Now let’s get a little more advanced: our room is now three-dimensional like a real room, except for being infinite, this time in the x- and z-directions. But the vertical direction, y, is bounded above and below by the ceiling and floor. As before, when a particle passes through the ceiling, it reappears instantly at the corresponding point on the floor. Three-dimensional space has been compactified to two dimensions. If the height of the room—in other words, the distance around the y-direction—were shrunk to a microscopic size, the space would be practically two-dimensional.
As I said, M-theory has no strings, only membranes. So what is its connection to String Theory? Imagine a ribbon, whose width is exactly the height of the room, with its width stretched from floor to ceiling. The length of the ribbon wanders about the room following some curve inscribed on the floor. The only rule is that the upper edge of the ribbon must lie exactly above the lower edge. In fact the ribbon no more has edges than the cylinder of paper did. But it is easiest to visualize a long ribbon snaking through the infinite room with its edges following the ceiling and floor.
By now you must have a pretty good idea of how the ribbon, itself a two-dimensional membrane, mimics a one-dimensional string. If the compact direction were so small that it couldn’t be seen without a microscope, the ribbon would, for all practical purposes, be a string. If the ribbon closed back on itself, it would be indistinguishable from a closed string: a Type IIa string, to be precise. That is the connection between M-theory and String Theory. Strings are really very thin ribbons or membranes that look more and more like thin strings as the distance around the y-direction shrinks. That’s not so difficult.
But things can get stranger. Now let’s go another step and compactify two dimensions: call them z and y. In order to visualize this, imagine the infinite room replaced by an infinite hallway. You have walls to the left and right of you and a ceiling and floor above and below. But if you look straight down the hallway, you can see forever in either direction. Again, if an object gets to the ceiling, it reappears at the floor. But what if it approaches one of the walls that bound the z-direction? You probably already know the answer: it appears at the opposite wall, directly across from the place where it touched the first wall.
Exactly the same trick can be done in the ten-dimensional space of M-theory, only this time the “hallway” extends indefinitely in eight of the ten spatial directions. As you might expect, when the width and height of the hallway get very small, the clumsy, large-scale observer thinks that he is living in a world of eight dimensions (plus one of time).
Now comes a shocking and bizarre consequence of String Theory. If the width and height of the hallway get smaller than a certain size, a new dimension grows out of nowhere. This new direction of space is none of the ones we started with. We know about it only through the indirect mathematics of String Theory. The smaller we make the original compact directions, the bigger the newly created compact direction becomes. Eventually, as the hallway is shrunken to zero height and width, the new direction grows infinitely big. Astoundingly, by shrinking away two of the space dimensions, we find nine, not eight large directions left over. This peculiar fact—that “ten minus two equals nine”—is one of the strangest consequences of String Theory. The geometry of space is not always what Euclid or even Einstein thought. Evidently at the smallest distances, space is different from anything physicists or mathematicians imagined in their wildest dreams.
By now you may be slightly confused by the precise distinction between String Theory and M-theory. String theorists are also confused (and confusing) about the terminology. For example, is the eleven-dimensional theory, which contains membranes but no strings, part of String Theory? Is the compactified version of M-theory, when it morphs into String Theory, still M-theory? I’m afraid the field has been rather imprecise on these matters. My own terminology is to call everything that grew out of the original theory of strings String Theory. That includes everything that is now called M-theory. The term M-theory I use when I want to emphasize the eleven-dimensional features of the theory.
The story of String Theory goes on in chapter 10, but now I want to take a time-out from the difficult technical aspects of String Theory and turn to an issue that deeply concerns every serious physicist. In fact it concerns anyone who has an interest in understanding nature at the very deepest level.
CHAPTER NINE
On Our Own?
