This is a continuation of "Spin Chronicles Part 43: Feeding GNS doll", where we have treated the GNS construction as if it was a lovely sweet doll. We played with it, and we have discussed three different ways of looking at it. The third method, called there Feed 3 is the most satisfactory, and in the following we will use it for the discussion of "time" emerging from space a'la Connes-Rovelli and Heller-Sasin. Do not worry now, we will get there soon, and all will be clear.
What if there is always a little bit of imaginary space and imaginary time?
Like a boat that floats on the surface, but is always embedded in water.
Perhaps there is a kind of "Archimedes Principle" here?
Why do I connect GNS to spinors? To answer this question let us first
take a look how spinors are defined when we want to connect them to
geometry, so that, for instance, they can be used also in curved
spacetimes. There are several standard ways of doing so. There is no
problem with getting the Clifford algebra and spin group inside it. In
General Relativity it will be the Clifford algebra of the tangent space
at a given point. But then what? One way is: take an irreducible
representation of this algebra. Call vectors of this representation
"spinors". Problem solved. Yes and no. There is unanswered question:
where do I get this representation from? Which one? To ease this
uneasiness we give the representation space a name "Clifford module". We
are dealing now with named objects, we feel better.
The second approach is: introduce "spinor structure". We postulate
its existence together with the covariant 2:1 map from spin frames to
orthonormal frames. But where does it come from? The answer is:
"somehow, does it really matter?". And then it is added: for some
spacetimes there may exist several inequivalent spin structures!
The third approach is: take a minimal left ideal in the Clifford algebra, perhaps take two or four of them - you will fit different spinors to different Fermions. Isn't it nice? Even better, the right action will shuffle the ideals. Left action corresponds to spacetime rotations, right action corresponds to "internal symmetry operations". It can even fit the Standard Model. For all practical purposes it can even work, but the question remains: which ideal and why this and not another one? I am not completely happy with this. We are onto something, but what is it, this "something"?
GNS construction associates representations to "states". If we follow
this philosophy, spacetime (or just "space") at a given point may be in
some "state". It fits my intuition. Once We have "state", we have the
associated representation. Pure states lead to irreducible
representations. Mixed states lead to reducible ones (we will discuss
this issue in the next post). This opens my memory bank. Pure states are
"extreme points" of the convex set of all states. They are at the
boundary. It is hard to get dynamically exactly to the boundary. Rather,
when we experimentally attempt to get exactly pure state, we will
obtain "almost pure" state, never "exactly pure". Pure states is an
extreme idealization. No state is in the Nature is exactly pure.
This opens a whole new perspective with possible physical consequences. I
like it. It leads to another idea: we think that our space is "real",
time is "real". Imaginary space and imaginary time are simply
mathematical tricks. But are they? What if "impurities" ("defects") are
important? What if there is always a little bit of imaginary space and
imaginary time? Like a boat that floats on the surface, but is always
embedded in water. Perhaps there is a kind of "Archimedes Principle"
here? In recent years Peter Woit develops a similar vision.
That is just "talk". Next post will be just "math". We will discuss "states dominated by other states", representations, cyclic vectors and irreducibility.
P.S. 07-02-25 8:42 Reading the News:
"In other words, the new U.S. administration is facing a growing avalanche of problems and the necessity of simultaneously solving the Riemann hypothesis, the Collatz conjecture, the Hodge conjecture, the Yang-Mills theory, the Navier-Stokes equation, the Birch and Swinnerton-Dyer conjecture, and the Goldbach problem—all while standing on their head on a rubber ball and performing loops on a roller coaster."
"Plane Carrying 10 People Goes Missing in AlaskaThe Cessna caravan was traveling from Alaska’s Unalakleet to Nome.:(...) “Ground crews have covered ground all along the coast from Nome to Topkok,” the fire department stated. The plane’s “exact location is still unknown. We continue to expand search efforts to as many avenues as possible until the plane is located.”
P.S. 09-02-25 19:21 I didn't finish my new post. Probably will finish tomorrow.
"What if "impurities" ("defects") are important?"
