Quaternions and Fundamental Physics

Transcript

[Music] can you tell me yeah I'm from from Munich just welcome everybody yesterday people were thanking the organizers I don't think this is enough I'm glad to be her but I explicitly include yesterday's dinner so well I'm among mathematicians I'm a physicists I beforehand I apologize for any imprecise terminology but actually what's the difference between mathematics and physics and in physics we have also beautiful mathematics we have Hilbert spaces and function spaces to describe quantum mechanics of the hydrogen atom and we have differential equations in electrodynamics and we have also beautiful differential geometry with curvature and torsion and so on and so on so actually what's the difference I think the main difference is fundamental constants and could be that physics is just math plus fundamental constants and think about what's the role of fundamental constants what do you imagine is it really essential for a description of reality and so we have for instance the gravitational constant H Planck's constant the speed of light and H zero is how the constants they it includes the age the age of the universe and and by the way the lung young lady yesterday said we are doing just for fun no I mean this is serious stuff we're talking about we're talking about properties of nature and one of these strange properties is something like the mass ratio of the proton on on there from the elementary constituents of the hydrogen atom or something very strange is the fine-structure constant and that would be the ratio of the speed of light to the ratio of the speed orbiting the electron orbits the nucleus so but what does that mean what are fundamental constants really and I'm going to question that concept and let me just remind you from a couple of quotes by Albert Einstein I cannot imagine a theory containing an arbitrary number which could have been chosen by the Creator just another way so this is all that Einstein talking about numbers like 137 points something the fine-structure constant and he said he made another interesting statement numbers arbitrarily chosen by God do not exist and their alleged existence relies on our incomplete understanding so this is an interesting point and as a matter of fact if you look at today's contemporary physics we have lots of lots of lots of unexplained numbers dozens hundreds they don't even count it anymore so if you look at the standard model of particle physics don't don't be afraid yeah you might say I don't understand this we have to explain but I'm sorry I don't understand it either so because I think a reasonable person for a reasonable person there is not very much to understand a reasonable person would say something like that if the Lord Almighty had asked me before embarking creation I should have recommended something simpler and now I go Google to was King Alfonso a sabbia and actually no joke he said that this in collision Portuguese language apologies again says in your todo Buddha Rose amici BCBG Durant is the embark on a creosote you derivative recommended oh I'll go my simplest and when did that happen that happened when he visited the library of Toledo and he saw these incredibly complicated model of medieval astronomy with all the epicycles and what would you could reasonably call a standard model of physics back then that lasted for one hundred five hundred one thousand five hundred years so we have to ask sometimes if we are doing physics in a similar way if we observe such a complication that I mentioned before and well let's get so I do not consider all the peasants and parameters of the standard model all the matrices I don't think this is real progress I don't think this these are very significant numbers but there are actually a few very significant numbers we have to deal with in physics and these are by the way not only not only numbers you know you can't calculate numbers like it's not like unfortunate it seems like P and Euler's number it would all be computable but this is not computable so it's a big riddle but some people also say Oh only dimensional only excuse me only dimensionless numbers are important would be a challenge no this is not true because it's true that the numerical value si of the gravitational constant is not essential because it depends on our number system and so on but as a matter of fact it tells you something very important about nature about the strength of the gravitational interaction and it's a matter of fact that it's not ten times or double times it has to precisely the value it has and it's our job of theoretical physicists to explain them okay so I would like to say I would call this constants of nature the gods of modernity because most physicists would believe okay these are just things we have to deal with but I don't think this is a good idea if you want to make fundamental progress fundamental progress means get rid of the guards and explain the numbers and also and then I'd like to state the law of nature which is based upon nothing more than the belief in simplicity that means comprehensibility of nature there are no betrayal constants that is to say nature is so constructed that it is possible logically to lay down such strongly determined laws which only contain logically deduced constants and I admit this is a very very I completely agree but this is a very ambitious program okay it's very ambitious and I'm not claiming I have found the solution but if you think in general terms about physics and what fundamental