Hyperspace (54 page)

This partial solution to the question of observation once again takes the best of both philosophies. On the one hand, this picture is reductionist because it adheres closely to the standard quantum-mechanical explanation of reality, without recourse to consciousness. On the other hand, it is also holistic because it begins with the wave function of the entire universe, which is the ultimate holistic expression! This picture does not make the distinction between the observer and the observed. In this picture, everything, including all objects and their observers, is included in the wave function.

This is still only a partial solution because the cosmic wave function itself, which describes the entire universe, does not live in any definite state, but is actually a composite of all possible universes. Thus the problem of indeterminacy, first discovered by Heisenberg, is now extended to the entire universe.

The smallest unit that one can manipulate in these theories is the universe itself, and the smallest unit that one can quantize is the space of all possible universes, which includes both dead cats and live cats. Thus in one universe, the cat is indeed dead; but in another, the cat is alive. However, both universes reside in the same home: the wave function of the universe.

Ironically, in the 1960s, the reductionist approach looked like a failure; the quantum theory of fields was hopelessly riddled with divergences found in the perturbation expansion. With quantum physics in disarray, a branch of physics called S-matrix (scattering matrix) theory broke off from the mainstream and began to germinate. Originally founded by Heisenberg, it was further developed by Geoffrey Chew at the University of California at Berkeley. S-matrix theory, unlike reductionism, tried to look at the scattering of particles as an inseparable, irreducible whole.

In principle, if we know the
S
matrix, we know everything about particle interactions and how they scatter. In this approach, how particles bump into one another is everything; the individual particle is nothing.
S
-matrix theory said that the self-consistency of the scattering matrix,
and
self-consistency alone
, was sufficient to determine the
S
matrix. Thus fundamental particles and fields were banished forever from the Eden of
S
-matrix theory. In the final analysis, only the
S
matrix had any physical meaning.

As an analogy, let us say that we are given a complex, strange-looking machine and are asked to explain what it does. The reductionist will immediately get a screw driver and take the machine apart. By breaking down the machine to thousands of tiny pieces, the reductionist hopes to find out how the machine functions. However, if the machine is too complicated, taking it apart only makes matters worse.

The holists, however, do not want to take the machine apart for several reasons. First, analyzing thousands of gears and screws may not give us the slightest hint of what the overall machine does. Second, trying to explain how each tiny gear works may send us on a wild-goose chase. The correct way, they feel, is to look at the machine as a whole. They turn the machine on and ask how the parts move and interact with one another. In modern language, this machine is the
S
matrix, and this philosophy became the
S
-matrix theory.

In 1971, however, the tide shifted dramatically in favor of reductionism with Gerard ’t Hooft’s discovery that the Yang-Mills field can provide a self-consistent theory of subatomic forces. Suddenly, each of the particle interactions came tumbling down like huge trees in a forest. The Yang-Mills field gave uncanny agreement with the experimental data from atom smashers, leading to the establishment of the Standard Model, while S-matrix theory became entangled in more and more obscure mathematics. By the late 1970s, it seemed like a total, irreversible victory of reductionism over holism and the S-matrix theory. The reductionists began to declare victory over the prostrate body of the holists and the
S
matrix.

The tide, however, shifted once again in the 1980s. With the failure of the GUTs to yield any insight into gravitation or yield any experimentally verifiable results, physicists began to look for new avenues of research. This departure from GUTs began with a new theory, which owed its existence to the S-matrix theory.

In 1968, when S-matrix theory was in its heyday, Veneziano and Suzuki were deeply influenced by the philosophy of determining the
S
matrix in its entirety. They hit on the Euler beta function because they were searching for a mathematical representation of the entire
S
matrix. If they had looked for reductionist Feynman diagrams, they never would have stumbled on one of the great discoveries of the past several decades.

Twenty years later, we see the flowering of the seed planted by the
S-matrix theory. The Veneziano-Suzuki theory gave birth to string theory, which in turn has been reinterpreted via Kaluza-Klein as a ten-dimensional theory of the universe.

Thus we see that the ten-dimensional theory straddles both traditions. It was born as a child of a holistic
S
-matrix theory, but contains the reductionist Yang-Mills and quark theories. In essence, it has matured enough to absorb both philosophies.

One of the intriguing features of superstring theory is the level to which the mathematics has soared. No other theory known to science uses such powerful mathematics at such a fundamental level. In hindsight, this is necessarily so, because any unified field theory first must absorb the Riemannian geometry of Einstein’s theory and the Lie groups coming from quantum field theory, and then must incorporate an even higher mathematics to make them compatible. This new mathematics, which is responsible for the merger of these two theories, is
topology
, and it is responsible for accomplishing the seemingly impossible task of abolishing the infinities of a quantum theory of gravity.

The abrupt introduction of advanced mathematics into physics via string theory has caught many physicists off guard. More than one physicists has secretly gone to the library to check out huge volumes of mathematical literature to understand the ten-dimensional theory. CERN physicist John Ellis admits, “I find myself touring through the bookshops trying to find encyclopedias of mathematics so that I can mug up on all these mathematical concepts like homology and homotopy and all this sort of stuff which I never bothered to learn before!”
⁶
To those who have worried about the ever-widening split between mathematics and physics in this century, this is a gratifying, historic event in itself.

Traditionally, mathematics and physics have been inseparable since the time of the Greeks. Newton and his contemporaries never made a sharp distinction between mathematics and physics; they called themselves natural philosophers, and felt at home in the disparate worlds of mathematics, physics, and philosophy.

