Last year, in a series of posts, I gave you a tour of quantum field theory, telling you some of what we understand and some of what we don’t. I still haven’t told you the role that string theory plays in quantum field theory today, but I am going to give you a brief tour of string theory before I do.

What IS String Theory? Well, what’s Particle Theory?

What is particle theory? It’s nothing other than a theory that describes how particles behave. And in physics language, a theory is a set of equations, along with a set of rules for how the things in those equations are related to physical objects. So a particle theory is a set of equations which can be used to make predictions for how particles will behave when they interact with one another.

Now there’s always space for confusion here, so let’s be precise about terminology.

“Particle theory” is the general category of the equations that can describe particles, of any type and in any combination.

“A particle theory” is a specific example of such equations, describing a specific set of particles of specific types and interacting with each other in specific ways.

For example, there is a particle theory for electrons in atoms. But we’d need a different one for atoms with both electrons and muons, or for a bottom quark moving around a bottom anti-quark, even though the equations would be of a quite similar type.

Most particle theories that one can write down aren’t relevant (or at least don’t appear to be relevant) to the real world; they don’t describe the types of particles (electrons, quarks, etc.) that we find (so far) in our own universe. Only certain particle theories are needed to describe aspects of our world. The others describe imaginary particles in imaginary universes, which can be fun, or even informative, to think about.

Modern particle theory was invented in the early part of the 20th century in response to — guess what? — the discovery of particles in experiments. First the electron was discovered, in 1897; then atomic nuclei, then the proton, then the photon, then the neutrino and the neutron, and so on… Originally, the mathematics used in particle theory was called “quantum mechanics”, a set of equations that is still widely useful today. But it wasn’t complete enough to describe everything physicists knew about, even at the time. Specifically, it couldn’t describe particles that move at or near the speed of light… and so it wasn’t consistent with Einstein’s theory (i.e. his equations) of relativity.

What is Quantum Field Theory?

To fix this problem, physicists first tried to make a new version of particle theory that was consistent with relativity, but it didn’t entirely work. However, it served as an essential building block in their gradual invention of what is called quantum field theory, described in much more detail in previous posts, starting here. (Again: the distinction between “quantum field theory” and “a quantum field theory” is that of the general versus the specific case; see this post for a more detailed discussion of the terminology.)

In quantum field theory, fields are the basic ingredients, not particles. Each field takes a value everywhere in space and time, in much the same way that the temperature of the air is something you can specify at all times and at all places in the atmosphere. And in quantum field theory, particles are ripples in these quantum fields.

More precisely, a particle is a ripple of smallest possible intensity (or “amplitude”, if you know what that means.) For example, a photon is the dimmest possible flash of light, and we refer to it as a “particle” or “quantum” of light.

We call such a “smallest ripple” a “particle” because in some ways it behaves like a particle; it travels as a unit, and can’t be divided into pieces. But really it is wave-like in many ways, and the word “quantum” is in some ways better, because it emphasizes that photons and electrons aren’t like particles of dust.

To sum up:

particles were discovered in experiments;

physicists invented the equations of particle theory to describe their behavior;

but to make those equations consistent with Einstein’s special relativity (needed to describe objects moving near or at the speed of light) they invented the equations of quantum field theory, in which particles are ripples in fields.

in this context the fields are more fundamental than the particles; and indeed it was eventually realized that one could (in principle) have fields without particles, while the reverse is not true in a world with Einstein’s relativity.

thus, quantum field theory is a more general and complete theory than particle theory; it has other features not seen in particle theory.

Now what about String Theory?

In some sense, strings also emerged from experiments — experiments on hadrons, back before we knew hadrons were made from quarks and gluons. The details are a story I’ll tell soon and in another context. For now, suffice it to say that in the process of trying to explain some puzzling experiments, physicists were led to invent some new equations, which, after some study, were recognized to be equations describing the quantum mechanical behavior of strings, just as the equations of particle theory describe the quantum mechanical behavior of particles. (One advantage of the string equations, however, is that they were, from the start, consistent with Einstein’s relativity.) Naturally, at that point, this class of equations was named “string theory”.

