Division algebras and physics (Video 1/14). First video ~ general audience
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- čas přidán 7. 10. 2017
- Note 1: More precisely, four algebras generalize the real numbers *in a particular way*. To see what we mean by *in a particular way*, please see Section 1.1 of arxiv.org/pdf/math/0105155.pdf
Note 2: no dividing by zero, of course.
Video 1 of 14 - first video is meant for a general audience
Series: Division algebras and the standard model
Some short videos filmed by Vincent Lavigne
Seminar by C. Furey,
Walter Grant Scott Research Fellow in Physics
Trinity Hall, University of Cambridge
damtp.cam.ac.uk/people/nf252/
arxiv.org/pdf/1611.09182.pdf
arxiv.org/abs/1603.04078
arxiv.org/abs/1405.4601
www.furey.space - Věda a technologie
Absolutely captivating series by Dr. Furey. I have an obsession with octonions and stumbled across the Dr.'s videos. Very good material, very good presentation, and very pleasant to watch.
Mind blowing. Congratulations Dr. Furey, I honestly do hope your research of Octonions leads to furthering our understating of our universe.
Illuminating lectures on Octonions. Thank you so much Dr. Furey for your tireless effort and all the progress that you have made!!!!!!
You're a great educator. Thanks for the video series. I appreciate that you get to write your first name on the blackboard time and time again as you explain your theories.
Thank you. That was excellent. Looking forward to seeing the following videos.
Beautifully presented!
Excellent and VERY clear
I think I got a crush.
Love your work! I'm a big fan!
I found my happy place. Thank you for this.
Dr. Furey, I want to understand this more. Can you be so kind as to give a suggested reading list that will get me up to speed with this material?
Some sort of prerequisistes.
Can you go into more detail on the relationship between Quaternions and Special Relativity ?
Also found out about surreal numbers, neat stuff
As for whether or not higher dimensional systems can be useful in describing nature, it doesn't need to be true that nature have more unseen dimensionality to it for these systems to be useful. You can describe anything which relies on x number of variables for the output with x numbered dimensional number systems. The octonions for example are useful it making sure 8 different radio towers send radio waves which always constructively interfere with eathother. But if it were 9 towers you could just add another dimension
Intriguing topic. Since studying the complex numbers for electronics, I had always wondered about adding more dimensions.
Are there numbers like this beyond Octonions? Can the system extend just like group theory? I would be keen to see more videos on this topic.
en.m.wikipedia.org/wiki/Sedenion
I like this argument. Similarly, should we believe that all that scale-space between Planck-length and subatomic particles is empty?
This is absolutely fascinating. Possible follow-up: Why is it that the 8 dimensional numbering system is the point at which algebraic operations break down?
maybe they don't. Maybe there's yet another system.
You suppose that it is hard to believe that nature uses R, C, and H, but not O (which is a compelling supposition). Is another way to say that this? Given that R is a special case of C which is a special case of H, then it seems reasonable to assume nature really uses O and what we have understood so far in nature is phenomena in terms of the first 3. This feels like the same kind of inspiration for Dirac deriving, well, the Dirac Equation. (BTW, I am a lay person who came here after reading the Quanta magazine article.)
en.m.wikipedia.org/wiki/Sedenion
Dr. Furey, As someone who finds almost everything interesting, but does not have much undergrad education in science and math I have learned a lot about many a things over the years in the subjects of Mathematics, Physics, The Standard Model, Quantum States, Genetics, Astronomy, Cosmology, Theoretical Physics....... and most recently about how the four number systems may help to explain the standard model (via Eric Weinstein) I have learned about all of these things via the Joe Rogan Experience podcast. Now, I didn't get to learn in depth about all those subjects by just listening to a single episode, but it was like a springboard to dive deep into the subject matter being discussed. Joe has world renowned scientists from all fields of expertise on the show. You would fit in perfectly as a guest. Joe is the best host because he actively listens and keeps asking deep questions in an effort to help himself and his audience not only learn, but understand esoteric concepts.
Please do the JRE podcast sometime soon,
A Fan
Seth
Am I correct to assume that in addition to quantum computers, we should also be able to develop quaternion and octonian computers?
Do I need to study quaternions before octonions?
