Between all the pros and cons one finds that it is a great way to share small pieces of information, fragments of ideas, snippets of thought, mini-discussions – and in fact, this can be quite efficient! Twitter builds networks of information streams by allowing you to follow people, whose tweets are then displayed directly to you. This being the primary way of communication management, some secondary tools have emerged like hashtagging, twitter groups and the habit of re-tweeting. A very immediate consequence of this is that interesting information can spread quite rapidly – and indeed, I often receive very interesting links to news, articles or blogs via twitter which I might have missed otherwise. What’s interesting though is defined by the twitter community, the mass of minds behind the millions of tweets.

Naturally, also science is entering the stream, which is definitively a good thing. Sharing ideas, discussing and delivering research results to the public are certainly part of the scientific responsibility. Some of the hashtags used by scientwists are e.g. #science (of course), #PRLit (for peer-reviewed literature) and #arxiv (for pre-prints).

Recently, people have become keen about the idea of using twitter for ranking information on the internet. Of course, interesting articles will be shared by tweeting their links. The more people regard some tweet as interesting, the more often it will be re-tweeted. Considering that usually only trustful sources are re-tweeted (let’s say so, at least), this gives a nice way of measuring the impact of some weblink (strictly speaking, since the link itself is the only information shared).

Not surprisingly also scientific articles have now been submitted to such a kind of ranking. On a daily basis the twitter stream is searched for arXiv-links, which provides the basis for a list of the week’s most popular pre-prints. While it is interesting to watch the (still enormously noisy) statistics – thanks, Robert, for creating the site! – rankings like this should also cause some uneasiness.

Science is subject to trends. Probably on a long term scale this is not a problem at all. Still, the mere existence of a mainstream can make it difficult for individuals to conduct their research outside that stream. Mainstream means popularity, popularity means high ranking, high ranking creates more popularity.

While this is a generic problem, a twitter-based ranking comes with even more, practical, problems: for example, a person with 20000 followers promoting a link will probably create a much larger number of re-tweets than a person with only few followers. Thus the ranking of a link will be correlated with the popularity of the tweeter – and it is quite obvious that there are ways to ‘optimize‘ your popularity. This means that papers and ideas can easily be actively promoted by individuals generating a distorted image of some research fields in the public. In addition possibly lower quality may be rewarded, since papers dealing with topics which are easily accessible to the public will gain higher impact than abstract technical papers with lots of formulas.

Still, I do not think that twitter should not be used in relation to science. Rather, we must become aware of the nature of twitter’s information stream and utilize it correctly. This way of information propagation is new, fast and developing quickly while providing many new possibilities. We must understand what a twitter based ranking measures, if anything. In the end, twitter is certainly not a filter which separates important from unimportant ideas. But it is a great way to link up, answer questions, engage in discussions and share ideas – if you want, follow my tweets.

When you start with a new theory, you probably can’t write down the full theory at once. Usually, one starts with some equations describing the basic principles of the idea. Then one puts these equations under stress and tests them in numerous ways, until one sees where the weak points are. Indeed, a theory is a bit like a device being built by an engineer: one adds, removes, modifies components step by step until it works (more or less) smoothly. For string theory it has turned out that many subtleties are much more complicated than anticipated, but that’s another story.

Five string theories

Soon early string theorists realized that there are a couple of ways how to formulate a fundamental theory of strings. The good news: there are not many ways (i.e. the theory is to a very large degree determined by consistency). But still, they came up with five different theories, all of them describing strings, but with tiny differences. They nicely arranged them in a picture like this:

by D. Crook

On this picture you see five circles with tags like IIA, IIB etc. These are the different versions of string theory. Remember, they were found merely by applying principles of mathematical consistency. If we had good experiments, it were possible to decide which one to work with. Since we don’t have such experiments we must work with all of them in order to keep all options.

But then something remarkable happened. When physicists studied these five string theories for quite some time they came to the conclusion that all these theories are not different theories, but they are aspects of one and the same theory! The interesting point is, there are connections between these string theories. In the picture they are indicated by arrows. Take IIA and IIB: between them there is an arrow denoted by T. This is short for T-duality. T-duality is a transformation of the geometric background, and what makes it a duality is the fact, that string theory is not sensitive to it. Let me give you a simple example.

