Honestly, they were not… or at least not totally. I cannot explain here all the details mainly because I will have to publish them first but there is a series of methods that can essentially show, not necessarily that string theory is wrong but that it is a form of wishful thinking or, better stated, a form of confusion. This will be a long post and will be changed in time so it should remain somehow “on the front view” of this blog. At this time, I can tell you where are the mistakes in the “way of thinking”. We may start with the basics because there are some problems even there. I discussed the problem with the fact that the theories are perturbative constructions without an underlying theory before. I don’t want to insist too much on the fact that string theory doesn’t predict anything. There can be theories that just are hard to be verified or are not finished yet so this should not be a problem, so I grant the fact that although it doesn’t predict anything now, it may be a good direction to go on. Unfortunately there are far larger fundamental problems there and not all of them can be put aside with a simple empirical argument. Let’s start (it won’t be easy and it won’t be nice but you’ll learn quite some things about the “ways of thinking”)

Take for the beginning a simple quantum field theory. If two fields are defined in points of the space-time that are separated by a space like distance then they should commute (or anti-commute for fermions). But what happens if one adds gravitation? Gravitation is a more interesting interaction… it affects the metric and it affects notions like distance or “separation” so, the information about the separation of the two points where we want to consider the fields is not well defined unless one solves the dynamical problem. See? It’s the usual kind of problem: “what happens with the information?” The essential question, better formulated: what is the rule for quantizing gravitation or a field theory that includes gravitation? Now, one should step back and think about a procedure called “renormalization”. This procedure is essentially defined for perturbative theories but it is, in the way it is constructed, a method that allows one to look a tiny bit beyond the perturbative regime. What am I talking about? Well, in essence, the need for renormalization tells you that you defined the parameters of your theory in a very bad way. It doesn’t tell you you cannot define them like that (and this should be noted) but it tells you that the information you are seeking will be hard to obtain if you stick to that choice of parameters. Renormalizability is a criterium that should be satisfied by a theory. What does this criterium tell you? Well, it says in principle that divergencies affect some integrals used to calculate physical objects (like cross sections) because some parameters you have chosen in your theory are ill defined in some regions of the integration. This means that you may have implicitly assumed something that is a bit in contradiction with the idea of getting finite values. Now, the thing is you can make a different choice that gives you finite values but that is still wrong or not consistent with what happens in reality. You see? Following this way of thinking you can never say that a specific outcome is what happens in reality. You can just say it gives finite values, which is ok, as long as you are interested only in that. The principle of renormalizability tells you that only some theories, where you can (starting from a perturbative expansion) perform such a transformation to a set of parameters that produce finite values of the predictions of your theory, should be allowed.

Now, * Gravitation is not a renormalizable theory!* So, what are you going to do? Eliminate gravity as “non-renormalizable”? Well… no, not so fast at least…

If you look more carefully you will see that this amounts that your theory should be defined at all values of the integration parameters over which you chose to integrate. Is that so? My personal belief is that the answer should be YES… but this is a personal belief. A group of people (those working in Quantum Loop Gravity) assume that this criterium is not necessary and in principle one may assume a discontinuous, lattice like structure of spacetime. I will discuss in what sense this idea has very much in common with the common way of thinking nowadays and with string theory probably in a future post, where I will show that the two “paths” are not essentially different. Another way of looking at the problem is that demanding renormalizability you practically ask the universe to be nice to you and not to make too much noise about the fact that you made a wrong choice. What tells you that you didn’t make a wrong choice that is so wrong you simply cannot renormalize the theory or you just cannot go to any domain where the theory is well defined in a “continuous way” from the place you are now??? That doesn’t mean there is a “lattice” somewhere in the universe. It just means that you were TERRIBLY wrong! So wrong that the universe simply doesn’t want to play with people like you, so incredibly naive and ill-inspired… that may be a metaphor but it is the closest one to what I too believe really happens…

Now, the general idea of quantization of a theory is well understood in most of the cases (mainly in all non-gravitational cases). All one has to observe is that one cannot speak of observables as simple functions or numbers but one has to extend this notion to operators. One observed that this was necessary directly from experiment. It was practically what was necessary to change to make a classical theory a correct statistical theory (yes, I choose the word “statistical” although quantum effects in statistics appear in another way. One has to consider that Quantum Mechanics can produce exclusively probabilistic results so one cannot define any quantum effect without speaking about probabilities, be they 1 or 0 as extremes)

Whenever you discuss about probabilities you must consider the whole of the system and all the possible outcomes and quantum mechanics adds a particularity to this: it says that one cannot ask any kind of question independently (and here again, appears the problem of information and our way of considering it). People had to understand that specific questions are “context dependent” or are framework dependent and some questions that can be asked independently in some framework cannot be asked in that way in another framework, the result being an interference of the outcomes. Now, one way of performing quantization is nowadays known as “path integral quantization”… you see, I speak about this method as “one way” and not as “the way”. It is a way imagined by Feynman originally but then improved by many others. The basic observation there was that one has to consider all possible paths of a particle in order to get the right probability of one outcome or another and to allow for interference in the way prescribed by the usual commutation relations. There are some funny tricks that have been done by Feynman here and these are extremely relevant for some of my observations. First, he replaced the operators with their commutation or anticommutation rules with some complex fields with no operatorial properties. The method by which he preserved the commutation relations in the standard formulation was by some “internal” variables… at the beginning they were the “time variables” and we had “time ordering” but this is not an exclusive choice. In Conformal Field Theories one can chose “radial ordering” and anyhow, the point is there must be something where you can impose an ordering upon, some sort of index or something like that. In phase space the choice is simple and one can see directly that the position and the momentum are the things that will not commute. In other situations the choices may be more interesting (in the better or in the worse). Ok, so now we end up with commuting or anti-commuting complex fields. Great! But, is the choice of fields we arrived at unique? No, of course not! Are the fields “real observable quantities”? No, of course not! Is the theory constructed in this way unique? No, of course not. We are dealing with constructions of our imagination. None is unique, real, physical or meaningful in any way. All we have is a theory that gives in the end some values. The values are very accurate and agree with reality but the whole process that leads there is dependent on some arbitrary assumptions that we made and the fact that we have chosen the parameters such that they match the experiment. Think differently: would an alien from a far away galaxy that has a brain with 6 lobes do the same assumptions as we did? Very unlikely. He may have a completely equivalent theory that gives the same results without even inventing notions like fields or distances or renormalization or probability… I define as “truth” or “reality” the aspects on which these two persons would agree if put into contact… and none of the above mentioned techniques are part of this “reality”