Farewell to Reality (30 page)

As a mathematician, Kaluza was used to dealing with abstract ‘dimensions' and was not overly concerned about how this fifth dimension should be interpreted physically. He simply found that solving Einstein's field equations in five dimensions instead of the usual four resulted in the natural and spontaneous emergence of Maxwell's equations for electromagnetism.

However, he did require the extra spatial dimension to be circular, such that a particle following a straight trajectory along this dimension would soon find itself back where it started.
^*
What appears as a straight line in our conventional, Euclidean geometry of up—down, left—right, back—front becomes a circle in Kaluza's additional dimension. Einstein wrote to him in April 1919: ‘The idea of achieving [a unified theory] by means of a five-dimensional cylinder world never dawned on me … At first glance I like your idea enormously.'
²

Einstein helped Kaluza to get his work published, although for reasons that are not clear, this took a further two years. Kaluza's ideas appeared in print in 1921.

Oskar Klein was by contrast a physicist. He had trained as a physical chemist under the great Swedish Nobel laureate Svante Arrhenius in Stockholm, before moving to Copenhagen in 1917 to work with Niels Bohr. He returned to his native Sweden to complete his doctorate in 1921. Five years later he arrived independently at Kaluza's idea but tried to give it at least some kind of physically realistic, and specifically quantum, interpretation. He also noted that if this dimension were to be rolled up small and tight, with the radius of the cylinder measurable only on sub-nuclear scales, then perhaps this was the reason we could not perceive it directly.

Such a ‘hidden' dimension forms what is known as a
compact set.
The process of transforming a dimension so that it curls up tightly in a closed structure or ‘manifold' instead of stretching off to infinity is called
compactification
.

Klein's reintepretation and modification of Kaluza's idea became known as Kaluza—Klein theory. Einstein was impressed, but was now also expressing some misgivings about the idea. ‘Klein's paper is beautiful and impressive,' he told his colleague, Austrian physicist Paul Ehrenfest, ‘but I find Kaluza's principle too unnatural.'
³
It seems he regarded it as somewhat curious to introduce a new dimension only to tie it up artificially and hide it away in order to account for the simple fact that we don't experience it.

Einstein went on to develop alternative theoretical structures that did not rely on compactification, but these were broadly unsuccessful.

For a time, Klein was encouraged by the thought that the extra spatial dimension might lie at the heart of all quantum phenomena. But the idea was largely forgotten as quantum theory and the Copenhagen interpretation became firmly established.

Of strings and superstrings

And so we turn to theories of strings and superstrings.

In this instance, I think that a broadly chronological tale will help us to understand the basis on which superstring theory has been constructed. Specifically, it will allow us to appreciate that, in seeking
to resolve problems associated with its applicability, the theorists have been obliged to pile assumption on top of assumption.

We will then be in a position properly to judge the theory for ourselves.

So, fast-forward some 42 years, from Klein's paper on a five-dimensional unification of gravity and electromagnetism to Gabriele Veneziano, a young Italian postdoctoral physicist working at CERN in the summer of 1968. Whilst puzzling over ways to describe the scattering of mesons (such as pions) using a theory of strong force interactions, Veneziano developed a mathematical relationship to calculate certain scattering ‘amplitudes'. These amplitudes represent the probabilities for the production (‘scattering') of two final mesons from two initial mesons at different collision angles. They became known as Veneziano amplitudes.

There was no strong theoretical basis for the formula that Veneziano had devised, but it was quite familiar. It was Swiss mathematician Leonhard Euler's ‘beta function', or Euler's ‘integral of the first kind'.

When young Yeshiva University professor Leonard Susskind heard about this relationship, he was struck by its simplicity and decided to try to understand where it had come from. As he later explained:

I worked on it for a long time, fiddled around with it, and began to realize that it was describing what happens when two little loops of string come together, join, oscillate a little bit, and then go flying off. That's a physics problem that you can solve. You can solve exactly for the probabilities for different things to happen, and they exactly match what Veneziano had written down. This was incredibly exciting.
⁴

At this time, the ‘strings' in question were identified with quarks tethered to anti-quarks
^*
by the ‘elastic' of the strong force. Similar discoveries were made independently by Danish physicist Holger Nielsen at the Niels Bohr Institute in Copenhagen and by Yoichiro Nambu in Chicago.

One thing led to another. What if, instead of treating elementary particles as point particles, we instead treat the elastic that holds them together as ‘fundamental'? What if the string's the thing?

In such a string theory the different particles would not be translated into different kinds of strings, but into different
vibrational patterns
in one common string type. The mass of a particle would then be just the energy of its string vibration, with other properties such as charge and spin being more subtle manifestations.

It was a breathtaking idea. Switching from abstract point particles to extended one-dimensional strings offered the promise that it might be possible to avoid all the problems with infinities and renormalization that had plagued quantum field theory from its inception. But this was an
idea
nonetheless.

Perhaps it would help to keep track of these, as we accumulate them.

Now, to be completely fair, we should acknowledge that quantum field theory, and hence the entire structure of the standard model of particle physics, is similarly founded on an (of ten unstated) assumption: that of point particles. The difference is that, within its recognized limits, we know that the standard model
works,
and this helps to justify the point particle assumption even though we know that in principle it can't be right, or at least it can't be the whole story. To justify changing the basis of this assumption to one of strings, we would need some confidence that this leads to a string theory that is demonstrably
better
than the standard model; a string theory that solves at least some of the standard model's problems.