Imagine, you have a point particle and for some reason it is forced to move on a circle of radius R. The movement of the particle will depend on the value of R – and vice vera, from the way the particle moves you can conclude what the value of R must be. E.g. you measure the speed of a particle and the time it needs for going once around the circle, then you can compute the circle’s radius R. Now repeat the same with a string. Again, R shows up in the string equations, but in a strange way. If you decide to start with a type IIA string at radius R and you change the radius until the new radius R’=1/R, then you realize that you suddenly find the IIB-type equations on your sheet of paper! In practice this means, you can describe one and the same string either by the IIA equations with radius R, or by the IIB equations with radius 1/R.

This is a more than remarkable phenomenon and could not have been anticipated from the beginning. In fact, this duality is a genuine ‘stringy’ duality, since no point particle theory could ever have such a property. T-duality is not limited to circles, but is shows up in various ways. A very prominent representative of it is mirror symmetry (maybe I blog later about it). The lesson to be learned here is, that string theory is to some extent ‘blind’ to geometry, and strings on possibly completely different background geometries could be related to each other by ‘stringy’ symmetries or dualities.

When you look at the picture again, you see another duality, indicated by S. By using T and S transformations all the five string theories can be connected to each other. And this is absolutely incredible: merely by mathematical consistency we did arrive at five solutions, which in the end turned out to be just different aspects of the same thing. If one really believes that nature is logical and consistent, the breathtakingly beautiful structure emerging from these dualities could indeed be a sign that we’re on the right track …

M-theory

Now, how does the M come into the game. Maybe you’re read this quotation before:

Nathan Seiberg, a colleague of Witten’s at the IAS, uses the analogy of blind men examining an elephant to explain the course of string theory until 1995. “One describes touching a leg, one describes touching a trunk, another describes the ears,” he says. “They come up with different descriptions but they don’t see the big picture. There is only one elephant and they describe different parts of it.” The Guardian

Of course, we could very well live with incomplete data by investigating each part of the elephant separately. It would be much nicer though and more satisfying if we could get the ‘big picture’. The ‘big picture’, the elephant, is denoted by M – M like mystery, mother, matrix … whatever (the inventor, Ed Witten, is not definite about this).

The proposition is the the following: M-theory is an 11-dimensional theory; since supersymmetric string theory is generally 10-dimensional, one dimension must be ‘removed’ in order to find the conventional string theories within M-theory. There are various ways how this can be achieved, each resulting in another of the known five string theories. It sounds of course weird to ‘remove a dimension’. What this means is to curl up one dimension in a small circle, so small that it can be neglected. In this sense the resulting theory does not depend on that particular dimension any more (and yes, you must practice thinking in higher dimensions!).

One problem of course is, that there are proofs that string theory must be 10-dimensional (I’m planning to write more on this, since this is not absolutely strict). So how can we expect to find an eleven-dimensional M-theory then? The answer is subtle: it could be shown that certain fields produced by a string can ‘arrange’ themselves in a way that it appear to us as another dimension – in the sense that it cannot be really distinguished from the other 10 dimensions we started with. This additional dimension is deeply hidden inside string theory. Another very interesting feature of M-theory is, that it not only reproduces the various string theories in a limit, but it also has an 11-dimensional supergravity limit (supergravity is a supersymmetric extension of the gravity theories, so it’s not a string theory). This is remarkable because it was shown that 11 dimensions are very special for supergravity – only in 11 dimensions it can be formulated consistently. So, this in fact very nicely closes the circle: when one assumes the existence of M-theory, everything seems nicely connected.

It’s not that simple, though.

M-theory as we know it is not a quantum theory. In other words, we know the classical limit of that theory, but we don’t have a clue how to quantize it. And remember, the fact that string theory includes field theory as well as gravity and can be quantized was the reason why scientists considered it as a reasonable theory. So, some ingredient, probably the most important ingredient in the M-theory soup is still missing. We’re hitting a bleeding edge here.

I often hear people say, M-theory is to replace string theory. This is of course nonsense. M-theory itself is a theory of strings (and of various objects which arise in string theory, like so-called M-branes). In the end, this issue is an issue of nomenclature. Some people prefer to use the term ‘string theory’ only on the five 10-dimensional limits of M-theory. However. Even when we assume that we can construct a quantum M-theory, this will still not be the final answer. Why? Because the final answer should be string field theory.

I am completely aware that this will complete your confusion. In fact, before I try to explain what a string field theory is, I should give a lecture on advanced quantum theory. However, let’s try to do it without.

