S2 = x2 + y2 + z2 + t2
but rather as
S2 = x2 + y2 + z2 - t2.
The minus sign that appears in front of t2 in the definition of space-time length, S, gives Minkowski space its special characteristics, and it is the reason our different perspectives of space and time when we are moving relative to one another are not simple rotations, as in the case of Plato’s cave, but something a little more complicated.
Nevertheless, in one fell swoop, the very nature of our universe had changed. As Minkowski poetically put it in 1908: “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.”
Thus, on the surface, Einstein’s Special Theory of Relativity appears to make physical reality subjective and observer dependent, but relativity is in this sense a misnomer. The Theory of Relativity is instead a theory of absolutes. Space and time measurements may be subjective, but “space-time” measurements are universal and absolute. The speed of light is universal and absolute. And four-dimensional Minkowski space is the field on which the game of nature is played.
The depth of the radical change in perspective brought about by Minkowski’s reframing of Einstein’s theory can perhaps best be understood by considering Einstein’s own reactions to Minkowski’s picture. Initially Einstein called it “superfluous learnedness,” suggesting that it was simply fancy mathematics, devoid of physical significance. Shortly thereafter he emphasized this by saying, “Since the mathematicians have invaded relativity theory, I do not understand it myself anymore.” Ultimately, however, as happened several times in his lifetime, Einstein came around and recognized that this insight was essential to understand the true nature of space and time, and he later built his General Theory of Relativity on the foundation that Minkowski had laid.
It would have been difficult if not impossible to guess that Faraday’s spinning wheels and magnets would eventually lead to such a profound revision in our understanding of space and time. With the spectacles of hindsight, however, we could have had at least an inkling that the unification of electricity and magnetism could have heralded a world where motion would reveal a new underlying reality.
Returning to Faraday and Maxwell, one of the important discoveries that started the ball rolling was that a magnet acts on a moving electric charge with an odd force. Instead of pushing the charge forward or backward, the magnet exerts a force always at right angles to the motion of the electric charge. This force, now called the Lorentz force—after Hendrik Lorentz, a physicist who came close to discovering relativity himself—can be pictured as follows:
The charge moving between the poles of the magnet gets pushed upward.
But now consider how things would look from the frame of the particle. In its frame, the magnet would be moving past it.
But by convention we think of an electrically charged particle at rest as being affected only by electric forces. Thus, since the particle is at rest in this frame, the force pushing the particle upward in this picture would be interpreted as an electric force.
One person’s magnetism is therefore another person’s electricity, and what connects the two is motion. The unification of electricity and magnetism reflects at its heart that uniform relative motion gives observers different perspectives of reality.
Motion, a subject first explored by Galileo, ultimately provided, three centuries later, a key to a new reality—one in which not only electricity and magnetism were unified, but also space and time. No one could have anticipated this saga at its beginning.
But that is the beauty of the greatest story ever told.
Chapter 6
* * *
THE SHADOWS OF REALITY
As they were walking along and talking together, suddenly a chariot of fire and horses of fire appeared and separated the two of them.
—2 KINGS 2:11
One might have thought that, in 1908, following the aftershock of the discovery of an unexpected hidden connection between space and time, nature couldn’t have gotten much stranger. But the cosmos doesn’t care about our sensibilities. And once again, light provided the key to the door of the rabbit hole to a world that makes Alice’s experiences seem tame.
While they may be strange, the connections unearthed by Einstein and Minkowski can be intuitively understood—given the constancy of the speed of light—as I have tried to demonstrate. Far less intuitive was the next discovery, which was that on very small scales, nature behaves in a way that human intuition cannot ever fully embrace, because we cannot directly experience the behavior itself. As Richard Feynman once argued, no one understands quantum mechanics—if by understand one means developing a concrete physical picture that appears fully intuitive.
Even many years after the rules of quantum mechanics were discovered, the discipline would keep yielding surprises. For example, in 1952 the astrophysicist Hanbury Brown built an apparatus to measure the angular size of large radio sources in the sky. It worked so well that he and a colleague, Richard Twiss, applied the same idea to try to measure the optical light from individual stars and determine their angular size. Many physicists claimed that their instrument, called an intensity interferometer, could not possibly work. Quantum mechanics, they argued, would rule it out.
But it worked. It wasn’t the first time physicists had been wrong about quantum mechanics, and it wouldn’t be the last. . . .
Coming to grips with the strange behavior of quantum mechanics means often accepting the seemingly impossible. As Brown himself amusingly put it when trying to explain the theory of his intensity interferometer, he and Twiss were expounding the “paradoxical nature of light, or if you like, explaining the incomprehensible—an activity closely, and interestingly, analogous to preaching the Athanasian Creed.” Indeed, like many of the stranger effects in quantum mechanics, the Holy Trinity—Father, Son, and Holy Ghost all embodied at the same time in a single being—is also seemingly impossible. The similarity ends there, however.