An aside: theories of non-relativistic strings have appeared in the literature, for instance in Gimon et al. from 2002. There must be earlier versions, but I haven’t found them. [Experts will think of light-cone gauge.]

A few more years of study of these equations led to a number of realizations about the simplest forms of string theory. Note my use of the vague term “simplest forms”. I’ve used this to alert you that what I’m about to say about string theory is not always true. It is true of what people knew about string theory back in 1985 or so. But keep in mind this is not the final word on what “string theory” means; it was, rather, just the first attempt. I’ll make this concept much less vague in my next post.

Anyway, here are some things that people learned about this theory in the 1970s and 1980s, presented in a somewhat ahistorical order.

Just as particles are ripples in fields, strings can be seen as a sort of ripple in string fields. (See Figure 1.) But unlike particle theory, which is actually incomplete unless one takes a field point of view, the mathematics of the string equations make it easier to understand strings on their own, without the use of string fields. And string field theory is very complicated, and still very poorly understood even today. This is part of why, for strings, people usually talk about “string theory” and not “string field theory”, while for particles, people usually talk about “field theory”, not “particle theory”.

In its simplest context, a theory of strings is equivalent to a theory of a huge number of fields and their particles. Roughly, even though there’s only one type of string, a string can move or vibrate in different ways. A string vibrating in one way will appear in an experiment as though it is one type of particle; a string vibrating in a different way will appear to be a different particle. In short, a single type of string, though so small it seems like a particle in current experiments, has many types of vibrations, and these would appear in current experiments to be many types of particles. See Figure 2. The masses and other properties of these particles, and the forces by which they interact with each other, are arranged in special patterns, a point I’ll return to in a moment.

The simplest string theories have only boson particles (particles which, like Higgs particles, photons and gravitons, have spin = 0,1,2…). The next simplest are superstring theories, which also have fermion particles (which have spin 1/2, 3/2, …; electrons and quarks have spin 1/2.) To describe our world, then, superstrings are necessary.

Moreover, for a theory of strings to be consistent and stable (remember a theory is a set of equations — it has to make mathematical sense, or you can’t use it to make predictions of any sort) the strings have to be superstrings, at least approximately.

In addition, for a theory of superstrings to be consistent, the strings have to [with a few subtle caveats] move in a space with 9 spatial dimensions. That is, adding on time as one more dimension, they can exist only in universes with a total of 10 space-time dimensions. That sounds bad… but it isn’t. It might be that only a few of those dimensions are like the ones we know; the others might be so short that we might not notice them, much as a sheet of paper appears two-dimensional if you don’t notice its thickness (Figure 3). So our world could be described by string theory as long as there are six unseen “extra dimensions”, which are too short for us to detect with current experiments.

In a (simple) superstring theory which describes a world similar to the one we live in, with three very large spatial dimensions, the pattern of masses for the particles in that universe would be something vaguely like that shown in Figure 4. There would be huge numbers of types of heavy particles, with masses so large that we aren’t even close to producing them in our experiments, and in the near term we have no hope of checking whether they can exist. Only a small number of types of particles, massless or with rather small masses, will be easily observable. These include (Figure 4)

particles and fields similar in type, though with details that may vary widely, to those that we find in the Standard Model of particle physics (the quantum field theory that describes the known particles and non-gravitational forces.) The equations that describe them would be those of quantum field theory.

a graviton: a spin-two particle that is a ripple in a gravitational field. The equations for this field are those of Einstein’s general relativity, in which gravity is an effect of the curvature of space and time. But these equations are generalized into a quantum mechanical form, which we call “quantum gravity”.

Let me say that again, because it’s a prediction — not a hugely impressive one, because it is vague and after-the-fact, but nevertheless, something deserving of the name.

String Theory’s First Prediction (Vague, and with Loopholes)

In its simplest forms, string theory, at least naively, predicts that in a universe whose basic objects are superstrings, one will probably observe a quantum version of something similar to (and possibly identical to) Einstein’s vision of gravity, and probably other very lightweight and massless particles not entirely dissimilar from the ones we observe, along with additional forces like the ones we observe… all of which can be described using quantum field theory.

Now this is a remarkable prediction of the theory [again, I remind you, of the theory in its simplest form, i.e. as it was understood in 1985], because this prediction agrees with data.