String field theory

After the invention of quantum mechanics people realized that there are quantum effects which are not easy to handle in this formalism. For example, the vacuum, which you might think is just empty space, has some very strange properties. Indeed, it was found that the vacuum creates and destroys particles (and anti-particles) all the time, and that these temporarily created (aka virtual) particles contribute to measurable effects. Moreover such particles do not satisfy the usual (quantum version of) equations of motions. Rather they take all possible paths through space and time (!), even those which are usually strictly forbidden. This way they capture all possible interactions between any particles. Since it is an extension of the usual quantum formalism, it has been named ‘second quantization’.

What’s the point of a second quantized theory? Imagine the vacuum has some kind of energy. Imagine we have a parameter which we use to change something in a theory (this could be a particle mass, for example) which influences that energy. Now, in order to set up a quantum theory, it is necessary to know the energetic minimum in dependence of that parameter, and the minimum is what one calls the vacuum. Usual quantum theory (i.e. first quantized theories) describe physics at the vacuum point or in the close vicinity – we call it a ‘perturbative theory’ because of this, since it captures only small perturbations. But maybe the energy globally looks like a mountain range with lots of separated valleys, each valley corresponding to a vacuum. You can use quantum theory to describe the physics at each of these vacua. But of course we want a theory that does the job for all vacua – we would call that a non-perturbative theory. And that is achieved by a (second quantized) quantum field theory.

In string theory vacua are much more exciting than in quantum field theory – and there are many many many many many many many more of them. Each vacuum has some special properties, e.g. maybe the masses of particles are a bit higher in one vacuum and a bit lower in the other vacuum. Maybe in one vacuum we find particles that don’t exist in another vacuum etc etc. I don’t go into depth here (I will do that elsewhere), just understand that it is interesting and important to study all those vacua. Just like for the case of quantum field theory, string field theory is the theory that teaches us how these vacua are connected. Technically it is a second quantized version of standard string theory. We also refer to it as an ‘off-shell’ theory. Off-shell means it does not restrict itself to particles that satisfy the classical equations of motion – in particular the energy of classical particles is confined by Einstein’s equation , or the more correct version (yes! The well known first equation is just a simplified version of the less well known second equation), which defines an ‘energy shell’. Particles satisfying this equation are therefore called ‘on-shell’. Second quantized virtual particles (and strings) are generically ‘off-shell’.

All the various string theories in the picture above as well as M-theory are perturbative theories with respect to the vacuum landscape. Still, all of them are supposed to be limits (valleys) of a second quantized string field theory. So, you might say, why don’t people put more emphasis on string field theory, but rather keep talking about the perturbative theories?

In fact, in the early days of string theory it appeared as the natural next step to go from string theory to string field theory. There has been much progress but no real break-through, while in other areas of string theory research much more spectacular results have been achieved. Hence over the years people’s focus shifted a bit, but of course string field theory has never been forgotten. A small revolution has taken place, initiated by Sen (1999) and later Schnabl (2005), who have described explicit new solutions of string field theory which interpolate between different vacua.

String field theory is technically very demanding. Other than M-theory, we know the equations of string field theory, and we know that it is a genuine quantum theory. From a conceptional point of view it is absolutely necessary to develop this theory. It has helped us gaining deeper insight into string theory in the past. I am sure it will contribute more to our understanding in the future and reveal new phenomena within string theory that are far beyond the scope of our present approaches.

All fundamental theories that have been formulated in the past were based on the concept of point-like particles. Take Newton’s laws – they work for point masses. If you use them on billiard balls you will soon seen that they need modification due to the presence of a finite (i.e. non-zero) radius of the balls. Of course, the motion of billiard balls can also be described by laws derived from Newton’s laws. But there will be differences: for example, balls can spin, points can’t.

The reason for formulating fundamental laws for point-particles rather than for those extended objects we see in daily life is probably the way we think of space. Space is nicely described by the mathematical concept of vector spaces. Vectors are basically a way to associate three numbers (x, y, z) to any position. For any such position we can ask if a particle can be found there or not. The straight-forward abstraction of a real object, which would cover infinitely many mathematical space points, is of course a point-like object. It turns out that it is indeed possible to formulate laws on the basis of such a microscopic point of view and derive from them the laws for macroscopic objects.

Using real numbers for the description of space locations is a very successful mathematical approach in physics (people are experimenting with other possibilities, too, of course). Therefore it has been used in all such theories like relativity, also quantum theory and quantum field theory (let me talk about wave-particle dualism another time). So, what’s the problem with point particles?

