Hyperspace (19 page)

Freund explains it this way:

Think of some imaginary people living in Lineland, which consists of a single line. Throughout their history, they believed that their world was just a single line. Then, a scientist in Lineland proposed that their world was not just a one-dimensional line, but a two-dimensional world. When
asked where this mysterious and unobservable second dimension was, he would reply that the second dimension was curled up into a small ball. Thus, the line people actually live on the surface of a long, but very thin, cylinder. The radius of the cylinder is too small to be measured; it is so small, in fact, that it appears that the world is just a line.
¹⁵

If the radius of the cylinder were larger, the line people could move off their universe and move perpendicular to their line world. In other words, they could perform interdimensional travel. As they moved perpendicular to Lineland, they would encounter an infinite number of parallel line worlds that coexisted with their universe. As they moved farther into the second dimension, they would eventually return to their own line world.

Now think of Flatlanders living on a plane. Likewise, a scientist on Flatland may make the outrageous claim that traveling through the third dimension is possible. In principle, a Flatlander could rise off the surface of Flatland. As this Flatlander slowly floated upward in the third dimension, his “eyes” would see an incredible sequence of different parallel universes, each coexisting with his universe. Because his eyes would be able to see only parallel to the surface of Flatland, he would see different Flatland universes appearing before him. If the Flatlander drifted too far above the plane, eventually he would return to his original Flatland universe.

Now, imagine that our present three-dimensional world actually has another dimension that has curled up into a circle. For the sake of argument, assume that the fifth dimension is 10 feet long. By leaping into the fifth dimension, we simply disappear instantly from our present universe. Once we move in the fifth dimension, we find that, after moving 10 feet, we are back where we started from. But why did the fifth dimension curl up into a circle in the first place? In 1926, the mathematician Oskar Klein made several improvements on the theory, stating that perhaps the quantum theory could explain why the fifth dimension rolled up. On this basis, he calculated that the size of the fifth dimension should be 10
⁻³³
centimeters (the Planck length), which is much too small for any earthly experiment to detect its presence. (This is the same argument used today to justify the ten-dimensional theory.)

On the one hand, this meant that the theory was in agreement with experiment because the fifth dimension was too small to be measured. On the other hand, it also meant that the fifth dimension was so fantastically small that one could never build machines powerful enough to prove the theory was really correct. (The quantum physicist Wolfgang
Pauli, in his usual caustic way, would dismiss theories he didn’t like by saying, “It isn’t even wrong.” In other words, they were so half-baked that one could not even determine if they were correct. Given the fact that Kaluza’s theory could not be tested, one could also say that it wasn’t even wrong.)

As promising as Kaluza-Klein theory was for giving a purely geometric foundation to the forces of nature, by the 1930s the theory was dead. On the one hand, physicists weren’t convinced that the fifth dimension really existed. Klein’s conjecture that the fifth dimension was curled up into a tiny circle the size of the Planck length was untestable. The energy necessary to probe this tiny distance can be computed, and it is called the
Planck energy
, or 10
¹⁹
billion electron volts. This fabulous energy is almost beyond comprehension. It is 100 billion billion times the energy locked in a proton, an energy beyond anything we will be able to produce within the next several centuries.

On the other hand, physicists left this area of research in droves because of the discovery of a new theory that was revolutionizing the world of science. The tidal wave unleashed by this theory of the subatomic world completely swamped research in Kaluza-Klein theory. The new theory was called quantum mechanics, and it sounded the death knell for Kaluza-Klein theory for the next 60 years. Worse, quantum mechanics challenged the smooth, geometric interpretation of forces, replacing it with discrete packets of energy.

Was the program initiated by Riemann and Einstein completely wrong?

Anyone who is not shocked by the quantum theory does not understand it.

Niels Bohr

IN 1925, a new theory burst into existence. With dizzying, almost meteoric speed, this theory overthrew long-cherished notions about matter that had been held since the time of the Greeks. Almost effortlessly, it vanquished scores of long-standing fundamental problems that had stumped physicists for centuries. What is matter made of? What holds it together? Why does it come in an infinite variety of forms, such as gases, metals, rocks, liquids, crystals, ceramics, glasses, lightning bolts, stars, and so on?

The new theory was christened
quantum mechanics
, and gave us the first comprehensive formulation with which to pry open the secrets of the atom. The subatomic world, once a forbidden realm for physicists, now began to spill its secrets into the open.

To understand the speed with which this revolution demolished its rivals, we note that in the early 1920s some scientists still held serious reservations about the existence of “atoms.” What couldn’t be seen or measured directly in the laboratory, they scoffed, didn’t exist. But by 1925 and 1926, Erwin Schrödinger, Werner Heisenberg, and others had
developed an almost complete mathematical description of the hydrogen atom. With devastating precision, they could now explain nearly all the properties of the hydrogen atom from pure mathematics. By 1930, quantum physicists such as Paul A. M. Dirac were declaring that
all of chemistry
could be derived from first principles. They even made the brash claim that, given enough time on a calculating machine, they could predict all the chemical properties of matter found in the universe. To them, chemistry would no longer be a fundamental science. From now on, it would be “applied physics.”

Not only did its dazzling rise include a definitive explanation of the bizarre properties of the atomic world; but quantum mechanics also eclipsed Einstein’s work for many decades: One of the first casualties of the quantum revolution was Einstein’s geometric theory of the universe. In the halls of the Institute for Advanced Study, young physicists began to whisper that Einstein was over the hill, that the quantum revolution had bypassed him completely. The younger generation rushed to read the latest papers written about quantum theory, not those about the theory of relativity. Even the director of the institute, J. Robert Oppenheimer, confided privately to his close friends that Einstein’s work was hopelessly behind the times. Even Einstein began to think of himself as an “old relic.”

