Monthly Archives: January 2014

How many times can you discover something for the first time…

now, now, the title sounds strange as discovering something should imply doing it for the first time… right? Again, WRONG! 

Some years ago (maybe three) we were told in the same way that THE MAGNETIC MONOPOLE HAS BEEN DISCOVERED! Back in Romania we had jokes with “radio Erevan” usually discussing about soviet censorship and how a honest news was deformed by the censors in order to fit with the soviet doctrine… fortunately this is not the case now… now, it’s scientists that change news in order to fit to the doctrine of the funding body… so, Three years ago they discovered a magnetic monopole which was in fact not a magnetic monopole, in a free state which in fact was a atomic lattice in a natural environment which in fact was magnetic ice… now, the subject is interesting and deserves some remote analysis… While doing second quantization (you know, go to a Fourier representation, take the coefficients, impose on them commutation and anticommutation rules, introduce a Hilbert space and start playing with them in all sort of interacting environments) what you actually do is define a vacuum… and impose something called “normal ordering”. This “normal ordering” prescription tells you in principle that all the creator operators are at the left of all annihilator operators. It is only a rule but it has important effects on the value of the “vacuum energy”… Whenever no gravitational interactions are included (and spacetime is flat) you can do that. The choice is not unique but the fact that you “renormalize the vacuum” means essentially you assign to it a finite value. Beware however that normal ordering is not a linear operation and normal ordering of a function of operators is not always well defined… but now I’m going into too much depth… The thing is that we now have operators that create excitations on our “vacuum”… However, in condensed matter this vacuum is not “the vacuum” whatever that means, but the set of particles you have there in your background… the movement of an excitation through this system (a.k.a. a collective excitation) can be quantized and can produce collective “particles” known as “quasiparticles”. They behave AS IF they were particles but in fact are collections of particles. Ok, the magnetic monopole in this case is just a specific configuration of the field generated by the spin structure in that object when the alignment did not reach a equilibrium. A hedgehog structure emerges for some time and it’s “pictured” as a monopole. Anyhow, a few days ago I started looking on facebook and people made me be careful and watch how they “discovered” the monopole… and of course, I said: “what? again?”… well, yes, this time not in a spin ice system but in a bose-einstein condensate… oh dear! I am about to fall of my chair… well… maybe not… 

Magnetic monopoles however, the true ones, the ones coming from vacuum don’t exist in nature… magnetic lines are always closed… well, Dirac was of course right: you can quantize a theory with magnetic monopoles and there is no problem or contradiction in doing so but they are not realized in nature for some reasons (about which I won’t talk now and here, as they reflect my own personal research and that has to be published first). String theory loves magnetic monopoles… not the ones discovered now, but the real ones that should be… around 1 per universe… so, keep searching… funding is assured! 

 

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Quantum mechanics…

I wrote several posts about this subject but I keep seeing nonsense around… so, here, another one… What is the connection between this evening and quantum mechanics… well, after reading two very interesting papers which I strongly recommend: this http://arxiv.org/pdf/0809.0305v2.pdf and reference [1] therein and I am thinking about writing a paper related to the subject myself as soon as I’ll have some time, I was bombarded on my facebook page with some nonsense involving the, by now ubiquitous “many world interpretation”… Now, you know, I get goosebumps whenever I hear about this “many worlds” nonsense… and I am absolutely not in the mood to explain here, again, and again, and again why the many world interpretation is equivalent to the Copenhagen interpretation and why it means essentially nothing, apart from the pointless media hype it always tends to get… On the other side, I strongly recommend the paper by Gukov and Witten and I also recommend you to read whatever you can find about geometric quantization. Also, the Kirillov quantization, also known as “orbit quantization” is pretty important and gives quite some insight into what quantization should be and what mathematical structures you need to understand before speaking about “quantization”… Let me tell you, quantization is a tricky thing to do… it is rather obvious if you know what you have to do, and this is the case in the 0 dimensional case… not that much in the 1 dimensional case (quantum field theory) and even less in the 2 dimensional case (string theory, sigma models, etc.). The basic prescription has many “holes” in it, places where arbitrariness is pretty important and places where a choice of a specific  structure may change results in a significant way. 

Unfortunately, most if not all “quantum fundamentalists” or “quantum computerists” nowadays have NO CLUE about all this “stuff” and keep doing things almost blindly. We see all the time some pointless nature and science papers saying they “teleported” whatever there might be, from a photon, a boson, an atom, energy, matter, me, you, him, her from here to there, where there is the table next to mine, the lab next door, the closest island, the closest satellite, the moon, etc. but NOTHING of any fundamental relevance… If you look at quantum chemists, which are at least 50 years behind the current knowledge you will be scared about how they calculate correlation… then you go to quantum computerists and fundamentalists who are at least 70 years behind and you will see they have even less contact to path integral quantization or quantum field theory than quantum chemists… what am I saying, they still debate about BELL STATES and CAT STATES! STONE AGE RE-ENACTED! 

