I decided that my introduction to string theory on this blog will be a gentle one. Fortunately I had the occasion to go through this subject in a gentle manner… I had 2 very bad courses on string theory very long ago and at that time I was very enthusiastic about the subject. Chance made it that now, when I could have the opportunity to actually and officially work on string theory to know about this subject more than many other people around… (it is just an objective statement, not a lack of modesty)… String theory can certainly attract an undergraduate student and probably after working in a formal environment (like a research department in a university, under the direction of a professor) that student would be able to publish several papers on this subject. I worked on string theory in a different manner, more like a hobby and like a person that does something in order to satisfy its own curiosity about something. What I wanted to know was how exactly looks nature at very small scales? What rules does it follow? I went far beyond the standard things known by a string theorist and learned quite a few other related subjects. When I worked on the AdS/CFT duality (yep, that I did in an “official” environment, but guess how much I cared about the environment? ) I started understanding the way string theorists think. In this way I ended up being able to predict the next “principle” to be “discovered” and now it is quite public (ER-EPR duality) or the following one (the oh so famous but practically inefficient Amplituhedron) and I can now predict even the next “discovery”… well, if I have some time I may even publish it someday in the future… It will be a correct improvement, done in an honest way, but I can tell you by now that I am not very impressed by it. Well, all these have been constructed following closely one single way of thinking. This way of thinking was developed in the 1980’s and was caught by all the other researchers in the field in an almost unconscious way. Now, back to the subject of this post: why string theory and Archimedes? Well, I explained in another post that we are used to think in a perturbative way and most of the methods of going beyond this are somehow related to “small quantities” that “unfortunately” cannot be made small enough. People generally managed to invent integro-differential calculus and that was seen as a major leap of knowledge… it allowed the trip to the moon and so on… but before this discovery people were dealing with the same problems in a more interesting way. What is now done via integration, or infinite sums and adding together small contributions to some values etc. was done back in the days in a more natural way: cutting shapes, weighting, measuring volumes, etc. They had a much more “hands on” approach to geometry and found out very deep laws by looking carefully at forms, shapes, etc. Archimedes was able to figure out which crown was fake only measuring volumes and weighting objects. There have been lots of advances in this domain… the largest part of the antique knowledge was of this kind, from the platonic solids to the theorem of Pythagoras. However, after the dawn of “modernity” we forgot this kind of way of thinking and relied almost completely on the new apparatus invented by Newton and Leibniz. Well, I cannot say that this is bad. It certainly wasn’t until we invented Feynman diagrams and started summing them up. Then we found out that it is not that easy to do so mainly because we have no clue how to identify them all in a natural way and to write them as a consistent sum. In most of the cases all we can do is a perturbative expansion and in fact this is what we are doing. The problem looked very geometrical so, some people came to the same idea the antiques like Archimedes came around a few centuries before: what about replacing the calculation of hard sums with the measurement of the properties of some objects that incorporate the information we need. The first idea (and the most simple one) was … well, let’s go from a point like fundamental object to a string like fundamental object. There are several advantages in doing this. First, most of the divergences disappear. Whatever is called UV divergency in the Feynman approach doesn’t even exist or is mapped to IR divergencies in the string approach (but now I am getting too far too fast). The point is that having an object with length instead of just a point allows one to automatically sum over whole sets of Feynman diagrams in an automatic way. One simply doesn’t have all the strange superpositions and internal loops and intersections and whatsoever that appear in the Feynman diagrams and that although have to be added together most of the time do not have a meaning independently and instead has a nice, beautiful expansion in the topological genus. Many things should become easier, isn’t it so? Well… actually no! Indeed most of the expansion terms in string theory are supposed (but not proved) to be finite but the theory one has started from was by no means fundamental. It was just an extension and re-ordering of a sum of some perturbative expansion… But, can we say we can start from whatever theory and get the topological expansion of string theory? Well, unfortunately no… String theory is, in this sense, a perturbative approximation of something that is not known or not uniquely defined. It was supposed to make the life easier but in fact it complicates it uselessly because, except the fact that one should assume some structure at the very low level of matter (high energies) one doesn’t have a clue what other principles should be added. Simply stated, it is as if you wanted to measure the volume of a dodecahedron and tried to define it as a perturbative approximation of a sphere. And you had no idea what is a dodecahedron. Many people started identifying problems in the theory, anomalies, inconsistencies etc. and imposed restrictions on natural parameters in order to avoid them. What came out was a bit odd, we all know about the 10-11 dimensions, Calabi Yau manifolds, fluxes, no go theorems, supersymmetry, etc. They all appeared in order to avoid inconsistencies in a perturbative expansion. Now, how much arrogance one has to have in order to impose restrictions on nature to correct anomalies of his/her perturbative expansion is left to the judgement of the reader… I discussed what I think about this before and I won’t do it again… Fortunately, lately, some smart people figured out what that means or came to some common sense (see L. Susskind who arrived at some interesting ideas about the way one has to think about nature… not necessarily ER-EPR which I still consider trivial, but the way of thinking is definitely the right one). The idea would be to explore some limit phenomena (black holes, etc.) and see what problems appear and how one can avoid them, derive new physical principles and try to guess what exactly an dodecahedron is?

Ok, this was the post of today, as I said, this is by far not the full story and it is obviously not the rigorous version of it but it can give some good ideas…

To be continued, especially with the next “leap” and some limitations of this “new” way of thinking…