physics might achieve it's it's important also to look at history i'm i'm stan is not my god i'm not gonna hear another ideology i think there is some pretty good evidence from history that free parameters have always in complication have always indicated something what has gone wrong and the brilliant philosopher Thomas Kuhn has written the book the structure of scientific revolutions and he pointed out the the connection between and normally something you do not understand and you will fix it with a free parameter which is the same thing as a concept of nature and but steadily you have this complication whenever there is an assumption and assumption at the places which is wrong okay if we have an anomaly that means we have to question something very basic we believe to be right but that might be wrong there is a simple measure of progress in fundamental physics again yeah elimination of free parameters has always been accompanied by really tremendous revolutionary insight in physics and I'm going to show you just a little history this is another difference I'm if I'm not I'm not presenting any proof sorry I just you physicists providing evidence but I think there is a pretty good evidence from the history of physics that getting rid of fundamental constants was always a big progress and I see a certain structure in this if we make real progress there is a visionary idea you have to have then there is an equation that gets rid of this parameter and you end up with one constant less and this this is the big progress but let's let's go to examples what's the visionary idea of Newton's gravity Wow celestial gravity and gravity add the same thing I mean you had to there was a bold assumption in in the 17th century right and of course you have then the the formulation which is not very difficult but if you look at it closely you have these inverse square law but you had you could say before okay local gravity is something like a constants of nature things are attracted with a certain acceleration okay that would be a constant of nature but if you have Newton's insight no it's not a constant of nature anymore our local gravity it's related to another more fundamental concept of nature the general gravitational constant so whenever we have progress we end up with less constants and maybe this is the best example of all the bold idea is light is an electromagnetic wave Wow I mean modern civilization would be unthinkable without that insight but at the very core of the discovery is just one simple equation you have the strength of the electric and the magnetic interactions which you can do in the laboratory and this is related to the speed of light this is sensational okay and of course you have to have not only an equation but you need also mathematical apparatus which in this case is Maxwell's theory in Maxwell's equations and all all this has to be consistent but at the very end you end up with one constant of nature less and this is the big progress this is the simplification that that we have to you have to reach last example I mean of course you have a beautiful structure I mentioned it hilbert spaces and and quantum mechanics and and differential equations you need all this apparatus to justify it but at the very end you are able to derive read Burke's constant which was measured by johann jakob Palma in 1885 he measured all the hydrogen spectra all these light lines yeah and it was you could say that were constants of nature and they were explained by such a simple formula and it was explained by a new constants of new constant of nature found ten years before by Max Planck and Albert Einstein gave it the right meaning and yeah in this case the posed idea have to mention this because Niels Bohr was really a visionary of physics he looked at this concept of nature and said okay this is known to be important in the photoelectric effect Planck's constant but it has the unit of an angular momentum so couldn't it be that the angular momentum in atoms is just H and this is the bold idea that led to another elimination of of a constants so I could go through the entire history of physics here but I have only thirty minutes so I will speed up a little bit let's and by the way I I said it was an ambitious project and I'm not going to tell you that I have the theory of and explain everything there are lots of lots of problems an important work to do the usual way is you can guess an equation like labor and next well did develop a mathematical theory and then at the end you're rewarded by this elimination of constants and there is another if you if you look at these three most important constant there is another very interesting thing I'd like to mention there is indeed a way to get rid of the gravitational constant it's a very interesting approach by Einstein in 1911 actually the very first idea he had when developing general relativity but unfortunately he stopped in Prague to develop this and and shows another differential geometric formulation but this is a variable speed of light theory and Einstein developed it and if you think if you develop that thoughts further you note very interesting relations to max principle we believed that the strength of the gravitational interactions is due to the size of the universe and there is a very interesting link to Dirac's hypothesis Dirac developed the cosmology he's famous for other things in quantum mechanics but he did very interesting work about cosmology in 1938 also shredding ahead and another cosmologists and he is the you