I'm no mathematician, but after reading the article on WIRED I instantly saw a correlation between octonions and other studies/theories which use an eight variable approach to understand behaviors in nature, human behavior, music & etc. Entertain me for a second but walk away from the complexity and math of octonions and replace some verbiage like "Law of Nature" with "Harmony". Like a scale in music, the 1st, 3rd & 5th notes make up the triad of harmony which is the principle of all music (well which has harmony that is..). Each of these notes have different properties as it relates to origin or key note "1" but basically a song spends more "Time" on the triad notes during a song/score. What if we took the quanternions (e1, e2 & e4) and changed them to be the triads (notes 1, 3 & 5) as it relates to the octonion peculiar theory/math. I would guess that once all of the 8 different unconventional elements are identified and further understood that they may have more in common with the laws of harmony than that of nature. Go ahead and laugh or call me crazy but sometimes it takes a complete different way of thinking about something to make so complex, simple enough to understand. As an example, over a decade ago, I read an article on technical analysis of the stock market (of all things) and explained the Elliot Wave Theory as a way to put "rules" or "laws" to the behavior of the market. At first I read for fun until I saw that the theory is based off an eight wave pattern of growth which also happens to be a fractal. Wouldn't you know it that the growth waves are wave 1, 3 and 5. I spent many years studying the stock market using the Elliot Wave Theory and I have to admit it is the most practical and best approach to understand the markets behavior. Mathematical harmony transcends so many different studies, fields that I would be shocked if we ever do achieve the great equations which takes gravity (I'll just call that the base cliff as foreshadowing) into account within our solid theories of physics today, that mathematical harmony would play a big part.
The forth number system(octonions) is for theory of everything I believe!!
🍎フラクタル的観察:
0次元の点は変分の
1/2の点で、
1/3の点で、
1/5の点で、
1/7の点で、
1次元の線・体積を埋め尽くす。
2次元の面・体積を埋め尽くす。
3次元の体積を埋め尽くす。
4次元の体積を埋め尽くす。
0±ε次元の体積を埋め尽くす。
そして0±ε次元に一致する。
点のネイピア数eは、
2次元から3次元を埋め尽くす。
点の円周率πは、
3次元から4次元を埋め尽くす。
3次元球面体同士の接触は
微小量の無限小面同士の接触で有るので、
そこは無限小面で扱うのが妥当だろう。
1/2は代数的に1/3,1/5,1/7を
生成する。
オイラーの公式、
e^iπ +1=0,
代数幾何学は
オイラーの公式を
普遍に保つことを
要請する。
臨界領域に於いては、
★
オイラーの公式が成り立つ物は、
これは極めて少ない。
★
オイラーの公式が成り立たない、
不道徳で有り得ない物が
満ちに満ちて
生成される。
1/2は2進法、
|↑>,|↓>,|←>,|→>,
メビウスの輪、
クラインの壷、
1/2+1/2=1=1/2 ÷ 1/2=2/2,
1/2は時空間を
尺取り虫
(a measeuring worm)の様に
変分を繰り返して移動する。
これは
ミクロなブラックホールと
ミクロなホワイトホールに
見える。
それを担うのが、
グルーオンであり、グルーオンの
体積空間の振る舞いが
尺取り虫(a measureing worm)に相当する。
正五角形はフリップをしながら、
軌道回転運動をする。
軌道角運動を獲得する。
正七角形はフリップして
軌道角運動をする。
厳密な理想の周回運動をすると
扱うなら、そうだろうが
実際は螺旋を描く！
原点から次第に離れる。
ハイパーインフレーション。
重力場を生み出すのは
1/7,7である。
ここから、
0,1,2,3,4,5,6,7,の
8元数が
量子力学と一般相対性理論を
統一する必要充分な元数である。
私はそう考える。
360°/7=7×51°+3°,
1周回度数360°ではなく
357°ならば
綺麗に周回する。
このとき時空間は歪む。
地球上の重力加速度程度ならば、
ほぼ平坦が保たれる。
剰余の3は原点の
無限小周りの時空間を、
決定する。
仮に
2π≡360×7= 2520,
⇔
π≡2520,
とするなら、
10以下の自然数で割り切れる。
空間を10次元に設定すると、
単純化された一個の
宇宙は
天球、プラネタリューム、
水晶球に、
あるいは
光点なり質点に
閉じ込めることになる。
天球の原点から極限的致命的
激烈な光子と素粒子が放出され
錯綜して
宇宙空間は射影される。
I came here to learn about Octonion after Joe rogan's podcast on eric weinstein
me to
so did i
Seems like the majority of people interested in this topic got this obsession because of that podcast and I am one of them :D
yeah me 2
Not me, Quanta
I'm no physicist, but I only know of talking about relativistic effects in terms of the Clifford algebra of Minkowski space, how do quaternions come into play here?
