The Larson Research Center

Not only does Lee Smolin write clearly and forcefully about the The Trouble With Physics, which he shows is ultimately about the age-old problem of reconciling the discrete (quantum mechanics) and continuum (general relativity) theories of modern physics, in light of the fact that nature obviously has no problem with the task, but, in the process, he manages to clearly explain why string theory has been, and still is, so compelling to theoretical physicists.

He begins his exposition, on page 103 of his book, by establishing the context in which string theory arose: He asserts that the idea that elementary particles might be vibrations of strings, instead of pointlike particles, is possibly another one of those rare insights, where the discovery of a missing element to a puzzle suddenly clarifies an otherwise perplexing physical mystery. He reminds us that it was the discovery of a hidden, or missing, motion that first led ancient astronomers to understand the motion of the planets. He writes:

The discovery of this third motion - the missing element - must have been one of the earliest triumphs of abstract thinking. We see two objects, the sun and the moon. Each has a period, known from earliest times. It took an act of imagination to see that something else was moving as well: the paths themselves. This was a profound step, because it required realizing that behind the motion you observe there are other motions whose existence can only be deduced. Just a few times since has science progressed by discovery of such a missing element.

The idea of a point particle in physics is not just a puzzle, but a puzzle wrapped in mystery inside an enigma, as they say. It is the mother of all physical mysteries. Interestingly enough, though, its definition is the physical counterpart to the definition of an irrational number, because, like the definition of an irrational number by mathematicians, the definition of a physical point by physicists is actually an ad hoc definition, based on the so-called Archimedean Axiom, where it is assumed that an infinitesimal quantity cannot be defined, indeed, must not be defined; that is, a quantity is an infinitesimal quantity, if it exists, yet, at the same time, it must be smaller than any division of 1 that can be imagined, and, since this is impossible, such a quantity cannot exist. In other words, for physicists, there is a need to limit how far the number 1, or the continuum, can be subdivided.

Of course, there is no limit to numbers, or limit to the elements of the set of natural numbers, to be a little more precise. Which is to say that there is no limit to n/1, and if there is no limit to n/1, it follows that there also is no limit to 1/n. However, this presents a problem in physics, when elementary particles, such as electrons, are regarded as point particles that cannot be subdivided. In his Lectures on Physics, Richard Feynman explains the problem as follows:

…the limit at r = ∞ gives no difficulty. But for a point charge we are supposed to integrate down to r = 0, which gives an infinite integral. [The equations say] that there is an infinite amount of energy in the field of a point charge, although we began with the idea that there was energy only between points charges…one way out of the difficulty would be to say that elementary charges, such as an electron, are not points, but are really small distributions of charge. Alternatively, we could say that there is something wrong in our theory of electricity at very small distances, or with the idea of the local conservation of energy. There are difficulties with either point of view. These difficulties have never been overcome; they exist to this day.

Thus, in physics, just as in mathematics, either an artificial cutoff of the infinitesimal is devised, or else the theories, whether classical or quantum mechanical, will fail due to fundamental incompatibilities of the momentum of charged particles and the energy of their associated fields. The only way out in physics is a solution that corresponds to the solution of the mathematical problem of the incommensurabilities in the rational and irrational numbers, the invention of an ad hoc concept. In mathematics, the ad hoc invention used is the invention of real numbers (see latest post in New Math).

In physics, the ad hoc invention used is the invention of renormalization. Feynman, one of the inventors of renormalization, who received a share of the Nobel Prize for it, concluded in his 1965 Nobel Lecture:

I don’t think we have a completely satisfactory relativistic quantum-mechanical model, even one that doesn’t agree with nature, but, at least, agrees with the logic that the sum of probability of all alternatives has to be 100%. Therefore, I think that the renormalization theory is simply a way to sweep the difficulties of the divergences of electrodynamics under the rug. I am, of course, not sure of that.

However, since then, LST physicists have become quite accustomed to Feynman’s renormalization theory and simply regard it as a useful technique in getting at the final solution. As Frank Wilczek explained in his 2005 Nobel Lecture:

The central observation that is exploited in renormalization theory is that, although interactions with high-energy virtual particles appear to produce divergent corrections, they do so in a very structured way. That is, the same corrections appear over and over again in the calculations of many different physical processes. For example, in quantum electrodynamics (QED), exactly two independent divergent expressions appear, one of which occurs when we calculate the correction to the mass of the electron, the other of which occurs when we calculate the correction to its charge. To make the calculation mathematically well defined, we must artificially exclude the highest energy modes, or dampen their interactions, a procedure called applying a cutoff, or regularization. In the end we want to remove the cutoff, but at intermediate stages we need to leave it in, so as to have well defined (finite) mathematical expressions. If we are willing to take the mass and charge of the electron from experiment, we can identify the formal expressions for these quantities, including the potentially divergent corrections, with their measured values. Having made this identification, we can remove the cutoff. We thereby obtain well defined answers, in terms of the measured mass and charge, for everything else of interest in QED.