ReplyDeleteThey usually are, and what usually happens is that they are "walled" into a sort of a closed domains, very similar to what happens in plasmas, magnetic field forms kind of a wall around an impurity, shadowing or screening its "excessive" charge over small distances around it, so not to affect and disrupt the rest of the collective in resonance.
In other words, someone basically right next to such a closed "magnetically" isolated domain would not even know there's a whole another realm in their vicinity. That's how efficient these screenings can be.
This kind of analogy brings forth an intriguing question;
Deletewhat would assume the role of the magnetic field as a realm border in plasmas, or in the context of the post, what would play the role of the surface on which boat floats?
Could it be Hamilton's forth scalar dimension?
Perhaps as 'time', imaginary as it would seem from one side, and real from the other side of the boundary?
"what would play the role of the surface on which boat floats?
DeleteCould it be Hamilton's forth scalar dimension?"
Border is always 1 dim less than the bulk that it circumvents. In view of this, 3d world ought to be a boundary of some 4d space. Then, it is natural to presume that most of the 3d world floats in the real subspace of 4d, but a little bit of it dips into the imaginary one.
The same can be true for the living beings. Most organic molecules are of left chirality, but some are of the right one. "Breaking of symmetry is the locomotive of evolution", as Freeman Dyson said, in my free paraphrase.
Thanks.
DeleteDo you entertain the hypothesis of entirely mindless random evolution ala Darwin?
By the way, as Alain explained in comments to previous post, and Ark noted at the beginning of this series, our pseudoscalar I as a trivector I=e1e2e3 has a dimension of a volume (oriented), so it could play the role of a boundary in a 4-dim scenario and maybe also assume the role of time, imaginary one at least.
DeleteJust dropping a thought that crossed my mind while thinking about what you wrote. FWIW.
Classically you can think of a bounded complex domain with the conformal group as its symmetry group. You would have our 4-dim real spacetime as the boundary and complex spacetime in the interior that you could in certain situations take shortcuts through. Hawking did this for his imaginary time to try to get around God but this whole structure could of course still come from "God".
DeleteThe conformal second time giving imaginary time is also a second time in the Kaluza Klein sense. In the Clifford algebra of a Clifford algebra sense I could see e1e2e3 as time and the time-like part of a volume form after dimensional reduction/symmetry breaking back down to 4-dim. Spinors might go from real to quaternion due to signatures of the Clifford algebras involved. Symmetry breaking makes things a bit wild since different people have different ideas about what is legal. When trying to fit someone's Lie group idea into to someone else's large Clifford algebra idea I have no idea how legal I am.
The surface on which the boat floats? The Dirac sea? I don't know if that speaks to a little bit of imaginary time or space, but, if it's filled by negative energy states and things like particles going backward in time then that may be.
Delete--
At the risk of being a scientific outlaw, I propose that imaginary be renamed to "irreducible" or "inaccessible" (save for by iteration). The square root of any positive or negative number, not just -1, ought to work for complex numbers, as a unit that cannot be factored or operated on, save for by exponentiation. All I think this would do is transform it to have a possibility for spiraling rotations instead of circles on the complex plane.
Then what's at issue is choosing with what numbers to work. I think this opens up some possibilities, and solves some problems. Does anyone have an opinion on this idea? This I think is a move from a specific case to the more general--does this create any problems, or utility?
The square root of any positive or negative PRIME* number... trying to perform that operation generates two transcendental halves you'll never be able to compute.
Delete"No one state is in the Nature is exactly pure."
ReplyDelete-> one "is" seems to be excessive
"Imaginary space and imaginary space are simply mathematical tricks."
-> one "space" is probably "time"
Thanks. I was thinking about third density or the material realm as a surface. Something in this direction. Impurities may be "widows" if the screening around them is for some reason weaker than usual.
DeleteIn that picture, material realm would be just the surface or let's say everything above the surface plus that "upper" side of the surface?
DeleteMy initial idea was that impurities or closed domains would be representatives of different "islands" of the material realm, or colloquially said different dimensions of material existence embedded into so called higher reality of the "space" of consciousness, or something to that extent in the lack of better terminology. Kind of different ideals as subsets or subspaces in the overall space or set of the whole algebra.