The problems arise when one uses point-particle laws as fundamental laws rather than effective laws for macroscopic objects.

Consider, for example, two balls and let’s say they are charged and attract each other. If you bring the balls closer together, the attractive force will increase. The smaller the distance, the larger the attraction. Until the balls touch each other.

When the balls’ surfaces come into contact, the force they exert on each other will oppose the attractive force between them, so that the system enters an equilibrium state. Any motion that has been caused by the attractive force has been stopped.

Imagine now the balls are very small. In this case they can come closer together before the motion stops, and also the attractive force will be larger. Now imagine that the balls have zero radius. The motion will not be stopped and the attractive force will grow and grow – and when the zero-radius balls approach zero distance, the force will be infinite. And that is a problem.

It is a problem because in reality, in nature, there are no infinite forces. The usual point of view taken is that such infinities are ‘unphysical’, are a mathematical relic of an inadequate description, and must be manually removed by a well defined procedure. Such procedures are available, they are well understood and work well, although they are quite complicated. In fact, the most successful and accurate theory ever, QED (quantum electro dynamics), relies heavily on them.

But there are situations with singularities (i.e. situations in which any kind of physical or unphysical infinities arise) which are not so easy to handle; even impossible to handle. Think of matter in extreme states like inside black holes (dramatic example, I know). We’d like to have a good idea how to deal with that. And if the standard infinities which arise inside any quantum theory were a little bit better behaved, this would also do its good. If we had just something which were a little bit less sensitive to singularities …

Wait! The reason why the particle, the zero-radius ball, is sensitive to point-like singularities is the fact that it is point-like itself. So, if we replaced the point-particle concept by something else … oh yes, a string!

A string does indeed have much less problems with singularities. The reason can be understood by common sense: since a string does not sit on a single point in space but is extended, it ‘sees’ a much larger portion of space than the point-particle. If a string hits a singularity, then only an infinitesimally small sector of the string will feel the infinite force, the rest will be unaffected. In this sense, the singularity is ‘smeared out’, the forces are averaged over. All this results in much a nicer behavior, mathematically. And, moreover, it paves the path to addressing problems in which ordinary quantum laws break down due to the presence on irrecoverable singularities. And this is not mere mathematical benefit, this is a physical application.

However, singularities are not the reason string were invented. Their history took another path, and I will talk about this another time, since it’s not really important for the basic understanding. Also, just being a little better behaved than point-particles is not the only attractive thing about strings – string are mainly interesting because they provide an easy conceptual modification of existing theories which with seemingly little effort unifies all the laws of nature we knew up to now. Big words, which will require a separate post to be discussed.

Let’s try to get the whole picture: point particles are great, but they are a mathematical abstraction. The simplest (yes, we like it simple!) and obvious generalization is replacing the point by a line or a string – historically this triggered an incredible boost of our understanding of spacetime, matter and forces. But why stop here? Why should we not look at two-dimensional objects, surface elements or membranes as fundamental objects?

Scientists have tried that of course. Honestly, it didn’t work too well. The fact that a string is one-dimensional is crucial, because only then there are reasonable mathematical tools available. If one just repeats the derivation of string theory for membranes, one creates a huge mathematical mess. Well, maybe we’re just bad mathematicians? Might be.

It turns out that the membranes and even higher-dimensional fundamental objects enter the stage again, but through the back door. This is subject of present research, so much is not yet clear to us. But it seems that string do indeed feel the presence of higher-dimensional objects, and they do interact with them. This means, such membranes and their generalizations are part of the theory and show up as fundamental objects, no matter if we like them or not.

What a surprise! We start with strings and find all kinds of higher-dimensional stuff – had we succeeded in starting with a theory of balls, we would probably have re-discovered strings in a similar way. This is one of the fascinating beauties of string theory. And you guess it, this is where everything is getting very very difficult. Only quite recently there have been propositions for a thorough mathematical description of such (quantum) membrane theories; many aspects are still unclear, though. And don’t get me wrong, they don’t replace strings – they are rather part of the whole theory. The ‘whole theory’ is an equally ill defined term. It is the far aim of string theorists to construct this theory. A very far aim as it seems. But we’re making progress and I will tell you about it.

Before we dive into the abyss of membranes, let me talk a bit about simpler but equally important aspects of string theory, in the next post.