Quaternions are an example of a Clifford algebra.
Not that I'm aware of. Although they are the even sub-algebra of the Clifford algebra of Euclidean 3-space.
I've often thought it interesting that black holes have three properties . . . three properties in common with quantum particles - spin, charge, and mass.
We should also note that quantum particles seem to have hidden mass. The closer you get to the self energy, the more energetic and more massive they become. The infinities are cancelled out by a renormalization process.
Well, could quantum particles be like broken symmetry black holes?
I've had this thought awhile ago; but, recently, the idea that dark matter could be primordial black holes . . . black holes created by the pressure from the Big Bang . . . leads me to take the above observation more seriously.
I was following quanta magazine on twitter for awhile. That account has been suspended for various reasons I think. They never quite tell you why. Anyways, I saw a quanta article months ago. And so, with the recent dark matter idea bringing my observations to my attention, I thought I'd point it out to you. It could be another, different clue to the unification of forces and particles.
Octonions are indispensable to the algorithm of quantum computing.
Visually, the chalkboard doesn't provide enough contrast to distinguish some of what is written. However, the sound of the chalk certainly provides a percussive contrast to her lecture. It would be vastly better if the writing were done via a projector, or, if the board was black and the chalk much brighter.
For gravity?
what practical and programmatic applications can we derive from these theorems?
can we find a way to utilize these theorems for algorithm generation?
I like MatLab for example, it is a programming platform [www.mathworks.com/products/matlab.html], my research did include trying to find ways to utilise these theorems.
It is possible if the Koide formula (en.wikipedia.org/wiki/Koide_formula ) is pulled from the maths, it will validate everything in one stroke
Geometric algebra seems to suggest that even quaternions are no longer very relevant (they are a special case of the geometric algebra)
key word "believe"
Why only division algebras? The sedenions seem like they may be a useful arena for physics, with their zero divisors and connection to G2.
Octonomial function: (11^n)(101^n)(73^n)(137^n)
Tetranomial function: (11^n)(101^n)
Binomial function: (11^n)
Three Pascalian triangles (11^n), (101^n), (73^n)(137^n).
Feynmans favorite number 137.
1 1 1
11 101 10001
121 10201 100020001
1331 1030301 1000300030001
Tetra--------------------------------
1
1 1 1 1
1 2 3 4 3 2 1
1 3 6 10 12 12 10 6 3 1
1 4 10 20 31 40 44 40 31 20 10 4 1
Octa-------------------------------
1
1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 7 6 5 4 3 2 1
1 3 6 10 15 21 28 36 42 46 48 48 46 42 36...
1*diagonals*Pascal
Fibonacci sequence, tetranacci sequence, octonacci sequence.
Palindrome Mountain------------------------n^2
1
121
12321
1234321
123454321
12345654321
1234567654321
123456787654321
I saw nothing about the photon. /I mean in all the videos/ Why?
Pretty awesome to make 'la' into a ligature.
einstein would have wanted it that way
Cohl Furey Labert Einstein?
please explain to me "What is Number"?
I get the feeling that there is LOT here that Einstein would have liked.
Please explain to me "What is Number" maybe after that I might be able to learn something
best
I am relearning maths for 3d graphics and programming. I have as much maths as high school students.
Is there a possibility that dual numbers could somehow fit in this? I'm thinking especially about dual-quaternions.
I take some exception to the language implying that number systems are discovered rather than created. Is there a reason that I might not be aware of that academics think of number systems, or any other dissection of experience, as having existed as such prior to realization, rather than that because of the way we experience, we create conventions that explain said experience?
Donald Kelly I wholeheartedly agree with you. It is silly to use words like "discovered" rather than words like "constructed" or "invented". My belief is that this sort of language tends to give mathematicians and their work an air of mysterious importance. Mathematics is a language, plain and simple. It is a tool invented by humans to help describe and understand the world around us.
