The God Equation

by Michio Kaku

  When visitors would come to visit Einstein’s house, he would ask them, “Does the moon exist because a mouse looks at it?” But no matter how much the quantum theory violated common sense, it had one thing going for it: it was experimentally correct. Predictions of the quantum theory have been tested to eleven decimal places, making it the most accurate theory of all time.

  Einstein would admit, however, that the quantum theory contained at least part of the truth. In 1929, he even recommended Schrödinger and Heisenberg for the Nobel Prize in Physics.

  Even today, there is no universal consensus among physicists concerning the cat problem. (The old Copenhagen interpretation of Niels Bohr, that the true cat emerges only because observation causes the wave of the cat to collapse, has fallen into disfavor, in part because with nanotechnology, we can now manipulate individual atoms and perform these experiments. What has become more popular is the multiverse, or many worlds, interpretation, where the universe splits in half, with one half containing a dead cat and the other containing a live cat.)

  With the success of quantum theory, physicists in the 1930s then turned their sights to a new prize, answering the age-old question: Why does the sun shine?

  Energy from the Sun

  Since time immemorial, the great religions of the world have exalted the sun, putting it at the very center of their mythology. The sun was one of the most powerful of the gods who ruled the heavens. To the Greeks, he was Helios, who grandly rode his blazing chariot across the sky every day, illuminating the world and giving it life. The Aztecs, Egyptians, and Hindus all had their version of the sun god.

  But during the Renaissance, some scientists tried to examine the sun through the lens of physics. If the sun were made of wood or oil, then it would have exhausted its fuel long ago. And if the vast reaches of outer space did not have air, then the sun’s flames would have been extinguished long ago. So the sun’s eternal energy was a mystery.

  In 1842, a grand challenge was issued to the scientists of the world. The French philosopher Auguste Comte, the founder of the philosophy of positivism, declared that science was indeed powerful, revealing many of the secrets of the universe, but there would be one thing that would forever be beyond the reach of science. Even the greatest scientists would never answer the question: What are the planets and the sun made of?

  This was a reasonable challenge, since the bedrock of science is testability. All discoveries of science have to be reproducible and tested in the lab, but it was clearly impossible to capture sun material in a bottle and bring it back to Earth. Hence, this answer would forever be beyond our grasp.

  Ironically, a few years after he made this claim in his book The Positive Philosophy, physicists met the challenge. The sun was primarily hydrogen.

  Comte had made a slight but crucial mistake. Yes, science must always be testable, but, as we’ve established, most science is actually done indirectly.

  Joseph von Fraunhofer was a nineteenth-century scientist who answered Comte by designing the most precise and accurate spectrographs of his time. (In a spectrograph, substances are heated up until they begin to glow with blackbody radiation. The light is then sent through a prism, where it creates a rainbow. Inside the band of colors, there are dark bands. These bands are created because electrons make quantum jumps from orbit to orbit, releasing and absorbing specific amounts of energy. Since each element creates its own characteristic bands, then each spectral band is like a fingerprint, allowing you to determine what this substance is made of. Spectrographs have also solved numerous crimes, by being able to identify where the mud of a criminal’s footprint came from or the nature of the toxins found in poison or the origin of microscopic fibers and hairs. Spectrographs allow you to re-create a crime scene by determining the chemical composition of everything present.)

  By analyzing the bands of light from the sun, Fraunhofer and others could tell that the sun was mainly made of hydrogen. (Strangely, physicists also found a new unknown substance in the sun. They named it helium, meaning “metal from the sun.” So helium was actually first found in the sun, rather than on Earth. Later, scientists realized that helium was a gas, not a metal.)

  But Fraunhofer made another important discovery. By analyzing the light from stars, he found that they were made of the same substances commonly found on the Earth. This was a profound discovery, since it indicated that the laws of physics were the same not just in the solar system but throughout the entire universe.

