by Michio Kaku
But where does this cosmological constant come from? At present no one knows. At the beginning of time, the antigravity force was perhaps large enough to cause the universe to inflate, creating the big bang. Then it suddenly disappeared, for reasons that are unknown. (The universe was still expanding during this period, but at a slower pace.) And then, about eight billion years after the big bang, the antigravity force resurfaced again, causing the galaxies to push out and causing the universe to accelerate once again.
So is it “impossible” to determine the ultimate fate of the universe? Perhaps not. Most physicists believe that quantum effects ultimately determine the size of the cosmological constant. A naïve calculation, using a primitive version of the quantum theory, shows that the cosmological constant is off by a factor of 10120. This is the greatest mismatch in the history of science.
But there is also a consensus among physicists that this anomaly simply means that we need a theory of quantum gravity. Since the cosmological constant arises via quantum corrections, it is necessary to have a theory of everything—a theory that will allow us to calculate not just the Standard Model, but also the value of the cosmological constant, which will determine the ultimate fate of the universe.
So a theory of everything is necessary to determine the ultimate fate of the universe. The irony is that some physicists believe that it is impossible to attain a theory of everything.
A THEORY OF EVERYTHING?
As I mentioned earlier, string theory is the leading candidate for a “theory of everything,” yet there are opposing camps on whether the string theory lives up to this claim. On the one hand, people like MIT professor Max Tegmark write, “In 2056, I think you’ll be able to buy a T-shirt on which are printed equations describing the unified physical laws of our universe.” On the other hand, there is an emerging band of determined critics who claim that the string bandwagon has yet to deliver. No matter how many breathless articles or TV documentaries are produced concerning string theory, it has yet to produce a single testable fact, some say. It’s a theory of nothing, rather than a theory of everything, claim the critics. The debate heated up considerably in 2002 when Stephen Hawking switched sides, quoting the incompleteness theorem, and said that a theory of everything might even be mathematically impossible.
It’s not surprising that the debate has pitted physicist against physicist, because the goal is so lofty, if elusive. The quest to unify all the laws of nature has tantalized and lured philosophers and physicists alike for millennia. Socrates himself once said, “It seemed to me a superlative thing—to know the explanation of everything, why it comes to be, why it perishes, why it is.”
The first serious proposal for a theory of everything dates back to about 500 BC, when the Greek Pythagoreans are credited with deciphering the mathematical laws of music. By analyzing the nodes and vibrations of a lyre string, they showed that music obeyed remarkably simple mathematics. They then speculated that all of nature could be explained in the harmonies of the lyre string. (In some sense, string theory brings back the dream of the Pythagoreans.)
In modern times nearly all the giants of twentieth-century physics tried their luck with a unified field theory. But, as Freeman Dyson cautions, “The ground of physics is littered with the corpses of unified theories.”
In 1928 the New York Times ran the sensational headline “Einstein on verge of great discovery; resents intrusion.” The news story helped spark a media feeding frenzy over a theory of everything that was whipped up to a feverish pitch. Headlines blared “Einstein is amazed at stir over theory. Holds 100 journalists at bay for a week.” Scores of journalists swarmed around his home in Berlin, maintaining a non-stop vigil, waiting to catch a glimpse of the genius and grab a headline. Einstein was forced to go into hiding.
Astronomer Arthur Eddington wrote to Einstein: “You may be amused to hear that one of our great department stores in London (Selfridges) has posted on its window your paper (the six pages pasted up side by side) so that passers-by can read it all through. Large crowds gather around to read it.” (In 1923 Eddington proposed his own unified field theory on which he worked tirelessly for the rest of his life, until he died in 1944.)
In 1946 Erwin Schrödinger, one of the founders of quantum mechanics, held a press conference to propose his unified field theory. Even Ireland’s Prime Minister, Eamon De Valera, showed up. When a reporter asked him what he would do if his theory was wrong, Schrödinger replied, “I believe I am right. I shall look like an awful fool if I am wrong.” (Schrödinger was humiliated when Einstein politely pointed out the errors in his theory.)
The harshest of all critics of unification was physicist Wolfgang Pauli. He chided Einstein, saying, “What God has torn asunder, let no man put together.” He mercilessly put down any half-baked theory with the quip: “It’s not even wrong.” So it is ironic that the supreme cynic Pauli himself caught the bug. In the 1950s he proposed his own unified field theory with Werner Heisenberg.
In 1958 Pauli presented the Heisenberg-Pauli unified theory at Columbia University. Niels Bohr was in the audience, and he was not impressed. Bohr stood up and said, “We in the back are convinced that your theory is crazy. But what divides us is whether your theory is crazy enough.” The criticism was crushing. Since all the obvious theories had been considered and rejected, the true unified field theory must be a dazzling departure from the past. The Heisenberg-Pauli theory was simply too conventional, too ordinary, too sane to be the true theory. (That year Pauli was disturbed when Heisenberg commented on a radio broadcast that only a few technical details were left in their theory. Pauli sent his friends a letter with a blank rectangle, with the caption, “This is to show the world I can paint like Titian. Only technical details are missing.”)