But how excited should we be about this success? The prediction is quite vague; it’s analogous to the fact that (as I explained here) the simplest forms of quantum field theory (before you choose a particular example) make only a few, very vague, predictions: that there will be particles in the world; these will be fermions or bosons; and any two particles of the same type will be literally identical. This prediction of string theory is so vague that its success is hardly convincing; one could imagine there are lots of other theories out there that predict the same thing. And just as complicated quantum field theories don’t always have particles, this simple prediction isn’t necessarily true when string theory gets complicated.

Furthermore, it’s a prediction of something we already knew about the world, which physicists sometimes call a postdiction. It is a lot easier to make a prediction about nature when you already know what the answer has to be! An example of a postdiction is Einstein’s calculation that his theory of general relativity, which he was developing at the time, predicted the small observed shift in Mercury’s orbit. The shift was already known from data, so he had a target to aim at… and that’s part of why people were mildly impressed (his theory could have failed at this step by giving the wrong answer) but hardly convinced. It was (and is) much more impressive that the theory correctly predicted things that had never previously been observed — the deflection of light by the sun, the gravitational slowing-down of clocks, the gravitational redshift of light, energy loss by radiation of gravitational waves, etc.

But historically and sociologically, this first prediction of string theory was a very important one. People had been trying for many years to find a quantum theory of gravity that would also be consistent with the quantum field theories used to describe other particles… or perhaps something more general that would contain both Einstein’s theory of gravity and the quantum field theory that describes the other forces and particles. It was a sort of “holy grail”, one that Einstein himself spent his last 30 years seeking fruitlessly. So you have to understand that it was quite remarkable that a theory of strings, once the equations were worked out carefully, simply dumped the holy grail on the table without anyone asking or looking for it to do so.

Of course, we didn’t and still don’t know it is the only holy grail; maybe there are others. But at worst, a simple form of superstring theory is a very interesting example of how a quantum theory containing both gravity and field theory might work!



Is Our World a World of Strings?

In short, back when physicists were still new to string theory, they found that simple forms of string theory could potentially describe universes much like the one we live in. And this fact generated a lot of optimism that maybe there is a string theory that describes our own universe. But if so, which one?

By 1983, it turned out that most string theories anyone had written down were mathematically inconsistent (or had to be very complicated to make them consistent), but not all of them. A short list of 5 simple superstring theories survived all mathematical requirements, and only one seemed relevant for our own universe. I’ll say a bit more about them in my next post. And since there was only one of these that seemed directly relevant, a huge wave of over-optimism swept across the string theory experts [I was an undergraduate, watching with some skepticism, at the time] that they were on the verge of figuring out the complete theory of elementary particles, forces and space-time.

I should note that at this same time the string theory folks took the unfortunate step of annoying all the other physicists, along with chemists, biologists, social scientists of all sorts, artists, writers, lawyers, and chefs by calling this potentially complete theory of particles, forces and spacetime a “Theory of Everything”. We won’t use that term here.

But This Was Just The Beginning of a Saga

Let me pause at this moment to make some important cautionary remarks.

Up to this point, the methods that people had used to study string theories were of a similar type used (as I described here) in the simplest quantum field theories. This is a method known as “perturbation theory”, which is a technique of successive approximation: the full calculation is written as a simple estimate, plus a small but more complicated correction to the estimate, plus a smaller but even more complicated correction to the correction… etc.

The problem is that — just as in quantum field theory — successive approximation only works when all relevant forces are relatively weak (in a technical sense — see this post). But there are some questions in string theory — most notably, how does the world end up with 3 large spatial dimensions rather than 9? and why don’t the strengths of all forces end up being zero? — for which the method of successive approximation is not good enough. When this was realized in the mid-1980s, progress in string theory slowed down markedly, and the goal of a complete theory of particles, forces and space-time receded for the moment.

Nevertheless, some experts continued to explore string theory, and made slow but steady progress. And then, spurred in part by advances in quantum field theory in 1993-1994, a major set of realizations occurred in string theory studies during the period 1994-1998. By the time that mini-revolution was over, what string theorists knew about this theory had dramatically expanded and changed. And that will be the subject of my next post in this series.