Windows would be "points" where the screening is weak or fails for some reason, possibly "connecting" one material island/realm as an impurity or domain with another one through the medium of consciousness or information that both islands are immersed into, so to say.
But as with any idea it needs a bit of preparation and proper cooking to be at least digestible if not really tasty for wider consumption.
Interestingly, the very similar considerations about surface or, generally, boundaries, i use when think about the origin of life. Protozoa cell is a closed capsule formed in a very thin layer at the ocean surface, its membrane separating two different worlds, with the entropy growing outside and self-organization occurring inside.
DeleteThere is something in common in our way of thinking...:)
A great perspective! Want to say a lot, but will formulate it after my yoga lesson. In the deepest thoughts the oldest parts of brain are involved, and even the body as a whole.
ReplyDeleteHave been thinking about what you wrote in the previous post about understanding consciousness, information and measurement.
ReplyDeleteMeasurement in its broad sense can be viewed as an observation, which presuposes an observer, that in its broad sense includes consciousness of some sort. Our other sources indicated that information arranged by truth becomes consciousness, which puts information as a foundational element of that set of concepts.
Leaving aside the usual approach to information like information theory or physical notion of entropy, went to see the etymology of that word, and arrived at its latin "components" which in short give the meaning like that which gives shape or form, or in its broad sense that which creates or simply "creator". And the "creator" meaning includes all sorts of things, it creates the existence of something, as in material and ethereal or abstract like an idea, and it also creates the existence of relations and relationships between or among these "created" things.
In a very real sense it can be said that information is a fibre of the whole Creation, the prima materia of the Existence. It does not need space or time to reside in, it creates space and time if needed. It is maybe a bit strange to comprehend because we are used to have something as a sort of a stage where things dwell in or on, and information does not need such a stage as it is the "material" of its own out of which everything is created, including various "stages".
Well, don't know if this is helpful in any way or not for better understanding, just thought to share it here. Maybe it can useful for something along the road in our quest here. FWIW.
'we are used to have something as a sort of a stage where things dwell in or on'
DeleteNot all of us are used to such a stage. Relationalism in more or less radical form has always lived side by side with the mainstream physics. Precedency of relations over objects, secondary nature of space and time, holism are the main relational principles. One of the most prominent representatives of modern relationalism is Lee Smolin, probably you know his works.
I don't like the word 'information', it is worn out yet not properly defined. You suggest taking 'information as a foundational element', this sounds close to the Thomas Goernitz ideas of AQIs (i mentioned him already). He is a very interesting person and eagerly communicates with adherents, so i think you can find understanding if you contact him.
@Anna
Delete"I don't like the word 'information', it is worn out yet not properly defined. You suggest taking 'information as a foundational element', "
I don't like the word ' foundational element', it is worn out yet not properly defined. You suggest taking 'foundational element as a foundational element'? :)))
Achimedes -our inspirer- expressed this "Give me a place to stand, and a lever long enough, and I will move the world"
Forgot to add that the world but not the whole world - because without this "place to stand"
He did not know the fixed point theorem? :)))
However, the genetic code is the foundation of biology, logos has a similar role in the field of intellect. Both are informational in nature :))))
With the comment above I've also clarified and explained, to myself primarily, some of the ideas and in part research done by Pierre Lescaudron, friend who prematurely, from our point of view, left us for greener pastures about a year ago.
DeletePierre used the term Information Field for describing the texture of the Universe and Creation, which at some point I was nitpicking about. At the end of the day, it basically comes down to the same things, with only maybe tiny differences in details. And as Ark points out repeatedly, the devil often hides in those details.
One of the Pierre's ideas and hypotheses that particularly rubbed me the wrong way was that we and other living beings connect to the Information Field by exchanging biophotons with it. With the view presented in the comment above, there would be no need for that, as we and everything else in the Creation, not just living beings, would be made of information or the Information Field to use Pierre's term.