might not know him by the physics he's famous for the discovery of the Cosmic Microwave Background and Robert Dicke had also reformulated another version of general relativity with this variable speed of light and you could get reasonably get rid of one constant G here I have written a book and about this but this is not my not my yeah it's just not my topic today we are among mathematicians right and this is still physics conventional physics if you want because in a way you guess as you guess an equation you try to develop a consistent scheme of mathematics that backs this and you end up with one constant list but if you get rid of the convent of the gravitational constant and you still have C and you still have Planck's constant H and this is very strange and here it's a real problem because there is no way of explaining C and age in a conventional manner it's impossible to to set up an equation like C equals blah blah blah are the constants of nature no it's impossible you would do circular reason why because you have just you have just two physical units meters in seconds and these are average arbitrary definitions time is some 91 billions of transition of a cesium atom okay is one second and and a meter is just defined by the speed of light or it was historically it was related to the circumference of the earth so there is no way to do this without being circular yeah now and this is the logical fallacy many physicists fall into now they say okay this is a matter of definition it's not important no it is important because it's very fundamental that these two constants occur in nature once we're doing experiment it's not trivial that you have this limiting speed and it's not trivial that if you go to microscopic scales you'll see all these strange phenomena related to quantum effects this is not reveal that means you can do it by the conventional way of setting up an equation that you have you still have to explain the occurrence of these numbers and so the question is is there a possibility that see and age are a consequence of mathematical properties okay and is it possible that C and H are anomalies in the kuhnian sense meaning that they showed up in physics once we started from a wrong assumption I mean but what what the sumption could possibly iran there was only Newtonian physics so and the only two they only two assumption arbitrary assumptions Newton did was there is a three-dimensional space and there is time so we have to question these two concepts once we want to understand the origin of C and H and once we on want to understand reality properly there is no reason why matter should not be accelerated beyond the speed of light there is no logical reason just try to set your mind and go back to Newton's thinking it's completely completely strange the very existence of thatis such a speed of light and of course it's also very strange that there is the Planck's constant H there is no reason for this continuous phenomena in classical physics and the notion of continuity and differentiability should be still well okay I mean by the way I think you mathematicians you should not tolerate this behavior of a physicist they say oh we have a Stokes theorem and now steering very nice we prove it and we use all this nice stuff for continued to the differentiability and well if we have the phenomena you have to include quantum phenomena and whatever that means why it has to be a reason for it so okay it's it's really becoming very difficult how possibly can we explain see and age it points to at the fact that they are related to meters and seconds to the division points to a problem with space and time but they must reflect qualitative mathematic properties okay there is no first-hand calculation that can give you some insight it has to be a property space and time are certainly the constant that are most easily accessible to human perception but think about there may be inappropriate when it comes to describe reality on a fundamental level and this is of course this kind of mind-blowing and it's kind of I mean destroying much of what has been the base was at what has been considered the basis of physics for 400 years if you think about that but we should never us try to find explanations also for see in the age and by the way I'll warn you what what's in my view is very wrong the paradigm which is which is dominating theoretical physics for one century put forward by Hermann Minkowski space by itself and time by itself are doomed to fight away into mere shadows and only a kind of Union was it who will survive in an independent reality no I think this is misguided and it's very nice to phrase a special relativity relativity in a four dimensional manner okay but space and time are not just four dimensions no okay because they have different phenomenology we have three spatial dimensions in one time you can't just glue it together if you're not literally departing from your senses it's something different which needs to be explained and Minkowski uses the speed of light oh it's just the conversion factor he he sweeps the problem under the under the rug but see is something we need to explain okay it's not just an unimportant conversion factor so I think this is the the wrong approach and the right approach here we are within Hamilton he knows that time is said to have only one dimension space to a three dimension to mathematics quaternion part takes both of these elements in technical language and so on he noticed that okay there is something very interesting in in quarter