CRITICISMS OF STRING THEORY
Today the leading (and only) candidate for a theory of everything is string theory. But, again, a backlash has arisen. Opponents claim that to get a tenured position at a top university you have to work on string theory. If you don’t you will be unemployed. It’s the fad of the moment, and it’s not good for physics.
I smile when I hear this criticism, because physics, like all human endeavors, is subject to fads and fashions. The fortunes of great theories, especially on the cutting edge of human knowledge, can rise and fall like hemlines. In fact, years ago the tables were turned; string theory was historically an outcast, a renegade theory, the victim of the bandwagon effect.
String theory was born in 1968, when two young postdocs, Gabriel Veneziano and Mahiko Suzuki, stumbled on a formula that seemed to describe the collisions of subatomic particles. Quickly it was discovered that this marvelous formula could be derived by the collision of vibrating strings. But by 1974 the theory was dead in its tracks. A new theory, quantum chromodynamics (QCD), or the theory of quarks and the strong interaction, was a juggernaut flattening all other theories. People left string theory in droves to work on QCD. All the funding, jobs, and recognition went to physicists working on the quark model.
I remember those dark years well. Only the foolhardy or the stubborn persisted in working on string theory. And when it became known that these strings could vibrate only in ten dimensions, the theory became the butt of jokes. String pioneer John Schwarz at Cal Tech would sometimes bump into Richard Feynman in the elevator. Ever the joker, Feynman would ask, “Well, John, and how many dimensions are you in today?” We used to joke that the only place to find a string theorist was in the unemployment line. (Nobel laureate Murray Gell-Mann, founder of the quark model, once confided to me that he took pity on string theorists and created a “nature preserve for endangered string theorists” at Cal Tech so people like John wouldn’t lose their jobs.)
Given that today so many young physicists are rushing to work on string theory, Steve Weinberg has written, “String theory provides our only present source of candidates for a final theory—how could anyone expect that many of the brightest young theorists would not work on it?”
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p; IS STRING THEORY UNTESTABLE?
One major criticism of string theory today is that it is untestable. It would take an atom smasher the size of the galaxy to test this theory, critics claim.
But this criticism neglects the fact that most science is done indirectly, not directly. No one has ever visited the sun to do a direct test, but we know it is made of hydrogen because we can analyze its spectral lines.
Or take black holes. The theory of black holes dates back to 1783, when John Michell published an article in the Philosophical Transactions of the Royal Society. He claimed that a star could be so massive that “all light emitted from such a body would be made to return to it by its own proper gravity.” Michell’s “dark star” theory languished for centuries because a direct test was impossible. In 1939 Einstein even wrote a paper showing that such a dark star could not form by natural means. The criticism was that these dark stars were inherently untestable because they were, by definition, invisible. Yet today the Hubble Space Telescope has given us gorgeous evidence of black holes. We now believe that billions of them could lurk in the hearts of galaxies; scores of wandering black holes could exist in our own galaxy. But the point is that the evidence for black holes is all indirect; that is, we have gathered information about black holes by analyzing the accretion disk that swirls around them.
Furthermore, many “untestable” theories ultimately become testable. It took two thousand years to prove the existence of atoms after they were first proposed by Democritus. Nineteenth-century physicists such as Ludwig Boltzmann were hounded to death for believing in that theory, yet today we have gorgeous photographs of atoms. Pauli himself introduced the concept of the neutrino in 1930, a particle so elusive it can pass through blocks of solid lead the size of an entire star system and not be absorbed. Pauli said, “I have committed the ultimate sin; I have introduced a particle that can never be observed.” It was “impossible” to detect the neutrino, so it was considered little more than science fiction for several decades. Yet today we can produce beams of neutrinos.
There are, in fact, a number of experiments that will provide, physicists hope, the first indirect tests of string theory:
The Large Hadron Collider (LHC) might be powerful enough to produce “sparticles,” or superparticles, which are the higher vibrations predicted by superstring theory (as well as by other supersymmetric theories).
As I mentioned earlier, in 2015 the Laser Interferometer Space Antenna (LISA) will be launched in space. LISA and its successor, the Big Bang Observer, may be sensitive enough to test several “pre–big bang” theories, including versions of the string theory.
A number of labs are investigating the presence of higher dimensions by looking at deviations from Newton’s famed inverse-square law at the millimeter scale. (If there is a fourth spatial dimension, then gravity should fall by the inverse cube, not the inverse square.) The latest version of string theory (M-theory) predicts there are eleven dimensions.
Many labs are trying to detect dark matter, since the Earth is moving in a cosmic wind of dark matter. String theory makes specific, testable predictions about the physical properties of dark matter because dark matter is probably a higher vibration of the string (e.g., the photino).
It is hoped that a series of additional experiments (e.g., on neutrino polarization in the south pole) will detect the presence of mini–black holes and other strange objects by analyzing anomalies in cosmic rays, whose energies can easily exceed those of the LHC. Cosmic ray experiments and the LHC will open a new, exciting frontier beyond the Standard Model.