I personally would not use the term field in relation to information as a fundamental structural fibre of the Creation, as it already presupposes an arrangement of some kind and relationships among the information/fibres making it, if viewed from the mathematical point of view, or a space where it resides if taken from scientific or physical perspective of a field. As explained in the comment, there would be no need for that also, and if the information would be "arranged" in some kind of a mathematical or algebraic structure like a field, which would presumably be the truth, then this "field" would already become consciousness, if we adopt the hypothesis that "information arranged by truth becomes consciousness".
In other words, the term Information Field would be kind of a misnomer for the Universal or Divine Consciousness or simply God, sort of "hiding" the fact that the Creation or Universe is conscious, and made of information which do not necessarily need to be arranged into just a field, but maybe even in some sort of a complex geometric algebra type of an arrangement.
Well, maybe I'm nitpicking again, but that's just my way of explaining things, primarily to myself, in an attempt to gain a little bit better understanding of the Reality we live in.
If interested, chapters from "Earth Changes and the Human Cosmic Connection", the book where Pierre in popular layman's manner discussed in part these topics mentioned above, were also translated into Russian, for example among other chapters there's the "Information theory and consciousness" one at:
https://ru.sott.net/article/1484-zemnye-izmeneniya-i-vzaimosvyaz-mezhdu-chelovekom-i-kosmosom-chast-35-teoriya-informatsii-i-chelovecheskoe-soznanie
If gravity is in some sense everything and 7th density is the light at the end of the tunnel then "field" applying everywhere in some sense seems possible. Bosons in general take you from one universe state to another and even for God there's probably the idea of going from one conscious universe state to another even if in the unbroken symmetry for God there aren't bosons as we know them. The 2^n dimensions of Clifford algebra are quite information-like.
Delete@Saša "Information arranged by truth becomes consciousness"
Delete"Creation or Universe is conscious"
These ideas are in remarkable resonance with Indian philosophy. Mircea Eliade, a historian of religion, compared the Universe with the Brahman cinema-house, where He produces the world on the screen of our consciousness.
Indians always respected mathematics and probably algebraic geometry as well, since the 8-dimensionality is the sacred number there, encountered every time when you touch Indian culture. Chess board is the most known example.
Physicists who are not adhere to the 'shut up and count' rule always felt this resemblance between physics and Indian philosophy. An excellent book to get the idea is the 'Tao of Physics' by Fritjof Capra for example.
Thanks for the recommendation, read the Tao during my college days.
DeleteThe ideas expressed before also have a touch of Sufism of Ibn al'Arabi, esoteric orthodox Christianity of B. Mouravieff, sort of a toltekian philosophy presented in works of C. Castaneda, and many other intelligent design let's say proponents. They are particularly informed by the Cassiopaean Experiment and modern revival of christianity in the form of Paleochristianity and Fellowship of the Cosmic Mind.
FWIW.
"read the Tao during my college days".
DeleteHow lucky you are. I was going to these ideas all by myself, alone, painfully and for many years, and only when was on the edge of despair found the book of Capra. Sent him a letter and got approval and encouragement for further reseach.
Well, other circumstances in my life at that time were very unsupportive of the ideas presented on this blog in general, which lead to being enured almost exclusively into basically pure materialistic paradigm of existence for almost completely selfish purposes as a meaning and motivation for being and living on this planet Earth. Kind of like materialistic service to self only way of life. So don't consider myself exactly as lucky because of being exposed to the Capra's Tao at that time, from my perspective it might have been even detrimental due to these other surrounding circumstances. If pressed to attribute luck to some events, finding the work of and meeting Laura and Ark and the rest of the family would surely be at the very top of that list.
Delete"I don't like the word 'information', it is worn out yet not properly defined."
DeleteSo, let's define it, find a suitable representation, and work our way to describing and representing consciousness, and even measurement, what Ark has invited us to do in the sense of better understanding these concepts in the previous blog post. Shall we? Maybe by doing that we can even help Ark.
For starters, I'm completely fine with what have written in previous comments about the information as a fabric of everything, that is as a creative fibre of Creation, in the broadest sense and all possible meanings.