But then that’s just another way of stating that the fundamental theoretical problem of how the energy of a particle’s electric field can be temporarily treated as finite, for the purposes of calculations (since it obviously is finite), until the calculations that depend on it are complete, and then we can safely ignore the theoretical implications of this procedure, because the calculations work exceedingly well. Clearly, Feynman’s characterization of this as “sweeping the difficulties…under the rug,” is more honest.

The truth is that the theoretical concept of elementary particles, as point particles, remains problematic. The problems are of the most fundamental nature too, regardless of the much touted success of the standard model of particle physics. The idea that the vibration of strings is a concept that “provides a plausible answer to several big problems of physics,” to put it in Smolin’s words, is a very powerful one indeed, for those who are keen to discover the true physical structure of nature and are not satisfied with just a practical approach to calculations. Some physicists are clearly not willing to let “the ends justify the means.”

Of course, the bigger picture is that while string theory “provides a plausible answer to several big problems of physics,” it also introduces a whole range of new problems at the same time, and, what is worse, these new problems are not as easily “swept under the rug,” as the original problems of infinities are. Nevertheless, and not withstanding this, the genie is now out of the bottle, and there is no going back. It seems that either there has to be an unknown way to solve the problems of string theory, so that we can return to science as we know it, where we can have confidence that the structure of the physical universe can be determined, as a consequence of the fundamental nature of reality, or else we have to give up the idea of science, as we know it, and accept the idea that the physical structure of nature can be anything at all, that there is an infinite number of possibilities, all of which are consistent with nature, but only one of which is consistent with the instance of nature that we observe, with no apparent way of ever discovering which possibility that might be.

Smolin clearly explains how each of these string theory possibilities corresponds to one of the ways in which 9 or 10 spatial dimensions can be compactified to fit in the 3 dimensions of space that we observe. Each way this compactification of “extra” dimensions proceeds determines a unique set of coupling constants (particle masses, charges, etc.) that characterizes the results. This lack of uniqueness is a major problem, but is characteristic of all approaches that incorporate extra dimensions, ever since Kaluza and Klein first tried it. In string theory, these extra dimensions are necessary in order to describe the several ways that strings can move, and this change of position (M₂) motion (see our Chart of Motion), requires extra constants, hundreds of them. Smolin explains:

[The use of extra dimensions] is how string theory solves the basic dilemma facing attempts to unify physics[, but even] if everything comes from a basic principle [like this], you [still] have to explain how the variety of particles and forces arises. In the simplest possibility, where space has nine dimensions, string theory is very simple; all the particles of the same kind are identical. But [in the case] when the strings are allowed to move in the complicated geometry of the six extra dimensions, there arise lots of different kinds of particles, associated with different ways to move and vibrate in each of the extra dimensions.

“So,” Smolin continues, “we get a natural explanation for the apparent differences among the particles…but there is a cost.” The cost is that the theory turns out not to be a single unified theory, but many different theories. So the challenge of unifying the discrete and continuous theories, which string theory promised to meet in the beginning, actually ended up severely exasperated by the concept of strings moving and vibrating in nine dimensions of space. But even then, Smolin admits, the theory would still be useful, if it had led to predictions of the twenty-some odd standard model constants:

…this scheme might have been compelling if it had led to unique predictions of the standard model. If by translating the standard model’s constants into constants denoting the geometry of the extra dimensions. we had found out something new about the standard model’s constants, and if these findings had agreed with nature, that would have constituted strong evidence that string theory must be true.

But, of course, “this is not what happened,” he writes. The given masses, charges, etc, constituting the constants of the standard model particles and forces were just replaced with the different geometries of string theory:

The constants that could be freely valued in the standard model were translated into geometries that could be freely valued in string theory. Nothing was constrained or reduced. And because there were a huge number of choices for the geometry of the extra dimensions, the number of free constants went up, not down.

This was bad enough, but it is not the end of the bad news. String theory equations are not able to produce the exact combinations of particles and forces seen in nature, but just the general features of fermions and gauge fields, and what’s even worse, they predict extra particles and forces of supersymmetry, something that, in spite of years of effort, has never been observed in nature.