Gauss, Riemann, and Poincaré all considered physics to be of the utmost importance as a source of new mathematics. Throughout the eighteenth and nineteenth centuries, there was extensive cross-pollination between mathematics and physics. But after Einstein and Poincaré, the development of mathematics and physics took a sharp turn. For the
past 70 years, there has been little, if any, real communication between mathematicians and physicists. Mathematicians explored the topology of
N
-dimensional space, developing new disciplines such as algebraic topology. Furthering the work of Gauss, Riemann, and Poincaré, mathematicians in the past century developed an arsenal of abstract theorems and corollaries that have no connection to the weak or strong forces. Physics, however, began to probe the realm of the nuclear force, using three-dimensional mathematics known in the nineteenth century.

All this changed with the introduction of the tenth dimension. Rather abruptly, the arsenal of the past century of mathematics is being incorporated into the world of physics. Enormously powerful theorems in mathematics, long cherished only by mathematicians, now take on physical significance. At last, it seems as though the diverging gap between mathematics and physics will be closed. In fact, even the mathematicians have been startled at the flood of new mathematics that the theory has introduced. Some distinguished mathematicians, such as Isadore A. Singer of MIT, have stated that perhaps superstring theory should be treated as a branch of mathematics, independent of whether it is physically relevant.

No one has the slightest inkling why mathematics and physics are so intertwined. The physicist Paul A. M. Dirac, one of the founders of quantum theory, stated that “mathematics can lead us in a direction we would not take if we only followed up physical ideas by themselves.”
⁷

Alfred North Whitehead, one of the greatest mathematicians of the past century, once said that mathematics, at the deepest level, is inseparable from physics at the deepest level. However, the precise reason for the miraculous convergence seems totally obscure. No one has even a reasonable theory to explain why the two disciplines should share concepts.

It is often said that “mathematics is the language of physics.” For example, Galileo once said, “No one will be able to read the great book of the Universe if he does not understand its language, which is that of mathematics.”
⁸
But this begs the question of why. Furthermore, mathematicians would be insulted to think that their entire discipline is being reduced to mere semantics.

Einstein, noting this relationship, remarked that pure mathematics might be one avenue to solve the mysteries of physics: “It is my conviction that pure mathematical construction enables us to discover the concepts and the laws connecting them, which gives us the key to the understanding of nature…. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.”
⁹
Heisenberg
echoed this belief: “If nature leads us to mathematical forms of great simplicity and beauty … that no one has previously encountered, we cannot help thinking that they are ‘true,’ that they reveal a genuine feature of nature.”

Nobel laureate Eugene Wigner once even penned an essay with the candid title “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”

Over the years, I have observed that mathematics and physics have obeyed a certain dialectical relationship. Physics is not just an aimless, random sequence of Feynman diagrams and symmetries, and mathematics is not just a set of messy equations, but rather physics and mathematics obey a definite symbiotic relationship.

Physics, I believe, is ultimately based on a small set of
physical principles
. These principles can usually be expressed in plain English without reference to mathematics. From the Copernican theory, to Newton’s laws of motion, and even Einstein’s relativity, the basic physical principles can be expressed in just a few sentences, largely independent of any mathematics. Remarkably, only a handful of fundamental physical principles are sufficient to summarize most of modern physics.

Mathematics, by contrast, is the set of all possible
self-consistent structures
, and there are vastly many more logical structures than physical principles. The hallmark of any mathematical system (for example, arithmetic, algebra, or geometry) is that its axioms and theorems are consistent with one another. Mathematicians are mainly concerned that these systems never result in a contradiction, and are less interested in discussing the relative merits of one system over another. Any self-consistent structure, of which there are many, is worthy of study. As a result, mathematicians are much more fragmented than physicists; mathematicians in one area usually work in isolation from mathematicians in other areas.

The relationship between physics (based on physical principles) and mathematics (based on self-consistent structures) is now evident: To solve a physical principle, physicists may require many self-consistent structures. Thus
physics automatically unites many diverse branches of mathematics
. Viewed in this light, we can understand how the great ideas in theoretical physics evolved. For example, both mathematicians and physicists claim Isaac Newton as one of the giants of their respective professions. However, Newton did not begin the study of gravitation starting with mathematics. By analyzing the motion of falling bodies, he was led
to believe that the moon was continually falling toward the earth, but never collided with it because the earth curved beneath it; the curvature of the earth compensated for the falling of the moon. He was therefore led to postulate a physical principle: the universal law of gravitation.

However, because he was at a loss to solve the equations for gravity, Newton began a 30-year quest to construct from scratch a mathematics powerful enough to calculate them. In the process, he discovered many self-consistent structures, which are collectively called
calculus
. From this viewpoint, the physical principle came first (law of gravitation), and then came the construction of diverse self-consistent structures necessary to solve it (such as analytic geometry, differential equations, derivatives, and integrals). In the process, the physical principle united these diverse self-consistent structures into a coherent body of mathematics (the calculus).

The same relationship applies to Einstein’s theory of relativity. Einstein began with physical principles (such as the constancy of the speed of light and the equivalence principle for gravitation) and then, by searching through the mathematical literature, found the self-consistent structures (Lie groups, Riemann’s tensor calculus, differential geometry) that allowed him to solve these principles. In the process, Einstein discovered how to link these branches of mathematics into a coherent picture.