Any narrowing of the "definition":
'that which gives shape, that which gives form, that which gives existence or simply that which creates',
seems to leave something out or to reduce the scope in the sense that something is "preceding" the information in the overall existence. But, of course, I might also be completely wrong in this reasoning.
What do you, or Bjab or other readers, or Ark as our host here, say about this "starting proposition"?
I wouldn't exactly call it an axiom, but if somebody likes that term better, it's fine with me as well.
To me this definition of information reads reads like the Christian mystical logos
DeleteIs that a bad thing or we could work with this definition?
DeleteIf that definition is acceptable, let's find suitable representation.
One possibility is to use the point as a representative. It's said that point is not-a-thing, which seems to perfectly suit the purpose for representing information as an atom or foundation of everything. It is something, but not exactly a thing to get hands on it, just like an information. Usually the "bit" is considered to be an unit of information, at that's fine from the point of view of quantifying it, but it's not particularly useful for representing it, especially from the perspective of geometry and arrangement if we want to go with the hypothesis that "information arranged by truth becomes consciousness".
A point has mathematical dimension 0, in a sense no additional meaning or information except of that of its existence. It just is, nothing else. There would be no need for anything else for it to exist and be, no need for space or time, in fact both space and time are made of points, and in a broad sense it can be said that everything is made of points. Seems to perfectly suit the purpose of representing information out of which everything is made, as the above definition says.
So, would these definition and representation be something we can work with in mathematical and geometrical manner?
Re: "P.S. 07-02-25 8:42 Reading the ..."
ReplyDeleteWhat a beautiful example of vectorized graphomania.
At the conversation level.
ReplyDeleteThe states of a quantum particle can be perceived as a probabilistic trap into which it falls, wandering through the vastness of space-time. The spinor state is a feature of a probability trap due to the true dimension of space. The vibrations of a metaphysical pendulum can serve as an excellent example to illustrate my thoughts.
https://www.researchgate.net/publication/325226826_Chaotic_dynamics_of_an_electron
@Igor, "quantum particle can be perceived as a probabilistic trap into which it falls, wandering through the vastness of space-time. The spinor state is a feature of a probability trap due to the true dimension of space".
DeleteAn interesting suggestion. But what if there is no space-time with its 3+1 dimensions at the level of quantum particles?The number of adherents of emerging space and time increasingly grows... A good description of levels of spacetime emergence is, e.g., here https://arxiv.org/abs/1807.04875
"But what if there is no space-time with its 3+1 dimensions at the level of quantum particles?"
DeleteAnd so it is. For the manifestation of quantum properties of a particle, the geometry of 4-dimensional Euclidean space is more essential. It is in it that the particle's trajectory has a spiral shape, which leads an observer in 3-dimensional Euclidean space to the need to use complex probability amplitudes and spinors. And the Minkowski metric is a higher-order creation.
Anna, thank you, of course, for the review article, but when you've had a clear picture of the universe in your head for a long time, you can only regret that other people don't see it.
Delete@Anna "A good description of levels of spacetime emergence is, e.g., here https://arxiv.org/abs/1807.04875"
DeleteThanks for the link. Many interesting references there. I like the term "geometrogenesis". Dodn't know this term before.
@Igor
Delete"For the manifestation of quantum properties of a particle, the geometry of 4-dimensional Euclidean space is more essential."
That is also Peter Woit Idea in his "Euclidean Twistor Unification" paper
https://arxiv.org/abs/2104.05099
Anna,
DeleteThanks for the link to the article. It's too abstract for me. I prefer concrete things. For example, the article talks about projective spaces, but there is no geometric representation of these spaces. In my case, the projective plane is represented as a torus stretched over a sphere without polar caps, and the projective line is represented as a circle folded in the form of an eight and stretched over a circle of half the radius.
Sorry, I got the recipient of the previous post mixed up. It turns out Arkadiusz recommended Peter's article to me.
Delete@Igor
Delete"In my case, the projective plane is represented as a torus stretched over a sphere without polar caps, and the projective line is represented as a circle folded in the form of an eight and stretched over a circle of half the radius."