  Once Einstein’s theories had gained traction, physicists like Hans Bethe put it all together to determine what fuels the sun. If the sun is made of hydrogen, its immense gravity field can compress hydrogen until the protons fuse, creating helium and the higher elements. Since helium weighs a bit less than the protons and neutrons that combine to form it, this means that the missing mass went into energy, via Einstein’s formula E = mc2.

  Quantum Mechanics and War

  While physicists were debating the mind-bending paradoxes of the quantum theory, war clouds were gathering on the horizon. Adolf Hitler seized power in Germany in 1933, and waves of physicists were forced to flee Germany, be arrested, or worse.

  One day, Schrödinger was witnessing Nazi brownshirts harassing innocent Jewish bystanders and shopkeepers. When he tried to stop them, they turned on him and began to beat him. They finally stopped when one of the brownshirts recognized that the person they were beating had won the Nobel Prize in Physics. Shaken, Schrödinger would soon leave Austria. Alarmed by the daily reports of repression, the best and brightest of German science left their country.

  Planck, the father of the quantum theory, was ever the diplomat, and even personally pleaded with Hitler to stop the mass exodus of German scientists, which was bleeding the country of its finest minds. But Hitler simply yelled and screamed at Planck, denouncing the Jews. Afterward, Planck wrote that “it was impossible to talk to such a man.” (Sadly, Planck’s own son tried to assassinate Hitler, for which he was brutally tortured and then executed.)

  For decades, Einstein was asked whether his equation could unleash fabulous amounts of energy locked inside the atom. Einstein would always say no, that the energy released by one atom is too small to be of practical use.

  Hitler, however, wanted to use German superiority in science to create powerful weapons that the world had never seen before, weapons of terror, like the V-1 and V-2 rockets and the atomic bomb. After all, if the sun was powered by nuclear energy, then it might be possible to create a superweapon using the same source of power.

  The key insight into how to exploit Einstein’s equation came from physicist Leo Szilard. German physicists had shown that the uranium atom, when hit by neutrons, could split in half, releasing more neutrons. The energy released by the splitting of a single uranium atom was extremely tiny, but Szilard realized you could magnify the power of the uranium atom via a chain reaction: splitting one uranium atom released two neutrons. These neutrons could then fission two more uranium atoms, releasing four neutrons. Then you would have eight, sixteen, thirty-two, sixty-four (and so on) neutrons—that is, an exponential rise in the number of split uranium atoms, eventually creating enough energy to level a city.

  Suddenly, the arcane discussions that divided the physicists at the Solvay Conference became a question of life-and-death urgency, with the fate of entire populations, nations, and civilization itself at stake.

  Einstein was horrified when he learned that in Bohemia the Nazis were sealing off the pitchblende mines that contained uranium. Although a pacifist, Einstein felt compelled to write a fateful letter to President Franklin Roosevelt, urging the United States to build an atomic bomb. Roosevelt subsequently authorized the largest scientific project in history, the Manhattan Project.

  Back in Germany, Werner Heisenberg, arguably the most prominent quantum physicist on the planet, was appointed to be the head of the Nazi atomic bomb
project. According to some historians, so great was the fear that Heisenberg might beat the Allies to an atomic bomb that the OSS, the forerunner of the CIA, hatched a plan to assassinate him. In 1944, a former Brooklyn Dodgers catcher, Moe Berg, was given the job. Berg attended a talk Heisenberg gave in Zurich, with orders to kill the physicist if Berg thought that the German bomb effort was near completion. (This story is elaborated in Nicholas Dawidoff’s book The Catcher Was a Spy.)

  Fortunately, the Nazi bomb project was considerably behind the Allied effort. It was underfunded, chronically late, and its base was also being bombed by the Allies. Most important, Heisenberg had not yet solved a crucial problem in making the atomic bomb: determining the amount of enriched uranium or plutonium necessary to create a chain reaction, an amount known as the critical mass. (The actual amount is roughly twenty pounds of uranium-235, which could be held in the palm of your hand.)