Einstein’s dream, we recall, was to create a universe made of “marble”—that is, pure geometry. Einstein was repelled by the relative ugliness of matter, with its confusing, anarchistic jumble of forms, which he called “wood.” Einstein’s goal was to banish this blemish from his theories forever, to turn wood into marble. His ultimate hope was to create a theory of the universe based entirely on marble. To his horror, Einstein realized that the quantum theory was a theory made
entirely of wood!
Ironically, it now appeared that he had made a monumental blunder, that the universe apparently preferred wood to marble.

In the analogy between wood and marble, we recall that Einstein wanted to convert the tree in the marble plaza to a marble statue, creating a park completely made of marble. The quantum physicists, however, approached the problem from the opposite perspective. Their dream was to take a sledge hammer and pulverize all the marble. After removing the shattered marble pieces, they would cover the park completely with wood.

Quantum theory, in fact, turned Einstein on his head. In almost every sense of the word, quantum theory is the opposite of Einstein’s theory. Einstein’s general relativity is a theory of the cosmos, a theory of stars
and galaxies held together via the smooth fabric of space and time. Quantum theory, by contrast, is a theory of the microcosm, where subatomic particles are held together by particlelike forces dancing on the sterile stage of space-time, which is viewed as an empty arena, devoid of any content. Thus the two theories are hostile opposites. In fact, the tidal wave generated by the quantum revolution swamped all attempts at a geometric understanding of forces for over a half-century.

Throughout this book, we have developed the theme that the laws of physics appear simple and unified in higher dimensions. However, with the appearance of the quantum heresy after 1925, we see the first serious challenge to this theme. In fact, for the next 60 years, until the mid-1980s, the ideology of the quantum heretics would dominate the world of physics, almost burying the geometric ideas of Riemann and Einstein under an avalanche of undeniable successes and stunning experimental victories.

Fairly rapidly, quantum theory began to give us a comprehensive framework in which to describe the visible universe: The material universe consists of atoms and its constituents. There are about 100 different types of atoms, or elements, out of which we can build all the known forms of matter found on earth and even in outer space. Atoms, in turn, consist of electrons orbiting around nuclei, which in turn are composed of neutrons and protons. In essence, the key differences between Einstein’s beautiful geometric theory and quantum theory can now be summarized as follows.

1. Forces are created by the exchange of discrete packets of energy, called
quanta
.

In contrast to Einstein’s geometric picture of a “force,” in quantum theory light was to be chopped up into tiny pieces. These packets of light were named
photons
, and they behave very much like point particles. When two electrons bump into each other, they repel each other not because of the curvature of space, but because they exchange a packet of energy, the photon.

The energy of these photons is measured in units of something called
Planck’s constant
(ħ ∼ 10
⁻²⁷
erg sec). The almost infinitesimal size of Planck’s constant means that quantum theory gives tiny corrections to Newton’s laws. These are called
quantum corrections
, and can be neglected when describing our familiar, macroscopic world. That is why we can, for the most part, forget about quantum theory when describing everyday phenomena. However, when dealing with the microscopic subatomic
world, these quantum corrections begin to dominate any physical process, accounting for the bizarre, counterintuitive properties of subatomic particles.

2. Different forces are caused by the exchange of different quanta.

The weak force, for example, is caused by the exchange of a different type of quantum, called a
W
particle (
W
stands for “weak”). Similarly, the strong force holding the protons and neutrons together within the nucleus of the atom is caused by the exchange of subatomic particles called π mesons. Both
W
bosons and π mesons have been seen experimentally in the debris of atom smashers, thereby verifying the fundamental correctness of this approach. And finally, the subnuclear force holding the protons and neutrons and even the π mesons together are called gluons.

In this way, we have a new “unifying principle” for the laws of physics. We can unite the laws of electromagnetism, the weak force, and the strong force by postulating a variety of different quanta that mediate them. Three of the four forces (excluding gravity) are therefore united by quantum theory, giving us unification without geometry, which appears to contradict the theme of this book and everything we have considered so far.

3. We can never know simultaneously the velocity and position of a subatomic particle.

This is the Heisenberg Uncertainty Principle, which is by far the most controversial aspect of the theory, but one that has resisted every challenge in the laboratory for half a century. There is no known experimental deviation to this rule.

The Uncertainty Principle means that we can never be sure where an electron is or what its velocity is. The best we can do is to calculate the probability that the electron will appear at a certain place with a certain velocity. The situation is not as hopeless as one might suspect, because we can calculate with mathematical rigor the probability of finding that electron. Although the electron is a point particle, it is accompanied by a wave that obeys a well-defined equation, the Schrödinger wave equation. Roughly speaking, the larger the wave, the greater the probability of finding the electron at that point.

Thus quantum theory merges concepts of both particle and wave into a nice dialectic: The fundamental physical objects of nature are particles, but the probability of finding a particle at any given place in space and
time is given by a probability wave. This wave, in turn, obeys a well-defined mathematical equation given by Schrödinger.

What is so crazy about the quantum theory is that it reduces everything to these baffling probabilities. We can predict with great precision
how many
electrons in a beam will scatter when moving through a screen with holes in it. However, we can never know precisely
which
electron will scatter in which direction. This is not a matter of having crude instruments; according to Heisenberg, it is a law of nature.

This formulation, of course, had unsettling philosophical implications. The Newtonian vision held that the universe was a gigantic clock, wound at the beginning of time and ticking ever since because it obeyed Newton’s three laws of motion; this picture of the universe was now replaced by uncertainty and chance. Quantum theory demolished, once and for all, the Newtonian dream of mathematically predicting the motion of all the particles in the universe.