Here I stay and think about what makes a discovery something important… It may be the way of thinking, it may be the result itself, it may be a new approach to some subject? In all cases it is not what is commonly described today “a discovery”. Indeed one cannot hope to discover much today keeping a certain way of thinking. I was looking at a nice movie about egyptian and greek mathematicians. Apparently they differ by a very important aspect and no, it is not related to the results but more to the way of thinking. The egyptians apparently invented numbers and operations with numbers but lacked some sort of ability to generalize. It took 4000 years for people to reach that level of abstraction to invent an axiomatic approach to mathematics. Nowadays however we think also in some patterns and one may consider we need another 4000 years to emerge from the current patterns of thinking. They are not “true” and science is not “additive” or “cumulative”. Cumulation in science keeps you in the same patterns of thinking. It is a radical shift from this pattern that gives you the possibility to see a bit of the “physics after the year 6000″…

# Monthly Archives: November 2013

# revolutions…

Is there a simple and direct way of thinking? That might be an interesting question. I am more and more interested in thinking about how one can think. In general there are incremental ideas, simple thoughts that are essentially related to other thoughts and so on. This forms the greater realm of “group thinking” and is to be avoided as much as possible. In this sense problems are more important than solutions although a balance between them is natural. Although one may not see it immediately knowledge advances through revolutions and not through simple adjustments of some ideas. It is often said that Einstein just enlarged the domain of Newton’s laws, or that Quantum mechanics just enlarged or improved classical physics but this is not so. Although any new idea must keep in some areas as some sort of special cases the reality as we see it, there might be much to reality that we do not see or to which we have no simple contact. In that case it is required to challenge and radically shift old patterns of thinking. Newtonian dynamics has very little to do with special or general relativity. It keeps some principles and brings some ideas together but the main body of relativity lies in simply trashing Newtonian dynamics and starting a new path. It may contain a special condition as Newtonian dynamics but the traditional way of thinking is turned upside down and failure of perception is for the first time introduced in the general way of thinking. This principle will become more general once one accepts the basics of quantum mechanics and one can go on and on with these ideas. In fact, at this moment we still are clueless about what is natural fact and what is imagination or the result of an abstract non-real formalism we use to describe nature. By this none of the assumptions of “generality” are to be accepted.

# Uniqueness!

So, what we learned today is that uniqueness in physics is a very difficult concept. To prove that a method is unique is a hard thing and it was not done, never! Also to say that X and Y are not related is something pretty hard. Subjects may be related in ways most of the people never noticed. They may have skipped this aspect not because they are not smart enough (although this is the case in many situations) but because they just do not have the information that makes a connection clearer. So, do not forget what Feynman said: To have a correct description of something you should have at least 10 different ways of describing that something! Uniqueness claims for a theory or another are in general nonsense or egomaniacal outbursts of this or that person. Claims that the “final answer” and “the theory of everything” are within reach are in general nonsense and should be avoided. There are very nice string theorists out there and I was in contact with a few of them. None of them was obsessed by string theory and considered it a “theory of whatever”… Some socio-cultural and political aspects have changed this situation in some places but there are quite some people that understand what is right and what is wrong with string theory. You can like ideas behind string theory. I do like some ideas there too but I will criticize whenever absolutist claims regarding some opinions are being made!

# Ask an alien!

Whenever in doubt about the physical validity or “sanity” of your assumptions imagine there is an alien in a galaxy 13 millions light-years away. Would that alien be forced by nature to think in the same way as you did? If the answer is NO then your theory is not fundamental, although it may produce accurate results… Then again: is there something really “fundamental”? Maybe…

# Is space time emergent?

The best answer to this question is “lol”… First, it is a point where string theory and LQG converge very easily. At least LQG said in a very naive way that the answer should be YES for a very very long time. But, in order to answer to such a question correctly we have to be in clear terms what is it that we are speaking about and this is where the math knowledge of string theorists goes down the hill… So, what means “space-time”? What means “emergent”? and now that I am asking questions : What means “is”?