The search for the fundamental principles of physics is a very risky business. It’s like any other exploration into the deep unknown. There is no guarantee of success and plenty of possibility of becoming hopelessly lost. The physicist’s guiding star has always been experimental data, but in this respect things are harder than ever. All of us (physicists) are very aware of the fact that experiments designed to probe ever deeper into the structure of matter are becoming far bigger, more difficult, and costlier. The entire world’s economy for one hundred years would not be nearly enough to build an accelerator that could penetrate to the Planck scale, i.e., 10–33 centimeters. Based on today’s accelerator technology, we would need an accelerator that’s at least the size of the entire galaxy! Even if future technology could shrink it to a more manageable size, it would still need a trillion barrels of oil per second to power it.
How then can we hope to succeed? Without experimental tests and new discoveries to keep us on the right track, it may be a futile enterprise. On the other hand, some grand leap, perhaps involving String Theory, might allow us to ignore the experimental difficulties and create a theory that so accurately describes the Laws of Physics that there will be no doubt of its correctness. The truth is that we just don’t know if this is possible. What we are attempting is so bold that there is no historical precedent for it. Some think it’s quixotic: a fool’s errand. Even those doing it are doubtful of eventual success. To divine the fundamental laws of nature that govern a world sixteen orders of magnitude smaller than any microscope will ever see is a very tall order. It will take not only cleverness and perseverance but also tremendous quantities of chutzpah.
Is the human race anywhere near being smart enough? I mean collectively, not individually. Are the combined talents of humanity sufficient to solve the great riddle of existence? Is the human mind even wired in the right way to be able to understand the universe? What are the chances that the combined and diverse intellects of the world’s greatest physicists and mathematicians will be able to divine the final theory with only the absurdly limited experiments that will be possible?
It was these questions that I wanted to explore with my colleagues that evening in 1995, at the physicists’ banquet. I feel they are also important to discuss in this book, if for no other reason than to give the reader an idea of the difficulties physicists will be up against in the twenty-first century. To get some perspective at that time, I indulged in a little conceit, a thought experiment. I tried to imagine a best-case scenario for how physics might have evolved if physicists of the twentieth century had been deprived of all experimental results after December 31, 1899. Most people will tell you that physics, or science in general, would quickly have bogged down. They may very well be right. But then again, they may be lacking in imagination.
The precise question I wanted to explore at the banquet was how much of twentieth-century physics could prodigiously smart theoretical physicists have discovered without any new experimental guidance. Might they have discovered all or most of what we know today? I did not claim that they would have succeeded but only that there were lines of argument that could have led them toward much of today’s physics. In the rest of this chapter, I’ll take you through my thinking.
The twin pillars of twentieth-century physics were the Theory of Relativity and quantum mechanics. Both were born during the first few years of the century. Planck discovered his constant in 1900, and Einstein interpreted Planck’s work in terms of photons in 1905. Planck’s discovery involved nothing more than the properties of heat radiation: the glow of ele
ctromagnetic radiation emitted by a hot object. Physicists call this radiation blackbody radiation because it would be emitted even by a perfectly black object if it is heated. For example, even the blackest of pots will glow red if heated to thousands of degrees. Physicists by the year 1900 were not only familiar with the problem of blackbody radiation, they also were deeply troubled by an apparent contradiction. The mathematical theory showed that the total amount of energy in blackbody radiation was infinite. The amount of energy stored in each individual wavelength was finite, but according to nineteenth-century physics, when it was all added up, an infinite amount of energy would reside in the very short wavelengths; hence, the term ultraviolet catastrophe. In a sense it was a problem of the same sort as the mother of all physics problems: too much energy stored in very short wavelengths. Einstein solved it (the problem of heat radiation) with the radical but very well-motivated hypothesis that light consists of indivisible quanta. No role was played by any twentieth-century experiment.
The year of the photon was also the year of the Special Theory of Relativity. The famous Michelson-Morley experiment that failed to detect the earth’s motion through the ether was already thirteen years in the past when the century turned.1 In fact it is not clear that Einstein even knew of Michelson and Morley’s work. According to his own reminiscences, the main clue was Maxwell’s theory of light, which dated from the 1860s. Einstein, master of thought experiments, asked himself at the age of sixteen—the year was 1895—what a light beam would be like to someone moving alongside it with the speed of light. Even at this early age, he realized that a contradiction would result. This, not new experiments, was the soil from which his great discovery sprung.