nians which has a three plus one dimensional structure but again a caveat I'm not can not explain you that criterions are a direct are directly representing a four-dimensional space-time no unfortunately it it does not work in such a simple way to identify because you still have four dimensions and they were all equivalent and space and time are different phenomena physically there must be some more subtle some more more difficult parallel and so but I think that the interesting properties of quarter notes they are almost all contained in the unit quarter earnings law all what you need to do afterwards is just stretching and shrinking but they the structure is contained in in a unit quaternion and I'm considering the unit or journey ins and again as a physicist we are very sloppy for me unit quaternion so as you - or even as three the three-dimensional universe all the same thing and well they're just very nice properties very interesting properties very rich properties because at the very end you're looking for something you're looking for something simple because simplicity is a good idea if you want to explain reality but at the same time it has to be a richer interesting structure okay by the way I forgot to mention of course everything we describe reality such as vector fields or complex numbers in quantum mechanics is somehow included in the Criterion's though these coordinates are very interesting candidate and you have very nice rich structures like the hopf fibration and well I shall should show you one particular interesting feature of of the of a three dimensional unit sphere it's related to so3 the rotations in three dimensional space we are talking about Newton's three dimensional space and the possible rotations of objects okay you can up rotate objects in any directions so it's a three dimensional manifold these numbers and so what happens if you rotate objects the best way to do this is with a book of algebraic topology but any book works so I'm trying to rotate this book in counterclockwise by a full turn this 2pi okay it's quite uncomfortable but I have the same position in three-dimensional space in conventional thinking this is again at the original possess this is identity my shoulders identity unfortunately that's not a contract to the path okay and you know you cannot do computer graphics and just like it so this is a real arm you would you would hurt it can you imagine that I can I continue to rotate huh that's important it seems impossible yeah yeah so I started here I rotate okay to Pienaar I continued you believe I continue to rotate you don't believe it I continue to rotate to PI and I end up with the same position and now we are in s3 in s3 this r2 this and this are just two different points and there is no need to contact the the line and but if you if you look at it for that reason s3 is a much simpler structure than the conventional rotation in three-dimensional space which gives you this uncomfortable situation if you try to if you try to perform a full turn okay so we have this simple connected net connectedness of the unit quaternion and by the way a category per element proof that the the s3 is the only is defined by this it's the most simple three dimensional manifold in this in this manner and it's interesting for that reason too and well I'm just until now I'm just saying well let's look at it it's an interesting structure and this is a beautiful picture beautiful visualization of the hopf fibration it's done by Niles Johnson a professor of mathematics and the question is can unit quaternion describe reality the peculiar 3 plus 1 dimensional form arises naturally in quaternion algebra you'll know this we have the real part in the vector for this is very interesting and the spin might be a consequence of the double cover I get to this but one of the most distinct features is non commutativity you think about I mean where do you have non commutativity it's a quite an interesting thing and of course in physics you have non commutativity if you think about angular momentum ok you try to we try to commute operators of x direction and y direction and it's not possible they do not commute why is this there is no a priori reason so non commutativity is a very interesting feature that makes you think there is some relation between quantum mechanics and this and this feature but I think the most striking example is stern-gerlach experiment in 1922 people have had discovered that electrons were orbiting the nucleus but LG electrons had also an intrinsic angular momentum they were rotating you have a spinning axis so she and gallic said ok let's do it an experiment let's let's take the electrons from a from a beam and this with this spin they would end up here and the opposite spin the wound would end up below but in between you have lots of different orientations edge they should all form a continuous picture here of the experiment but the outcome was sensational there were just two and that means there is something fundamentally wrong in our assumption our understanding of rotations in three-dimensional space and elementary particles there is some very strict subtle doubling of states okay and this is of course what reminds you very strongly from from the double cover you're talking about rotations in three-dimensional space and immediately you have this doubling of of of appearances so you might ask well what's wrong with how do we how can we explain four dimensions maybe we don't need four dimensions if you move on it if you move on the