  After the war, the world began to slowly learn that the arcane, obscure equations of the quantum theory held not only the key to atomic physics, but also perhaps to the destiny of the human race itself.

  Physicists, however, began to slowly return to the question that had puzzled them before the war: how to create a complete quantum theory of matter.

  4

  THEORY OF ALMOST EVERYTHING

  After the war, Einstein, the towering figure who had unlocked the cosmic relationship between matter and energy and discovered the secret of the stars, found himself lonely and isolated.

  Almost all recent progress in physics had been made in the quantum theory, not in the unified field theory. In fact, Einstein lamented that he was viewed as a relic by other physicists. His goal of finding a unified field theory was considered too difficult by most physicists, especially when the nuclear force remained a total mystery.

  Einstein commented, “I am generally regarded as a sort of petrified object, rendered blind and deaf by the years. I find this role not too distasteful, as it corresponds fairly well with my temperament.”

  In the past, there was a fundamental principle that guided Einstein’s work. In special relativity, his theory had to remain the same when interchanging X, Y, Z, and T. In general relativity, it was the equivalence principle, that gravity and acceleration could be equivalent. But in his quest for the theory of everything, Einstein failed to find a guiding principle. Even today, when I go through Einstein’s notebooks and calculations, I find plenty of ideas but no guiding principle. He himself realized that this would doom his ultimate quest. He once observed sadly, “I believe that in order to make real progress, one must again ferret out some general principle from nature.”

  He never found it. Einstein once bravely said that “God is subtle, but not malicious.” In his later years, he became frustrated and concluded, “I have second thoughts. Maybe God is malicious.”

  Although the quest for a unified field theory was ignored by most physicists, every now and then, someone would try their hand at creating one.

  Even Erwin Schrödinger tried. He modestly wrote to Einstein, “You are on a lion hunt, while I am speaking of rabbits.” Nevertheless, in 1947 Schrödinger held a press conference to announce his version of the unified field theory. Even Ireland’s prime minister, Éamon de Valera, showed up. Schrödinger said, “I believe I am right. I shall look an awful fool if I am wrong.” Einstein would later tell Schrödinger that he had also considered this theory and found it to be incorrect. In addition, his theory could not explain the nature of electrons and the atom.

  Werner Heisenberg and Wolfgang Pauli caught the bug too, and proposed their version of a unified field theory. Pauli was the biggest cynic in physics and a critic of Einstein’s program. He was famous for saying, “What God has torn asunder, let no man put together”—that is, if God had torn apart the forces in the universe, then who were we to try to put them back together?

  In 1958, Pauli gave a talk at Columbia University explaining the Heisenberg-Pauli unified field theory. Bohr was in the audience. After his talk, Bohr stood up and said, “We in the back are convinced your theory is crazy. What divides us is whether your theory is crazy enough.”

  This began a heated discussion, with Pauli claiming that his theory was crazy enough to be true, while others said his theory was not crazy enough. Physicist Jeremy Bernstein was in the audience, and he recalled, “It was an uncanny encounter of two giants of modern physics. I kept wondering what in the world a non-physicist visitor would have made of it.”

  Bohr was right; the theory presented by Pauli would later be shown to be incorrect.

  But Bohr had actually hit upon something important. All the easy, obvious theories had already been tried by Einstein and his associates, and they all failed. Therefore, the true unified field theory must be radically different from all previous approaches. It must be “crazy enough” to qualify as a true theory of everything.

  QED

  The real progress in the postwar era was made in developing a complete quantum theory of light and electrons, called quantum electrodynamics, or QED. The goal was to combine Dirac’s theory of the electron with Maxwell’s theory of light, thereby creating a theory of light and electrons that obeyed quantum mechanics and special relativity. (A theory combining Dirac electrons with general relativity, however, was considered to be much too difficult.)