Common sense also tells us that light cannot be both a wave and a particle at the same time. However, in spite of what common sense suggests, and whether we like it or not, experiments tell us it is so. Unlike the Creed, developed in the fifth century, this fact is not a matter of semantics or choice or belief. So we don’t need to recite quantum mechanics creeds every week to make them seem less bizarre or more believable.
One hears about the “interpretation of quantum mechanics” for good reason, because the “classical” picture of reality—namely the picture given by Newton’s laws of classical motion of the world as we experience it on human scales—is inadequate to capture the full picture. The surface world we experience hides key aspects of the processes that underlie the phenomena we observe. So too Plato’s philosophers could not discover the biological processes that govern humans by observing just the shadows of humans on the wall. No level of analysis would be likely to allow them to intuit the full reality underlying the dark forms.
The quantum world defies our notion of what is sensible—or even possible. It implies that at small scales and for short times, the simple classical behavior of macroscopic objects—baseballs thrown from pitcher to catcher, for example—simply breaks down. Instead, on small scales, objects are undergoing many different classical behaviors—as well as classically forbidden behaviors—at the same time.
Quantum mechanics, like almost all of physics since Plato, began with scientists thinking about light. So it is appropriate to begin to explore quantum craziness by starting with light, in this case by returning to an important experiment first reported by the British polymath Thomas Young around 1800—the famous “double-slit experiment.”
Young lived in an era that is hard to appreciate today, when a brilliant and hardworking individual could make breakthroughs in a host of different fields. But Young was not just any brilliant hardworking individual. He was a prodigy, reading at two, and by the age of thirteen he had r
ead the major Greek and Latin epic poems, had built a microscope and a telescope, and was learning four different languages. Later, trained as a medical doctor, Young was the first to propose, in 1806, the modern concept of energy, which now permeates every field of scientific endeavor. That alone would have made him memorable, but in his spare time he also was one of the first to help decipher the hieroglyphics on the Rosetta stone. He developed the physics of elastic materials, associated with what is now called Young’s modulus, and helped first elucidate the physiology of color vision. And his brave demonstration of the wave nature of light (which argued against Isaac Newton’s powerful claim that light was made of particles) was so compelling that it helped lay the basis of Maxwell’s discovery of electromagnetic waves.
Young’s experiment is simple. Let’s return to Plato’s cave and consider a screen placed in front of the back wall of the cave. Place two slits in the screen as shown below (as seen from above):
If the light is made of particles, then those light rays that pierce the slits would form two bright lines on the wall behind these two slits:
However, it was well known that waves, unlike particles, diffract around barriers and narrow slits and would produce a very different pattern on the wall. If waves impinge on the barrier, and if each slit is narrow, a circular pattern of waves is generated at each slit, and the patterns from the two slits can “interfere” with each other, sometimes constructively and sometimes destructively. The result is a pattern of bright and dark regions on the back wall, as shown below:
Using just such an apparatus, with narrow slits, Young reported this interference pattern, characteristic of waves, and so definitively demonstrated the wave nature of light. In 1804, this was a milestone in the history of physics.
One can try the same experiment that Young tried for light on elementary particles such as electrons. If we send a beam of electrons toward a phosphorescent screen, like the screen in old-fashioned television sets, you will see a bright dot where the beam hits the screen. Now imagine that we put two slits in front of the screen, as Young did for light, and aim a wide stream of electrons at the screen:
Here, based on the reasoning I gave when I discussed the behavior of light, you would expect to see a bright line behind each of the two slits, where the electrons could pass through to the screen. However, as you have probably already guessed, this is not what you would see, at least if the slits are narrow enough and close enough. Instead, you see an interference pattern similar to that which Young observed for light waves. Electrons, which are particles, seem to behave in this case just like waves of light. In quantum mechanics, particles have wavelike properties.
That the electron “waves” emanating from one slit can interfere with electron “waves” emanating from the other slit is unexpected and strange, but not nearly as strange as what happens if we send a stream of electrons toward the screen one at a time. Even in this case, the pattern that builds up on the screen is identical to the interference pattern. Somehow, each electron interferes with itself. Electrons are not billiard balls.
We can understand this as follows: The probability of an electron’s hitting the screen at each point is determined by treating each electron as not taking a single trajectory, but rather following many different trajectories at once, some of which go through one slit and some of which go through the other. Those that go through one slit then interfere with those that go through the other slit—producing the observed interference pattern at the screen.