We kno what a projective space is. What we do not know is: a precise mathematical relation of the definition to your representation. Can you you provide a thorough proof that your projective space is the same as the projective space from the definition? I would really appreciate such an explanation! And do not say "I do not have time for this". This would be great service to all humanity!
Аркадиуш, давайте без пафоса. Насчёт восьмёрки вроде бы не должно быть вопросов. Возьмите резинку и скрутите раками в 8, сложите в 0 и пальцами расщепите две соседние точки. После этих манипуляций вы сразу же увидите, что ваши руки держат противоположные точки исходной окружности. Что касается проективной плоскости, то тут действуем по аналогии. Берём тор (произведение двух окружностей) и превращаем каждую из его задающих окружностей в проективную прямую. Визуализация проективной плоскости это уже второй вопрос. В моём случае это тор, натянутый на сферу без полярных шапок. На мой взгляд, самое интересное это появление комплексных амплитуд вероятностей.
DeleteI cannot verify it, but I think it may be отличная работа... So, I can доверяй, but cannot проверяй.
DeleteIncidentally, I'd never heard of Parson's magneton (per your first reference) a.k.a. toroidal ring model, so, I thank you for that.
However, there IS a "question about the 8": don't you intend rather to have a trefoil? You say you, "...the hierarchy of leptons could be
explained by a bunch of trefoils", in the paper. Should one spread the crossover at the middle of the 8 into a trefoil?
I apologize doubly: once for anonymous posting, as I cannot sign in here, and once for double posting; since by posting anonymously I cannot edit.
DeleteIf one twists twice over the middle to make their 8, of course then a trefoilis is there after spreading.
The trefoil is also a (2,3)-toric node, so, probably, we can just talk about the rational winding of the torus. Actually, I'm still "licking" this paper. I'm afraid I'll get tired of it soon, or I'll make a hole in it.
DeleteI had, "...the unitary group of isometries of a torus on a sphere tells us that the nature of the electron is somehow related to the geometry of the sphere’s winding", then "...the electron mass - with the pseudo-Euclidean length of the closed winding of the sphere has the shape of a (2,3)-torus knot" just before prompt me to see the (2,3)-torus.
Delete++
I don't think you've "got it licked" (all figured out) yet.
Why a metaphysical penduluum? Is that supposed to be like an harmonic oscillator? Can toroidal windings map all the electron orbitals? What is driving the chaotic dynamics?
I don't know if you're "taking the piss" (making a joke), though, either. The quotes here are around English slang phrases or colloquialisms (with interpretation); I must do better writing.
- -
I hope the following repost sticks and has relevance here as it would've in part 43. I posted it there twice and once here (now with fair use disclaimer) but it does not show on either page (and so an intended segue is missing in my writing).
EXCERPT COVERED UNDER FAIR USE AS EDUCATIONAL MATERIAL, names removed
from: https://www.math.columbia.edu/~woit/wordpress/?p=13152
P.O. says:
"I expect you already know this, P, but there is also a classical version of spin-1/2. It’s a little subtler than integer angular momentum, but works just as well. This a charge on a sphere in the presence of a static monopole. This yields a Wess-Zumino action for the unit vector (or element of S^2), and upon quantization, this vector’s components become the Pauli matrices. A more general model (due to Tamm) is a charge anywhere in R^3, in the presence of a static monopole at the origin (which has wave functions which change sign under a rotation).
There is even a nice relativistic generalization. This is a slightly more complicated Wess-Zumino term, depending on two unit vectors in 𝑅4, one of which is parallel to the four velocity, and the other which is orthogonal to the first (so the configuration space is 𝑆3×𝑆3/𝑈(1). This gives Dirac fermions after quantization (in Euclidean space. To get to Minkowski space, the 𝑆3’s must be replaced by hyperboloids). One vector is the velocity divided by its norm. When quantized, this vector’s components become the Dirac gamma matrices. The other unit vector’s components become the rho matrices."
P.W. says:
"Thanks P,
That’s a interesting physical variant of the general idea of taking 𝑆2 as phase space and quantizing, but using the monopole to give the quantization line bundle.
