People often ask me why we work on a theory based on the idea of strings. What is the reason strings are so interesting? Why do we like them so much? This is because: strings are not points.
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.