  Back in 1930, Robert Oppenheimer (who would later lead the project to build the atomic bomb) realized something profoundly disturbing. When one tried to describe the quantum theory of an electron interacting with a photon, one found that the quantum corrections actually diverged, yielding useless, infinite results. Quantum corrections were supposed to be small—that had been the guiding principle for decades. So there was an essential flaw in simply combining the Dirac equation of electrons and Maxwell’s theory of photons. This haunted physicists for nearly two decades. Many physicists worked on this problem, but little progress was made.

  Finally, in 1949, three young physicists working independently, Richard Feynman and Julian Schwinger in the United States, and Shin’Ichiro Tomonaga in Japan, cracked this long-standing problem.

  They were spectacularly successful, able to compute things like the magnetic property of the electron with enormous accuracy. But the way they did it was controversial and still causes physicists some unease and consternation even today.

  They started with the Dirac equation and Maxwell’s equation, where the mass and charge of the electron are given certain initial values (called the “bare mass and bare charge”). Then they calculated the quantum corrections to the bare mass and charge. These quantum corrections were infinite. This was the problem found earlier by Oppenheimer.

  But the magic occurs here. If we assume that the original bare mass and charge were actually infinite to start with, and then calculate the infinite quantum corrections, we find that these two infinite numbers can cancel each other out, leaving a finite result! In other words, infinity minus infinity equals zero!

  This was a crazy idea, but it worked. The strength of the magnetic field of the electron could be calculated using QED to an astonishing accuracy—that is, one part in one hundred billion.

  “The numerical agreement between theory and experiment here is perhaps the most impressive in all science,” Steven Weinberg noted. It is like calculating the distance from Los Angeles to New York to within the diameter of a hair. Schwinger was so proud of this that he had the symbol for this result carved on his gravestone.

  This method is called renormalization theory. The procedure, however, is arduous, complex, and mind-numbing. Literally thousands of terms have to be computed exactly, and they all have to cancel precisely. The tiniest error in this thick book of equations can throw off the entire calculation. (It is no exaggeration to say that some physicists spend their entire lives calculating quantum corrections to the next decimal place using renormalization theory.)

  Because
the process of renormalization is so difficult, even Dirac, who helped to create QED in the first place, did not like it. Dirac felt that it seemed totally artificial, like brushing things under the rug. He once said, “This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it turns out to be small—not neglecting it just because it is infinitely great and you do not want it!”

  Renormalization theory, which could combine Einstein’s special relativity with Maxwell’s electromagnetism, is indeed supremely ugly. One has to master an encyclopedia of mathematical tricks in order to cancel thousands of terms. But you cannot argue with results.

  Applications of the Quantum Revolution

  This, in turn, paved the way for a remarkable set of discoveries, which would bring about the third great revolution in history, the high-tech revolution, including transistors and lasers, and thus help create the modern world.

  Consider the transistor, perhaps the pivotal invention of the last hundred years. The transistor ushered in the information revolution, with a vast network of telecommunications systems, computers, and the internet. A transistor is basically a gate that controls the flow of electrons. Think of a valve. With a slight turn of a valve, we can control the flow of water in a pipe. In the same way, a transistor is like a tiny electronic valve that allows a small amount of electricity to control the much larger flow of electrons in a wire. Thus, a small signal can be amplified.

  Similarly, the laser, one of the most versatile optical devices in history, is another by-product of the quantum theory. To create a gas laser, start with a tube of hydrogen and helium. Then inject energy into it (by applying an electric current). This sudden injection of energy causes trillions of electrons in the gas to jump to a higher energy level. However, this array of energized atoms is unstable. If one electron decays to a lower level, it releases a photon of light, which hits a neighboring pumped-up atom. This causes the second atom to decay and release another photon. Quantum mechanics predicts that the second photon vibrates in unison with the first. Mirrors can be placed at either end of the tube, magnifying this flood of photons. Eventually, this process causes a gigantic avalanche of photons, all vibrating back and forth between the mirrors in unison, creating the laser beam.