I realized there’s another way to think of this, to argue that the spin-1/2 degree of freedom is what you get when you look not at classical mechanics, but at pseudo-classical mechanics (your phase space variables are anti-commuting) in the simplest non-trivial (d=3) case. I wrote up the details of this in chapter 30 and 31 of the book on QM (http://www.math.columbia.edu/~woit/QMbook/qmbook.pdf). "
C.R.says:
August 7, 2020 at 1:55 pm
"Hi P,
Are you sure that geometric quantization isn’t the best way to understand the spinor degree of freedom? It seems like a good way to go, based on a lower-dimensional example. Quantize space with a honeycomb lattice, and spin 1/2 pops right out. See this paper in Physical Review Letters: https://arxiv.org/abs/1003.3715."
• P.W.it says:
"C.R.,
There are lots of ways to get a spinor degree of freedom out of a much more complicated structure. But spinors are really an extremely simple story about two by two matrices. What I’d like to advertise is this simplicity. If you want the simplest possible way to explain where spinors come from, twistor geometry is very compelling, giving spinors tautologically (a point in space time is exactly the spinor space)."
Tautological bundle is nice. But how to get the REAL compactified Minkowski space seems to me to be a problem.
Delete@Ark,
DeleteCollapse all pseudo-Euclidean planes of the Minkowski space so that all isotropic lines turn into projective lines (that is, X-->8-->0) and it is automatically compactified.
"Why a metaphysical penduluum? Is that supposed to be like an harmonic oscillator? Can toroidal windings map all the electron orbitals? What is driving the chaotic dynamics?"
DeleteA harmonic oscillator assumes a certain force of return, and here we consider the free movement of a pendulum with evolutionary attenuation. As for the chaotic dynamics of an electron, it is also a free movement in the form of a random walk through a compactified Minkowski space. And to connect the calibration interactions, it is necessary to extend the model to a doublet of Minkowski spaces.
Small side remark.
ReplyDeleteAfter checking the details, it turned out that the choice f=p=(1+n)/2 is not a valid one, because then f(1)=(p,1)=(p,1p)=1/2 which obviously does not satisfy the condition for f being a state, i.e. f(1)=(f*,1)=(f½,1f½)=1.
The closest valid choice for f that gives f(1)=1 would be f=u1=(1+n) for u1 from exercises in Part 30, and probably also including the u2=iu2×n part, i.e. the "real" left ideal.
In case of f=u1, the positive square root of f or of u1 would be f½=u1½=(1+n)/√2 that would satisfy the condition f(1)=(u1½,1u1½)=1.
FWIW.
You are right. We had essentially the same problem in:
Deletehttps://ark-jadczyk.blogspot.com/2025/01/in-part-40-we-used-gns-construction.html?showComment=1738089021181#c8543987329607165897
In light of that comment, if we choose f=u1, then the density matrix rho becomes in fact p, our non-trivial hermitian idempotent, rho=f/2=½u1=(1+n)/2, which seems very nice and convenient.
DeleteIf we also include u2 then we maybe have a material for a cyclic vector from u1½ as we have the ideal in A, right?
Our u2=iu2×n is a "funny little fellow" as it's a (complex) vector that's perpendicular to itself, meaning that its scalar product and so also its norm is 0,
Delete|u2||^2 = ScalarPart[(u2*u2)] = -i(u2×n)·u2 = 0.
In the comment to Part 40 it was said,
"In this case norm zero can be considered as not giving us anything useful.",
but in this case it seems it would be needed so to have the "complete" left ideal in A, and thus a cyclic vector
Ω = (1+n)/√2 + u2,
which would again be a left ideal in A, as described in Part 32.
At least that's my current understanding of the situation.
What's bothering me more at the moment is that our choice of u2 from Part 30, u2 = e1 - ie2, does not give the norm ||u2||^2=0, but
||u2||^2 = ||e1 - ie2||^2 = ScalarPart[(e1 - ie2)*(e1 - ie2)] = ScalarPart[(e1 + ie2)(e1 - ie2)] = 2,
while it's square (e1 - ie2)(e1 - ie2)=0.