But early versions of string theory did not look at all promising, and consequently did not attract much attention. The theory was derived from mathematical relationships established for the scattering of ‘scalar' particles, such as mesons, with zero spin. It could deal only with strings that describe bosons. The theory also required as many as 26 spacetime dimensions, 25 dimensions of space and one of time.

Worst of all, the theory also predicted the existence of tachyons, hypothetical particles that possess imaginary mass and travel only at speeds faster than light. Tachyons wreak havoc on the principles of cause and effect and are anathema to any theoretical structure with pretensions to describe the real world. The presence of tachyons is a sign that something, somewhere, has gone horribly wrong.

Most theorists turned their attentions away from string theory. But a few persisted, enamoured of its potential and reluctant to throw the baby out with the bathwater. By 1972, one of the big problems of early string theory had been solved, by French theorist Pierre Ramond, working at the National Accelerator Laboratory in Chicago, and by American John Schwarz and French theorist André Neveu at Princeton University.

You can probably figure out for yourself roughly how this was done. How do you get fermions into a theory that describes only bosons? By assuming a fundamental symmetry between fermions and bosons, such that for every boson there is a corresponding fermion. In other words, by assuming that the string universe is supersymmetric.

Now this is a greatly oversimplified way of looking at the development of superstring theory; as is typically the case, the true history is not quite so neat. Ramond had written a paper concerned with non-interacting fermionic strings; Schwarz and Neveu were working on the theory of bosonic strings when they realized that these could be brought together. As Schwarz later explained:

So we got ahold of [Ramond's] paper just about the time we were finishing ours, and we were struck by the mathematical similarity of what we had done and what he had been doing. Then we realized that Ramond's fermionic strings could be made to interact with our bosonic strings and it was all part of a consistent theory. Over the years, we've used different names for this theory, and it has evolved into what we today call superstrings.
⁵

SUSY was emerging at around the same time, and some have argued that it was actually derived from early string theory. It doesn't matter much precisely how it all came about, but we do need to acknowledge another important layer of assumption. Although we encountered the supersymmetry assumption in the last chapter, it is important to repeat it here in the context of string theory.

Schwarz found that the number of spacetime dimensions required for superstrings to vibrate in was no longer the 26 demanded by string theory, it was just ten: nine dimensions of space and one of time. Although this was still a lot more than the four dimensions of our everyday experience, it was at least a step in the right direction.

And — good news — the tachyons were now gone.

Superstrings and the graviton

Despite these advances, there was no great rush to embrace superstring theory. In 1971, Gerard't Hooft and Martinus Veltman showed that quantum field theories incorporating the Higgs mechanism could in general be renormalized. Two years later, the notion of asymptotic freedom was demonstrated to be compatible with quantum field theory, and this gave birth to what was eventually to become QCD. Within a few years, the structure of the standard model of particle physics was established.

There seemed little to be gained from further work on superstrings as a theory of the strong nuclear force. The few hundred or so theorists who had worked on early string theory moved on to other things.

But Schwarz was still not ready to let go. At Princeton he had collaborated with two French theorists, André Neveu and Joël Scherk. Schwarz's work on superstrings had not earned him a tenured professorship at Princeton University, in itself a sign that the theory was not highly regarded. He accepted the offer of a research associate position at the California Institute of Technology (Caltech) in Pasadena. Scherk visited Caltech in January 1974 and they continued their collaboration. Schwarz explained why:

I think we were kind of struck by the mathematical beauty [of superstring theory]; we found the thing a very compelling structure.
I don't know that we said it explicitly, but we must have both felt that it had to be good for something, since it was just such a beautiful, tight structure. So, one of the problems that we had had with the string theory was that in the spectrum of particles that it gave, there was one that had no mass and two units of spin. And this was just one of the things that was wrong for describing strong nuclear forces, because there isn't a particle like that. However, these are exactly the properties one should expect for the quantum of gravity.
⁶

If the challenge to provide a theoretical description of the strong nuclear force had been met, then the challenge to provide a quantum theory of gravity had not. It seemed that superstring theory not only promised to accommodate all the elementary particles of the standard model then known; it also predicted a particle with the properties of the graviton.

Superstring theory was not a theory of the strong force. It seemed that it was potentially a theory of
everything
.

Superstrings come in two forms: open and closed. Open strings have loose ends that we can think of as representing charged particles and their anti-particles, one at either end, with the string vibration representing the particle carrying the force between them. Open strings therefore predict both matter particles
and
the forces between them. However, the theory also
demands
closed strings. When a particle and anti-particle annihilate, the two ends of the string join up and form a closed string.

But if there are closed strings, there are also gravitons and the force of gravity. Rather than trying to shoehorn quantum theory into general relativity, or vice versa, superstring theory appeared to be saying that all nature's forces are just different vibrational patterns in open and closed strings. In superstring theory these forces are
automatically
unified.

Superstring theory appeared to offer great promise. It seemed that all the elementary particles then known — their masses, charges, spins, the forces between them and all the standard model parameters that could not be derived from first principles — could be subsumed into a single theory with just two fundamental constants. These were the constants that determine the tension of the string and the coupling between strings.