sphere this is a two-dimensional example if you move from that you you always have a tangent space but couldn't it be that the sequence of tangent spaces just creates the illusion of an extra dimension okay so time could be not something really four dimensional but just the sequence of tangent spaces on a three dimensional manifold and if you have time later I show you this because it's a very nice video and it's just I'm just mentioning it because it shows a very nice interesting property of of rotations of multiplications in quaternion algebra and well emergence of see that on each phase could be related to light count observations at distance it's just an approximation you're not sure every every information you get is light but you're not sure you get it from the manifold it could be that you get it from the tangent space and there is another interesting observation light has angular momentum so we imagine photons we imagine light propagating in a straight space for some subtle reason there is an inherent twist but it could be the other way around that light is actually something that propagates straight but is in an inherently twisted space like s3 okay so this is another interesting feature and I said that the double cover I already mentioned I already mentioned non commutativity and there is something interesting of left and right multiplication if you you introduce the screw sense I can show you the video later but you have this property in elementary particles you you have a screw sense okay regardless of orientation in space I can do it the right-hand side org I couldn't do it the left-hand side okay and this is something introduced also by deep by the quaternion multiplication then what to summarize your Aleutians and physics has always illuminated arbitrary numbers a complete theory of reality must do without any constants of nature C&H are anomalies that casts doubt on the basis of Ujjain physics space and time we must explain me there must be a reason for this peculiar 3 plus 1 dimensional structure and well you now you invite it to speculate I unfortunately I cannot give you a good model of reality but I think it's just terrifically interesting and you might speculate about if you talk about fiber bundles you might not want to replace only the fiber but also the not only the bandit that also the fiber so how to classify all these maps from s 3 to s 3 would be an interesting problem I think and space and time are in proprietary ality a sequence I mentioned that the sequence of three dimensional manifold may create the illusion and I also mentioned the rest could be a consequence of the tank by seas by the way related to light and h2 merit the two basic phenomena and well you have things like you know I shouldn't stir you make you might think well we're doing interesting things here and it might be application in physics but I think most of the applications are really not interesting and do the right thing somehow patterns are fundamental building block of the universe and I apologize here because it's not exactly a new trend in physics so if you want to look at my publications there's one in another physic which all started then there is a book about bankrupting physics which is a translation of that book that got the award for the science book of the year in Germany and I have a critique of particle physics and this is about the idea of cosmology and eliminating G and this is my last book and you note here the cerebral Attic projection of quarter nians here okay thank you very much what do you need if well there is I mean as I said I mean the solid model the physics community is based on the standard model and many people are questioning it and could it be the right thing that that's that's a feeling but on the other hand there are no there no really fundamental good ideas yeah but it's certainly what I'm proposing is certainly a heretic point of view yeah yeah I don't believe in gods and in constants of nature you didn't get from supersymmetry yes yeah I know I mean there there's a beautiful mathematics I don't want to discourage you and there are applications but you have to realize that all these supersymmetry and all these all these group theoretical arguments are just tools to sort out to sort all the mass of the particles of the standard models so they do not really question why we do have these 35 parameters so it's it's interesting mathematics but on top of the wrong assumptions from a physical point of view it's much more direct it's much more important what you're doing with quaternions they might have a much more direct relation to reality that's the take-home message effort before time is an illusion yeah you have an idea of illusion it is now to be honest yeah yeah to be to be honnest I can just give you a glimpse this is no developed theory so it's true that there are other people like Julian Barbour has written interesting books at the end of time in 1996 these his ideas are a little bit related but at the very end also also different there are things in common and there are different things yeah yeah I mean it unfortunately as I said you can you can create the illusion of four dimensions by just looking at the sequence of tangle tangent spaces but unfortunately it let's not work in a very direct way because you have closed paths and what's closed path and time I don't know so there must be some some inherent process maybe flows I think flows are an interesting concept but I can't I can't give you a good answer here thank you very much [Music]