When Cardano invented the Complex numbers or when Hamilton decided to spend years of his life constructing the Quaternions they needed to create some numerical system (with their associated operations) capable of fulfill certain strict properties.
The discovery resides in the fact that, among the infinite possibilities of systems that could be constructed, only a few (or just one) fulfill those properties. Find those systems is not an easy task.
So, in my opinion, the use of the verb "to discover" is quite correct and more precise than "to invent" or "to construct".
Thank you. Your perspective is one I was not able to consider on my own. I like your section about "among the infinite".
My own paradigm is one of infinite paradox. I believe that of the "infinite possibilities" you mention, each possibility has an infinite number of sub possibilities. Thinking of fractals helps make sense of my point. The point being that if we mean infinite when we say infinite, that the number of systems that can (and paradoxically cannot) be constructed to express/model the behavior under observation is also infinite, particularly when we adjust our perspective (of which there are also infinite). I think it would be senseless to argue for infinite systems but not infinite models. The reason is that our ability to model precisely is limited by the number of variables we can account for. We can not measure infinity precisely. Therefore, what is assumed beyond what we can measure is an infinite number of variations of our model. Creating a more precise model slowly but surely ad infinitum is a recursive process imo that always leaves us with an infinite number of solutions and an infinite number of inaccurate models. Maybe by this reasoning each workable model is both inaccurate and a solution?
I think I concede though that there is some whittling of infinity, as weird as that is to consider. I suppose this paradoxically makes sense of both invent and discover?
furthermore, at the first level of systems that could be constructed that fulfill the properties we observe, I would also argue that when the broadest possible breadth of knowledge of the universe is considered, there is still infinite unknown in each specialty (of which there are infinite), and that as such there are an infinite number of systems for each specialty and of the delimitation under observation that are solutions at this scale are also infinite. (where solutions are models that empower us to manipulate and reason about a delimitation). Above this paragraph I speak almost strictly to the infinite subset solutions found within a given solution at the highest level of expression.
Thank you Kenneth. I think Codex makes a very interesting point. Please see if our discussion computes something we haven't yet considered.
Thought I had a thread going here already. Picturing 8 spinning balls?
Like the Stargate, six points + a point of origin, but with an exit...3d lattice?
Susskind at PiTP 2018, part 1 has a clue.
At the center of a Mandala network? 🌻 www.nature.com/articles/srep09082
Re-entry per Edelman: division-> product? czcams.com/video/aTNuZAdzo6k/video.html
Shout out to D-Wave! Check out @dwavesys’s Tweet: twitter.com/dwavesys/status/1103055930848104452?s=09
How do we know that there are only four number systems? Couldn't the number of number systems also be infinite? It almost sounds like the discovery of number systems is happening in half the time of human existence previously. Sort of a Moore's law of numbers? (just babbling, please ignore the man behind the curtain)
I think that she refers to the fact that there are only four types of numbers that have in common the property of being added, subtracted, multiplied and/or divided among them while verifying certain important mathematical properties like Diophantus' identity, Lagrange's identity and Euler's four-square identity.
We owe the mathematical proof to Adolf Hurwitz. Who, in 1898, proved that any algebra that allows these four operations uses one of those four class of numbers.
Here is the work, if you can read German -> www.digizeitschriften.de/dms/img/?PID=GDZPPN002498200
You may also find this article from Wikipedia interesting -> en.wikipedia.org/wiki/Composition_algebra
Thank you for the thoughtful answer. Question: she says that there are "only" four number systems that share those properties. My thought was that is all we are aware of as of today. Wouldn't the possibility exist that there might be other number systems that exist that have not been discovered yet?
Not according to Hurwitz. His work (1898) postulated that any composition algebra that is unital, real (and not-necessarily associative) and has a quadratic form (positive definite) can only happen in 1, 2, 4 or 8 dimensions (making them isomorphic to R, C, H, and O). Later, the proof of this restriction of dimensionality was independently demonstrated by other mathematicians like Radon, Eckmann and Lee.
Nature hasn’t forgotten about the fourth number system (i.e. the Octonions), we will be needing them desperately to look for a more fundamental theory (a true Theory of Quantum Spacetime, if you like) that will eventually unify General Relativity and Quantum Mechanics.
reference ?
A
Nature does not use number systems at all. We use them to describe nature. Nature is not obliged to use any of our mind constructs.