In fact, the idea of supersymmetry is inextricably connected with string theory today. It is an extension of the standard model, where the names of sparticles are modified names of particles, etc., doubling the number of entities and greatly expanding the forces in the model. What binds string theory to the supersymmetric version of the standard model is a particular way to compactify the extra dimensions of string theory, called a Calabi-Yau space, which enables one to correlate the constants of the supersymmetric standard model to the geometries of the extra dimensions, compactified as Calabi-Yau spaces. This would have been a good thing, except for the fact that there are practically an infinite number of ways to construct these spaces. Smolin explains:

This was great progress. But there was an equally great problem. Had there been only one Calabi-Yau space, with fixed constants, we would have had the unique unified theory we yearned for. Unfortunately, there turned out to be many Calabi-Yau spaces. No one knew how many, but Yau himself was quoted as saying that there were at least a hundred thousand. Each of these spaces gave rise to a different version of particle physics. And each space came with a list of free constants governing its size and shape. So there was no uniqueness, no new predictions, and nothing was explained.

Soon, though, it was shown that there were many more ways to construct Calabi-Yau spaces, as many as 10⁵⁰⁰ ways to do it! Certainly, this fact was very discouraging, but it was later discovered, to the absolute delight of string theorists, that there were only five string theories that needed to be defined in these spaces, and, moreover, that extending the extra space dimensions from 6 to 7, enabled physicists to unify these five theories, through a principle of duality, even though the unified string theory, dubbed M-theory, has not been successfully formulated yet, and, if it exists at all, it is only dimly understood at this point in time.

Nevertheless, the discovery of Calabi-Yau space and the duality of the five string theories, with its promise of a unified string theory, in an eleven-dimensional version of some unspecified theory of strings, represents a great deal of progress, in the minds of many, that seems to clearly indicate that the whole enterprise will eventually lead to a theoretical unification of the discrete and continuous magnitudes observed in nature. However, to many others, including Smolin, replacing the paradoxical notion of point particles, given their inexplicable properties of mass and charge, with vibrating strings, given their inexplicable properties of extra dimensions, only shifts the mystery of the physical structure of nature from one challenge to another, from how a point of space can have the necessary properties of elementary particles, such as mass and charge, to how vibrating strings can have the necessary properties of motion in ten or eleven dimensions.

The reason for this is found in the definition of motion. Without the different definitions of motion in the chart of motion (CM), the only known definition is change of position (M₂) motion, but this type of motion requires a fixed background of point locations to be defined, or a given geometry, which would constitute the one background of M-theory that is suitable for all five string theories that are supposed to be part of it, with all their different geometries that can be compactified in one to ten dimensions of space. As Smolin explains it:

[All these geometries] provide backgrounds for strings and branes to move. But if they are part of one unified theory, that theory cannot be built on any one background, because it must encompass all backgrounds.

Thus, we have come full circle. A concept of magnitudes of motion, in the form of vibrating strings, was invented to overcome the problematic concept of point magnitudes of force, a property of motion. This seems to work much better, in some ways, but the necessary magnitudes of motion can only be defined in terms of different sets of points that satisfy the postulates of a given geometry, requiring one to ten dimensions. So, the fundamental question has evolved from how a single point of space can have the necessary physical properties to explain nature, to how a single set of points can have the necessary properties to explain nature. As Smolin writes:

The key problem in M-theory, then, is to make a formulation of it that is consistent with quantum [or discrete] theory and background independence. This is an important issue, perhaps the most important open question in string theory. Unfortunately, not much progress has been made on it. There have been some fascinating hints, but we still do not know what M-theory is, or whether there is any theory deserving of the name…

Unfortunately, M-theory remains a tantalizing conjecture. It’s tempting to believe it. At the same time, in the absence of a real formulation, it is not really a theory - it is a conjecture about a theory we would love to believe in.

The most fascinating aspect of this modern drama is the light it throws on the subject of fundamental concepts of magnitudes of geometry and magnitudes of motion.  In the CM, we find that magnitudes of space are actually manifestations of past motion, but this motion can take the form of three types of motion, M₂, M₃, and M₄ motion, and only M₂ motion requires a fixed background, or a set of points, satisfying the postulates of Euclidean geometry. The M₄ motion of the CM actually defines these points, and it does so in a completely background free manner, which, of course, it would have to do, because the space of geometry cannot be defined in terms of itself, which, in the final analysis, is the tautological concept that string theory leads to.

Nevertheless, the LST community’s fascination with string theory’s ability to automatically provide

1. Unification of all elementary particles
2. Gauge fields of electromagnitism and nuclear forces
3. Gravitons and unifications of gravity with other forces
4. Unification of particles and forces

will not permit them to abandon it anytime soon, even with all the trouble it’s revealing with the attempt of theoretical physicists to cope with the incompatibility of discrete and continuous aspects of nature.  However, with the advantage of the RST, and the light it sheds on the fundamental nature of space and time, we can see that the effort to formulate a physical theory on the basis of vibration, or motion, is not misguided, just naive and uninformed.