And there are some physicists who hold out the possibility that the big bang was so explosive that perhaps a tiny superstring was blown up into astronomical proportions. As physicist Alexander Vilenkin of Tufts University writes, “A very exciting possibility is that superstrings…can have astronomical dimensions…We would then be able to observe them in the sky and directly test superstring theory.” (The probability of finding a huge, relic superstring that was blown up during the big bang is quite small.)
IS PHYSICS INCOMPLETE?
In 1980 Stephen Hawking helped to spark interest in a theory of everything with his lecture entitled “Is the End in Sight for Theoretical Physics?” in which he said, “We may see a complete theory within the lifetime of some of those present here.” He claimed that there was a fifty-fifty chance that the final theory would be found in the next twenty years. But when the year 2000 arrived and there was no consensus on the theory of everything, he changed his mind and said that there was a fifty-fifty chance of finding it in another twenty years.
Then in 2002 Hawking changed his mind once again, declaring that Gödel’s incompleteness theorem may suggest a fatal flaw in his original line of thinking. He wrote, “Some people will be very disappointed if there is not an ultimate theory that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind…Gödel’s theorem ensured there would always be a job for mathematicians. I think M-theory will do the same for physicists.”
His argument is an old one: since mathematics is incomplete and the language of physics is mathematics, there will always be true physical statements that are forever beyond our reach, and hence a theory of everything is not possible. Since the incompleteness theorem killed off the Greek dream of proving all true statements in mathematics, it will also put a theory of everything forever beyond our reach.
Freeman Dyson said it eloquently when he wrote, “Gödel proved the world of pure mathematics is inexhaustible; no finite set of axioms and rules of inference can ever encompass the whole of mathematics…I hope that an analogous situation exists in the physical world. If my view of the future is correct, it means that the world of physics and astronomy is also inexhaustible; no matter how far we go into the future, there will always be new things happening, new information coming in, new worlds to explore, a constantly expanding domain of life, consciousness, and memory.”
Astrophysicist John Barrow summarizes this logic this way: “Science is based on mathematics; mathematics cannot discover all truths; therefore science cannot discover all truths.”
Such an argument may or may not be true, but there are potential flaws. Professional mathematicians for the most part ignore the incompleteness theorem in their work. This is because the incompleteness theorem begins by analyzing statements that refer to themselves; that is, they are self-referential. For example, statements like the following are paradoxical:
This sentence is false.
I am a liar.
This statement cannot be proven.
In the first case, if the sentence is true, it means it is false. If the sentence is false, then the statement is true. Likewise, if I am telling the truth, then I am telling a lie; and if I am telling a lie, then I am telling the truth. In the last case, if the sentence is true, then it cannot be proven to be true.
(The second statement is the famous liar’s paradox. The Cretan philosopher Epimenides used to illustrate this paradox by saying, “All Cretans are liars.” However, Saint Paul missed the point entirely and wrote, in his epistle to Titus, “One of Crete’s own prophets has said it, ‘Cretans are always liars, evil brutes, lazy gluttons.’ He has surely told the truth.”)
The incompleteness theorem builds on statements such as “This sentence cannot be proven using the axioms of arithmetic” and creates a sophisticated web of these self-referential paradoxes.
Hawking, however, uses the incompleteness theorem to show that a theory of everything cannot exist. He claims that the key to Gödel’s incompleteness theorem is that mathematics is self-referential, and physics suffers from this disease as well. Since the observer cannot be separated from the observation process, it means that physics will always refer to itself, since we cannot leave the universe. In the final analysis, the observer is also made of atoms and molecules, and hence must be an integral part of the experiment he is performing.
But there is a way to avoid Hawking’s c
riticism. To avoid the paradoxes inherent in Gödel’s theorem, professional mathematicians today simply state that their work excludes all self-referential statements. They can then circumvent the incompleteness theorem. To a large degree, the explosive development of mathematics since Gödel’s time has been accomplished simply by ignoring the incompleteness theorem, that is, by postulating that recent work makes no self-referential statements.
In the same way it may be possible to construct a theory of everything that can explain every known experiment independent of the observer/observed dichotomy. If such a theory of everything can explain everything from the origin of the big bang to the visible universe that we see around us, then it becomes academic how we describe the interaction between the observer and observed. In fact, one criterion for a theory of everything should be that its conclusions are totally independent of how we make the split between the observer and the observed.
Furthermore, nature may be inexhaustible and limitless, even if it is based on a handful of principles. Consider a chess game. Ask an alien from another planet to figure out the rules of chess simply by watching the game. After a while the alien can figure out how pawns, bishops, and kings move. The rules of the game are finite and simple. But the number of possible games is truly astronomical. In the same way the rules of nature may also be finite and simple, but the applications of those rules may be inexhaustible. Our goal is to find the rules of physics.
In some sense we already have a complete theory of many phenomena. No one has ever seen a defect in Maxwell’s equations for light. The Standard Model is often called a “theory of almost everything.” Assume for the moment that we can shut off gravity. Then the Standard Model becomes a perfectly sound theory of all phenomena besides gravity. The theory may be ugly, but it works. Even in the presence of the incompleteness theorem, we have a perfectly reasonable theory of everything (besides gravity).