I am quite lazy to write the answer to these questions now but it should be funny…

# how could string theorists be so wrong?

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”

# Numbers

There is an interesting concept humans used during almost the whole of their written history, at least starting from Sumer: numbers… But there is a quite large difference in how we perceive numbers today and how numbers were seen 4000 years ago or more. Let’s start with the beginnings. Numbers were used to count things. A sumerian farmer had to know how many cows and sheeps he has, how to sell them, what they require as food etc. All these concepts were encoded via symbols for whole (natural) numbers. This is why the concept of “natural” number was so … natural… and was found everywhere in nature. If a farmer had a cow and an ox they were not as surprised to say they have 2 animals and the number may even go up to 3, 4, etc. There was no need for a too deep philosophical understanding for why there are these jumps (quantum leaps) in the total number of animals the farmer possesed. Of course, after some progress in science and technology one found out that a new animal somehow evolves from small cells that organize themselves until they grow up as another animal so the evolution appeared to be somehow “continuous”. So, one had to invent fractionary numbers and then, with the discover of the circle one found out that some question regarding the circle have answers that cannot be represented using a finite number of digits. After all these extensions one has arrived today at several types of numbers: natural, integer, rational, irrational, real, complex, etc. What one has forgotten is that they do not encode “nature”. They encode possible answers to some questions related to nature. So, the sumerian farmer was not very surprized while seeing natural numbers in nature… well… don’t worry, the modern student in quantum mechanics, after learning complex analysis is very surprized to see that some questions have answers that are encoded solely as natural numbers. In order to understand this better the modern student has to unlearn modern analysis in one-year courses of general topology and algebraic topology. After this, our student finally should understand that there is no problem or contradiction between having some quantities giving a continuum of values and others a discrete set of values… How many times do you have to turn around a circle to come to the same place? Any integer number of times. The trajectory is continuous and if the question is “what is the trajectory one goes on when turning on a circle” the answer will obviously involve some real or complex numbers described in a continuous way. That doesn’t change the fact that when the question is “how many times do you have to turn around a circle to get to the same place” the answer will be an integer… see, integers and reals can live peacefully together…

# Ways of thinking

I was reading some comments about mathematics as being “an art”. Well, physics is an art in some sense too. The art comes from the various ways of thinking that one can approach in order to describe reality. There is this choice that is not at all obvious but is essential in doing anything relevant. If there is something that interests me most then it is this: How should we think about reality? Are there some new rules we have to discover? Do we have the chance of discovering them? I am not interested in this or that approximation that allows one to compute something faster or more accurate. I am interested in the connection between reality and our way of thinking. I see the underlying ways of thinking that stay behind various theories and this makes me see that in fact there are very few ways of thinking from where most of the ideas emerge. For example I can see that string theory and QLG are based essentially on the same way of thinking. Remember when I discussed about the two ideas: perturbative approaches and lattice approaches and that both are based on the same way of thinking, namely, how to represent reality in terms of something “small enough” but not “infinitely small”… Both ideas come from a mix between geometry and perturbative approaches. If brought to the limit they should produce the same thing… some sort of “QLG-ST” diabolic mix. It is funny though that string theorists have managed to make these two ideas look like “opposing”… they differ in the way we choose to look at them but they are not that different from the perspective of the ways of thinking.

# The perimetric way of thinking

One special way of thinking one can go on is the perimetric one. As for any perimeter, it is a closed way, i.e. following it for too long brings you where you started. This way of thinking is what I call the “perimetric way”. Except the fact that it brings you to the same place it has some other features that can help you identify it when you see it in a research article. The main idea is to see what happens if nature were not to be as it is. Otherwise stated it is the Wittgensteinian question “why did people think the sun turns around the earth? Why, because it looks as if it does so. Then how should it have had to look if it were to look the other way around?”. Of course, this way of thinking has many roundabaouts of this type. The perimetric way of thinking asks itself how the world would have looked like if… and here it comes: if the graviton were a spin 7/2 particle, if the gravitational waves were not existing, if the space were 0 dimensional, if c were not a constant of nature, if the electron were not point-like, if space-time were not minkovski (which usually it is not) and so on. The general answer to these questions is : “bad” … yep, the universe would be in a very bad shape if these things were not so. There is another good reason to call this way “perimetric”: it always keeps a decent distance from the core of the problem while still being able to generate publications. I will not show here the desastrous shape of the universe if for example special relativity were wrong. I do not even show why quantum mechanics has to be right. I just show a way of thinking that is relatively common and simple to go on. Whenever you want to publish something just assume some thing you learned during the first year is wrong. You can certainly persuade someone to pay you for this research and you may improve the technological abilities : new experiments, new space-probes, etc. So, don’t be shy… it’s just about some new “perimetric” innovative ideas…

# doing the math

First you find out you cannot do the math. Then you learn how to do the math. Then you feel extremely proud about the fact that you can do the math. Then you find out that what you do is not math…