mathematically impossible…

There is a phrase that I sometimes hear from string theorists saying that any other option is “mathematically impossible”… really? is it? And I go back to the discussion about “interactions” in mathematics… no kidding? Really impossible? Let me give you an example: in spectroscopy, a field that I find rather boring unless someone discovers the electron is an ellipsoid, there is something called “selection rule”. In principle it says that if the symmetry groups to which the various parts of a matrix element do not fit together (have no common irreducible representation) the matrix element is zero and there is no transition allowed… Well, this never happens in reality. There is NEVER in reality a situation in which the symmetry is exact or the selection rule is perfectly applied! Every student after he/she (or it) finishes the 1st year of physics knows that selection rules should be taken with a pitch of salt and pepper and that in principle corrections to perfect symmetries are always there: the operator in the middle has corrections, the wavefunctions are not perfect etc. Why am I giving now such a trivial example? I mean: spectroscopy??? Well, because the same thing happens in string theory: you cannot simply say you have a specific structure, say a monoid and whatever goes beyond that structure and is induced by who knows what “does not exist”… because that is NONSENSE!  

Holly crêpe!

Am I defending string theory? No way! This is why this will be a short post… Some people started saying that string theory “predicts” that gravity should be different at who knows what scales and because we measured gravity and it fits to what we know string theory is wrong! I am very short on this: THAT ARGUMENT IS TOTALLY WRONG! 

axiomatization?

So, you may ask yourself why the last post about axiomatization? Well, because it shows some forms of generalizations that can be made… If you look at a quantum field theory and at a sigma model you will see that string theory is the generalization of QM in 2 dimensions… Simple quantum mechanics is 0-dimensional. Line quantum mechanics or quantum field theory is 1-dimensional quantum mechanics and string theory is 2-dimensional quantum mechanics. Now, for some very odd reasons string theorists consider that the only possible generalization is geometrical (or topological) and certainly only dimensional… But think again, what is the axiomatization of quantum mechanics? And what is the axiomatization of path integral quantization? Are you sure you cannot compensate a dimensional generalization by other mechanisms? Because I am certainly not that sure! We live probably in the “age of dimensional generalization of geometry” but if mathematics had stopped at that we would have no group theory, no sheafs, cohomologies, stacks, spaces, modules, rings, topoi etc. strings are JUST A WAY of looking at reality and certainly not the best…

Axiomatization

Today, another important discussion… of course, what I say here is rather trivial for mathematicians but it is so unknown to physicists that I start asking myself what physicists actually do? Well, the discussion is about a procedure called “axiomatization”. In essence it is the way in which a notion is reduced and by that generalized to its main properties! To give you an example: what does the derivative, the Poisson bracket and the commutator have in common? Well, if you axiomatize derivation you end up with the fact that the Poisson bracket and the Lie bracket and the commutator are in fact derivatives. There is also the Lie derivative which is another name for our well known commutator in physics. You see, they all share some common axioms and a set of theorems that unify their properties. Believe it or not, many objects and operations we use are in fact at their origin the same thing and have properties that are “covering” all of them. So, the main idea of this post is to think not only in terms of direct applications of some notions you learned but also in terms of how you define them. Extra information is hidden more often than not in the axiomatic definition of the concepts you are dealing with.

And now that the idea of axiomatization is more or less known to my fellow readers, I can end by quoting my favorite childhood book:

“Why is a Raven like a writing desk?”

Alice in Wonderland, Charles Lutwidge Dodgson (aka Lewis Carroll)

proofs…

For some unknown reason, or probably because of my early education, I am closer to the mathematical way of proving than to the physical way of proving. It is only now that I see that physicists have a rather odd and silly way of proving things. Of course, I totally disagree with the physicist’s way of proving things because I see it as very limitative. On the other side, maybe there is a point… what’s the difference: well, usually when I bring some arguments based on some proved theorems and go from one result to the other choosing some adequate theorems that make the connections for me the proof is obvious… Physicists are used to calculate (and beware, have the impression that mathematics means making lots of calculations on some models). THIS IS WRONG! Most of the time it is best to avoid doing ANY calculations at all and just to bring logical arguments that lead you from one proposition to the next. Sometimes you have to verify something directly, true, but that is the UGLIEST part of the endeavor. If I tell you that following from this and that theorem, this or that structure induces a group structure via this or that map to this or that space and the proof that this happens is there I don’t have to explicitly VERIFY it again for a specific example. Mainly because the SPECIFIC example will be a particularization with a FAR LOWER proving power than the general theorem! When I compute something by applying a theorem to a specific case I reduce the power of that theorem to a specific case. THAT IS UGLY! 

So, not the models are those that create knew knowledge! They just create new examples. It is the ETERNAL proof that is important and lasts forever and it is there you have to invest most of the time to make it as rigorous as possible! 