By the end of the nineteenth century, physicists had begun the exploration of the microscopic world of electrons and the structure of matter. The great Dutch theoretical physicist Hendrik Antoon Lorentz had postulated the existence of electrons, and by 1897, the British physicist J. J. Thomson had discovered and studied their properties. Wilhelm Conrad Roentgen had made his dramatic discovery of X-rays in 1895. Following up on Roentgen’s discovery, Antoine-Henri Becquerel discovered radioactivity a year later.
But some things were not known until years later. It took Robert Millikan until 1911 to determine the precise value of the electron’s electric charge. And until Ernest Rutherford devised a clever experiment to probe the atom, the picture of electrons orbiting a tiny nucleus was not known, although some speculation along those lines did go back to the nineteenth century.2 And of course the modern idea of the atom goes way back to John Dalton, in the early years of the nineteenth century.
Rutherford’s discovery of the “planetary” structure of the atom—light electrons orbiting a heavy, tiny nucleus—was key. It led, in just two years, to Bohr’s theory of quantized orbits. But was it absolutely necessary? I doubt it. I was surprised to learn recently that Heisenberg’s first successful attempt to create a new quantum mechanics did not involve the atom at all.3 His first application of the radical “matrix mechanics” was to the theory of simple vibrating systems, so-called harmonic oscillators. In fact the Planck-Einstein theory was understood as a theory of the harmonic oscillation (vibration) of the radiation field. That the energy of an oscillator comes in discrete jumps is the analog of Bohr’s discrete orbits. It seems unlikely that Rutherford’s atom was essential to the discovery of quantum mechanics.
But still there was the problem of the atom. Could its solar system-like structure have been guessed? Here I think the key would have been spectroscopy, the study of the same spectral lines that Hubble used to determine the velocity of galaxies. There was a huge amount of nineteenth-century spectroscopic data. The details of the hydrogen spectrum were well known. On the other hand, the idea that the atom consists of electrons and some object with positive electric charge had been in the air for a few years by the time 1900 rolled around. I recently learned from a Japanese friend that the first speculations of a planetary atom (electrons orbiting a nucleus) were due to a Japanese physicist, Hantaro Nagaoka. There is even a Japanese postage stamp with a picture of Nagaoka and his atom.
Nagaoka’s paper, available on the Internet, dates from 1903, eight years before Rutherford’s experiment. Had the idea come a few years later, when more was known about quantum theory, the story might have been different. Given the wealth of spectroscopic data, the quantum behavior of oscillators, and Nagaoka’s idea, would a brilliant young Heisenberg or Dirac have had the necessary eureka moment? “Ah ha, I’ve got it. The positive charge is at the center, and the electrons orbit around it in quantized orbits.” Perhaps Bohr himself would have done it. Physicists have made much larger leaps than that: witness the General Theory of Relativity or, for that matter, the discovery of String Theory from the spectroscopy of hadrons.
And what about the General Theory of Relativity? Could it have been guessed without a twentieth-century experiment? Certainly! All that was required was Einstein’s thought experiment that led to the equivalence principle. Reconciling the equivalence principle with special relativity was the path taken by Einstein.
No serious theoretical physicist today is content with two apparently incompatible theories. I am referring, of course, to quantum mechanics and the General Theory of Relativity. In the late 1920s a very similar problem existed—how to reconcile quantum mechanics with special relativity. Physicists of the caliber of Dirac, Pauli, and Heisenberg would not, and did not, rest until they saw the Special Theory of Relativity reconciled with quantum mechanics. This would entail a relativistic quantum theory of the electron interacting with the electromagnetic field. Here I really don’t have to speculate. The early development of Quantum Electrodynamics was motivated by nothing other than Dirac’s desire for such a synthesis of quantum mechanics and special relativity. But would he have known that his Dirac equation was correct?