The straight-forward way to get hands on natural laws is to conduct experiments, measure the outcome and fit the results into a neat formula. Indeed, this is Galileo Galilei’s approach and his definition of science (his bottom-up approach is opposing the predominant top-down point of view of those days, where religion or philosophy were considered as ultimate truth and any law was derived from it; or rather fitted to it. See Ernst Mach’s book on the history of mechanics for an excellent discussion). While this sounds like a good idea, there is one problem with it: it does not work. At least not perfectly.

Let’s do a couple of Gedankenexperiments and assume we’re a curious person in a non-technical society where there is no science and where even ideas about existence of laws behind natural processes is a philosophical novelty.

Imagine that you want to study the laws of a falling body. Take a stone, lift it, drop it. Imagine you have all the devices you need to measure the position and velocity and whatever other parameter of that stone might be interesting. Then you will see that each experiment indeed gives a slightly different result. Oh well, you might say, that’s because of the influence of all kinds of disturbances, the wind for example. And you’re right.

So, let’s design the experiment in a way that the stone is not exposed to the wind. What else could influence its motion? The sun maybe? Air humidity? Time of the day? Stellar configurations? The number of birds in the tree next door? Sure, that sounds ridiculous. But imagine yourself in a situation where you do not know anything about science – then, how can you be sure which parameters are important for the law you want to find, and which are not?

Well, that can be investigated: you would probably repeat your measurements at various places, at various times, under various changing conditions. As you collect more and more data you will get more and more convinced that the laws you found for that stone are universal. Well done! A universal natural law is just what we were looking for.

How do you know that you have revealed a universal law? After all those successful experiments with the stone you might be shocked when you repeat them with a feather. Because the feather does obviously not obey same law as the stone. What now.

Today of course we all know that a feather does indeed obey the same gravitational law as a stone. The only difference is that a feather is really sensitive to the presence of air. Its flow resistance is very high compared to the stone’s, and that’s what makes it falling down much slower. Maybe you have seen the same experiment conducted in a vacuum tube, where stones and feathers are falling exactly in the same way, with the same speed. But in those days when Galileo made his experiments, people did not even have an idea about what a vacuum is.

Many say that this is the instance where the genius is evident: understanding that lots of partially contradictory data are unified under the same law when ideal conditions are imposed – which by the way requires a definition of ‘ideal condition’ on the fly. Or, as I like to re-phrase it: bootstrapping.

So we have seen that in order to find and formulate a universal law of nature we need two things: experiments and a genius (and probably someone who can do the maths, if the genius can’t). I might sound picky, but still there are fundamental flaws in the details: once we’ve understood the falling process we could ask where would a stone fall to? It falls until it hits the ground. And what if there’s no ground? Will it keep falling until the end of the universe? Will the laws of falling be the same 2 million light years away? No way to check this. Will the stone fall in exactly the same way when we repeat the experiment in 10 billion years? Or did it fall in the same way, 10 billion years ago? We can’t tell.

So, what’s the upshot. Hunting for the fundamental laws of nature is extremely exciting. A great part of the job though is the art of dealing with the unknown. Those laws that we theoretical/mathematical physicists uncover in order to describe the universe or anything else, are “relatively true”: they are certainly true with respect to the range of parameters for which they were designed. And they are certainly true with respect to our understanding.

What’s the worst case scenario? Let’s say, after having conducted hundreds of stone falling experiments we have written down a law for it and saw that it doesn’t work for the feather. We realized that this is because the feather’s parameters don’t lie in the valid range for the stone’s law and formulated another law for the feather. If there’s right now no genius available who would realized that both laws are the same in the vacuum, the worst thing that happens is that we would assume both laws to be separately fundamental. Which is not wrong, just ugly maybe. We could have worked with a set of laws for stones and a set of laws for feathers for centuries maybe, until someone had discovered vacuum or space flight and re-considered those old experiments in the new light – allowing us to directly measure that stones and feathers are gravitationally identical. All that would not have been a scientific show stopper, only a major ‘inconvenience’. Let’s be glad history has not taken that path.

On the other hand we’re not sure how many scientific short cuts we have missed in history, just because no genius was available at the right time. And we do not know how many established laws of nature could be easily unified in a maybe single beautifully simple fundamental law if we were not constantly hindered by incomplete information and our own ignorance. That’s what I call the “fog of science“.

Of course this should not stop us from working on fundamental natural laws. And it does not. String theory is by far the most successful approach in history in that respect. I will deal with it and with alternative theories in subsequent posts.