A model is just a model… it brings nothing fundamental to knowledge. It may bring some applications but that is the engineer’s work… 

Interactions

Today comes a very important topic… No, I mean it, it is one that starts a discussion like those questions children use to ask… like “where are babies coming from?”… No, really, it couldn’t be more important… the subject of this evening in called “interactions”… Why is it so important? Well, physicist should know it: interactions make things real. What could you tell about a particle with no interaction whatsoever? Well, that it doesn’t exist! As long as it is a physical object it must represent some form of energy and if that is the case it must interact at least gravitationally… spacetime, what am I saying, geometry itself is a manifestation of interactions. Physicist should know that better than anyone else, right? WRONG! Unfortunately physicists tend to look at interactions in terms of physical objects interacting via some other physical objects… this is rather unfortunate… because this way of looking at reality hides some very important interactions, the logical interactions… the possible results of the coexistence of mathematical structures… and these are widely and wildly ignored by todays physicists… Let me be more specific: at the famous CERN LHC there are (as the name says) hadrons (strongly interacting particles) that collide… when the collision energy is high enough all hell breaks lose! You get interactions that generate all sort of jets and particles and there will be virtually everything, photons, gluons, neutrinos, Higgs’s, particles mediated by weak or strong or e.m. decay and so on, and so on… However, this fact makes physicists rather arrogant… they tend to believe that everything that exists in nature appears because of tiny somethings that interact with one another through tiny other “somethings”… This is utterly incomplete… It is as if you said the universe is a lattice or other nonsense like that… In reality the interactions we see are just an epiphenomenon of other interactions, those between mathematical structures. There, just put a classifying space, a group and an isomorphism together and all hell breaks lose! We wouldn’t even notice that there is lots of jittering at the level of mathematical structure if there were not for something called “quantization prescriptions”… These, especially when involving gravity, allow us to make a point about how mathematical structures interact at a level otherwise unseen… and believe me, the are less restrictions than one could imagine! It is very unlikely that you simply get to restrict the possibilities of mathematics. The information is encoded in the most unexpected ways and it is transmitted from one structure to another by things that also encode some form of information. Even the possible ways in which a structure is mapped into another one can give hints about the desired information. 

What happened to me today? Well, a colleague in my office today spoke to the phone and had to spell the letters of a pretty difficult name. She used the standard method of contextualizing the data: L from Lima, I from Italy etc. I thought: what an amazing thing!!! How does this reflect into some mathematics? Isn’t it amazing that a letter is by itself badly received while put into a known context it becomes a mean of information transmission? Isn’t it amazing that by adding extra structure you facilitate and accelerate the communication in noisy environments? What you actually do is to encode the information of an entity in the structure of a larger object of known properties. Every letter from LIMA makes you think at the first letter L… if you change the first letter: say P instead of L you get PIMA which makes no sense… It is not only a topological information coding… it is more than that… I still have to understand how this reflects on quantum mechanics… 

Isn’t it amazing how a seemingly irrelevant fact can make me think at very interesting things? Well, this is the beauty of the human mind! Not the repetitive performance of tasks! 

path integral quantization

I keep hearing, mostly in the areas of “expertise” of some uneducated quantum fundamentalists that “path integrals are not well defined”… my dears… this is so wrong! It would be better to say: path integrals can be so badly defined that they make no sense at all… this may be true but it is not the fault of the path integrals. The main “problem” with path integrals is the definition of their measures. Obviously, when integrating over a functional space the measure may not exist in a classical sense but then again, why should it? The equivalence classes or homology and cohomology classes could do a equally fine job in defining the results. Whenever you define your integral over cohomology classes the problems disappear (provided you define these last ones correctly, of course)… So, reminder: don’t tell me again “the path integral is ill defined” … it is so only when you make it so!  

How fundamental is the holographic principle?

Really really… really fundamental… it is the most fundamental discovery since the discovery of socks… and maybe doughnuts…

Well, it is in some sense fundamental because it is based on much common sense. In principle, in its better form it says that a volume of space can communicate with the outside world via its boundary and not otherwise. It also imposes a limit on the degrees of freedom accessible in a volume, mainly related to the area around that volume… sure, there might be a problem if the topology is not that of a sphere, which most likely it is not but hey, we don’t like problems here, in string theory… But then, let’s go to the applications. One of these applications is well known to everyone who went to a high-school course of general physics… What? did you never compute the electrostatic forces acting on an object? Ya’ know, the old F_{e}=1/4pi*epsilon (q1*q2)/r^2 ? Well, where do you think comes the r^2 or the 4*Pi factor? Never bothered asking? Well, it’s because of the holographic principle of course: the interaction between domains happens to occur through the area that encompasses the charge… Ok, now, now, don’t look that surprised… you will see that Gauss’ theorem can be rewritten in a nice and complete way in the form of antisymmetric differential forms and that Stokes theorem invented in a primitive form by Lord Kelvin is a generalization of Gauss-Ostrograsky theorem and in the end the generalization of all these in a currently fashionable form is given by the oh so acclaimed “Holographic principle”…

Thing is, this is not completely correct… Why? Because space-time is not what we used to believe… many properties of a normal (meaning not so crazy) space must be abandoned in order to even hope to construct a correct theory for quantum gravity and among the properties that shake quite a bit are also these theorems all implying orientable stable manifolds…

So, don’t get too excited about the holographic principle or the AdS/CFT conjecture… They are cute but rather trivial…