we could say, in a sense, that the other spectators aren’t real—you only get to confirm their reality (or presence) by focussing on each of them in turn.