Here Pauli makes his dramatic entrance with the exclusion principle. What Pauli was motivated by was chemistry: the periodic table and how it was built up by successively placing electrons in atomic orbits. In order to understand how the electrons fill the atomic orbits and block other electrons from already filled orbits, Pauli had to invoke a new property of electrons, their so-called spin. And where did the idea of spin come from? Not from new, twentieth-century experiments, but rather from nineteenth-century spectroscopy and chemistry. The addition of the new spin degree of freedom meant that Pauli could place two electrons in every orbit, one with its spin pointing down and the other with its spin up. Thus, in helium two electrons fill the lowest Bohr orbit. This was the key to Mendeleyev’s periodic table. Pauli’s idea was a guess based on nineteenth-century chemistry, but Dirac’s relativistic theory of the electron precisely explained this new, mysterious property of spin.
But Dirac’s theory did have one serious problem. In the real world the energy associated with every particle is a positive quantity. At first Dirac’s theory seemed inconsistent—it had electrons, which carried negative energy! Particles with negative energy are a very bad sign. Remember that in an atom, electrons of higher energy eventually “drop down” to orbits of lower energy by emitting photons. The electrons seek out the lowest-energy orbit that isn’t blocked by the Pauli exclusion principle. But what if an infinite number of negative energy orbits were available to the electrons? Wouldn’t all the electrons in the world start cascading to increasingly negative energy, giving off enormous amounts of energy in the form of photons? Indeed, they would. This potentially damning feature of Dirac’s idea threatened to undermine his whole theory—unless something could prevent the electrons from occupying the negative energy states. Again Pauli saves the day. Pauli’s exclusion principle would rescue Dirac from disaster. Just suppose that what we normally call vacuum is really a state full of negative-energy electrons, one in every negative-energy orbit. What would the world be like? Well, you could still put electrons into the usual positive-energy orbits, but now when an electron gets to the lowest positive-energy orbit, it is blocked
from going any farther. For all intents and purposes, the negative-energy orbits might as well not exist, since an electron is effectively blocked from falling into these orbits by the presence of the so-called Dirac Sea of negative-energy electrons. Dirac declared the problem solved, and so it was.
This idea soon led to something new and totally unexpected. In an ordinary atom an electron can absorb the energy of nearby photons and be “kicked up” into a more energetic configuration.4 Dirac now showed his real brilliance. He reasoned that the same thing could happen to the negative-energy electrons that fill the vacuum; photons could kick negative-energy electrons up to positive-energy states. What would be left over would be one electron with positive energy and a missing negative-energy electron—a hole in the Dirac Sea. Being a missing electron, the hole would seem to have the opposite electric charge from the electron and would look just like a particle of positive charge. This, then, was Dirac’s prediction: particles should exist identical to electrons, except with the opposite electric charge. These positrons, which Feynman would later interpret as electrons going backward in time, Dirac pictured as holes in the vacuum. Moreover, they should be created together with ordinary electrons, when photons collide with enough energy.
Dirac’s prediction of antimatter was one of the great moments in the history of physics. It not only led to the subsequent experimental discovery of positrons, but it heralded the new subject of quantum field theory. It was the forerunner of Feynman’s discovery of Feynman diagrams and later led to the discovery of the Standard Model. But let’s not get ahead of the story.
Dirac wasn’t thinking about any experiment when he discovered his remarkable equation for the relativistic quantum mechanics of electrons. He was thinking about how the nonrelativistic Schrödinger equation could be made mathematically consistent with Einstein’s Special Theory of Relativity. Once he had the Dirac equation, the way lay open to the whole of Quantum Electrodynamics. Theorists studying QED would certainly have found the inconsistencies that were papered over by renormalization theory.5 There was no obstruction to the discovery of modern quantum field theory. And physicists would have puzzled endlessly over the enormous vacuum energy and why it didn’t gravitate. We might question whether theorists would have been willing to carry on without experimental confirmation of their ideas. We might question whether young people would want to pursue such a purely theoretical enterprise. But I don’t think we can question the possibility of physics progressing up to this point. Moreover, the thirty-five-year history of String Theory suggests that as long as someone will pay them, theoretical physicists will continue to push the mathematical frontiers until the end of time.
The Cosmic Landscape Page 28