Put more bluntly, one cannot say the electron goes through either one slit or the other, as a billiard ball would. Rather it goes through neither and at the same time it goes through both.
Nonsense, you insist. So you propose a variant of the experiment to prove it. Put an electron-measuring device at each slit that clicks when an electron passes through that slit.
Sure enough, as each electron makes its way to the screen, only one device clicks each time. So each electron apparently does go through one and only one slit, not both.
However, if you now look at the pattern of electrons accumulating at the screen behind the slits, the pattern will have changed from the original interference pattern to the originally expected pattern—with a bright region behind each of the two slits, just as if one were shooting billiard balls or bullets and not waves toward the screen.
In other words, in attempting to verify your classical intuition, you changed the behavior of the electrons. Or, as more commonly asserted in quantum mechanics, measurement of a system can alter its behavior.
One of the many seemingly impossible aspects of quantum mechanics is that there is no experiment you can perform that demonstrates that in the absence of measurement the electrons behave in a sensible classical way.
This strange wavelike nature of objects that would otherwise be considered to be particles—such as electrons—is mathematically expressed by assigning to each electron a “wave function,” which describes the probability of finding that electron at any given point. If the wave function takes on non-zero values at many different points, then the electron’s position cannot be isolated in advance of accurately measuring its position. In other words there is a non-zero probability that the electron is not actually localized at just some specific point in space in advance of making a measurement.
While you might imagine that this is a simple problem of not having access to all the information we need to locate the particle until we make a measurement, Young’s double-slit experiment, when updated for electrons, demonstrated that this is most certainly not the case. Any “sensible” classical picture of what is happening between measurements is inconsistent with the data.
• • •
The strange behavior of electrons was not the first evidence that the microscopic world could not be understood by intuitive classical logic. Once again, in keeping with the revolutionary developments in our understanding of nature since Plato, the discovery of quantum mechanics began with a consideration of light.
Recall that if we perform Young’s double-slit experiment in Plato’s cave with light rays, we get the interference pattern on the wall that Young discovered, which demonstrated that light was indeed a wave. So far, so good. However, if the light source is sufficiently weak, then if we try to detect the light as it passes through either of the slits, something strange happens. We will measure the light beam as traveling through one slit or the other, not both. And as with electrons, in this case the pattern on the wall will now change, looking as it would if light were particles and not waves.
In fact, light also behaves like both a particle and a wave, depending on the circumstances under which you choose to measure it. The individual particles of light, which we now call photons, were first labeled quanta by the German theoretical physicist Max Planck, who suggested in 1900 that light might be emitted or absorbed in some smallest bundle (although the idea that light might come in discrete packets had earlier been floated by the great Ludwig Boltzmann in 1877).
I have come to admire Planck even more as I have learned about his life. Like Einstein, he was an unpaid lecturer and was not offered an academic position after completing his thesis. During this time he spent his career trying to understand the nature of heat and developed several important pieces of work in thermodynamics. Five years after defending his thesis, he was finally offered a university position, and he then quickly rose up the ranks and became a full professor at the prestigious University of Berlin in 1892.
In 1894 he turned to the question of the nature of light emitted by hot objects, in part driven by commercial considerations (the first example I know of in the story I have been telling where fundamental physics was commercially motivated). He was commissioned to explore how to get the maximum amount of light out of the newly invented lightbulbs while using the minimum amount of energy.
We all know that when we heat up an oven element it first glows red, and then, when it gets hotter, it begins to glow blue. But why? Surprisingly, the conventio
nal approaches to this problem were unable to reproduce these observations. After struggling with the problem for six years, Planck presented a revolutionary proposal about radiation that agreed with observations.
Originally there was nothing revolutionary about his derivation, but within two months he had revised his analysis to accommodate ideas about what was happening at a fundamental level. In a quote that has endeared him to me since I first read it, he wrote that his new approach arose as “an act of despair. . . . I was ready to sacrifice any of my previous convictions about physics.”
This reflects to me the fundamental quality that makes the scientific process so effective, and which is so clearly represented in the rise of quantum mechanics. “Previous convictions” are just convictions waiting to be overturned—by empirical data, if necessary. We throw out cherished old notions like yesterday’s newspaper if they don’t work. And they didn’t work in explaining the nature of radiation emitted by matter.
Planck derived his law of radiation from the fundamental assumption that light, which was a wave, nevertheless was emitted only in “packets” of some minimum energy—proportional to the frequency of the radiation in question. He labeled the constant that related the energy to the frequency the “action quantum,” which is now called Planck’s constant.
The Greatest Story Ever Told—So Far Page 7