Can anybody please help me find the error in the above and reconcile these two seemingly discrepant implications/things?
@Saša
DeleteScalarPart[(u2*u2)] = -i(u2×n)·u2 ->
ScalarPart[(u2*u2)] = -i(u2*×n)·u2
@Saša "while it's square (e1 - ie2)(e1 - ie2)=0"
DeleteI can only join to the question. From student years, felt uncomfortable about the fact that the square of a complex number is not equal to the square of its modulus. So inconvenient!
@Anna
DeleteHere we have an extra reason for surprise due to the fact that e1 and e2 anticommute. That is how these pesky nilpotents are born. On the other hand, I speculate, without them there would be no light and no nothing :)
@Saša,
Deletei'am lost in halves. In Part 32, you showed that E1=(1+n) and E2 = u2 span the left ideal of A. But now you propose to take a "complete" cyclic vector in the form:
Ω = (1+n)/√2 + u2
i.e. as a sum of E1=(1+n)/√2 and E2 = u2.
Which variant is right?
Thanks Bjab, so the square of u2 would be 0, but not the norm, just like for our choice u2 = (e1-ie2).
DeleteAnna, I think what's presented in Part 32 is the correct version of things, in the above comment here I made a hasty and incorrect deduction, that is jumped to conclusion based on not exactly understanding what's going on.
If you all keep sharpening this spinor construction I think you may poke a hole in something.
Delete@Ark "without them there would be no light and no nothing :)"
DeleteThis may be in line with the de Broglie's fusion theory (as Vadim told me recently) according to which fermions of spin 1/2 are fundamental, and bosons are their tensor products of even degree. According to de Broglie, light (photon) is a combination of two neutrinos. This is the famous (and forgotten) neutrino theory of light by de Broglie, which, by the way, no one has ever been able to refute.
"I think you may poke a hole in something".
ReplyDeleteThis week, our Skolkovo Science Park is hosting an exhibition of works by deaf-blind people. I can't imagine how, without sight and hearing, they act out plays by Shakespeare and Chekhov, write poetry, discuss scientific problems. It reminded me of our attempts, when, like blind kittens, we try to penetrate dimensions that are physically inaccessible to us, while some higher beings with super-sight and super-hearing watch us indifferently...)
@ARK Dear Ark (and others), can you have a look at this little text ? I find things in algebra ... funny.
ReplyDeletehttp://rencontres-science-conscience.com/documents/Fermionic%20Oscillator.pdf
Sure, it's a draft ;=)
ReplyDeleteAlain, Finite fields is not my domain.
Delete'Sure, it's a draft ;=)'
Delete@Alain, it should be YOURs draft, do i guess right?
Thank you for reminding us about Norbert J. Wildberger. He is also one of my favorite virtual teachers. Some time ago i was astonished of how easily he derived Maxwell's equations right from 3d space geometry. Then watched his excellent course of algebraic geometry and topology for students on YouTube...
The idea of dihedron algebra looks curious. But i cannot see properly - where do you want to get with it?
@Anna Thank you very much ... Please don't neglect my little jobs.
DeleteI'm away for a few days, I don't have much time to explain.
I want just to construct a very "simple" chiral *-algebra from stratch. I am convinced that God is constructing the World as he construct the Fibonacci sequence. 0 1 1 2 3 and so on...
I'm convinced, as Wildberger, that Real Number is not the good Field ! (there is to much problems with infinity ... and infinitesimal)
We have to construct from a very simple Field. F2 {0 1} seems the simplest to me. So I'm happy to see how the Peter Woit's chapter 27 is close of this idea (using just 0 and 1 in 2x2 matrix).
I'm convinced that all matter is made of electrons and positrons. So to explain matter (fermionic) I need only to describe electron or positron. That's all ;=)
I'm convinced that Universe is very close to a Quantum Computer with a 3-qbit system (as the Universal Toffoli Gate). Our 3D of space correspond to this 3-qbits gate (8d matrix).
Dihedrons are just Split-Octonions, but over a finite field or not finite. I'm surprised Wildberger didn't see it. Dihedrons are algebra for 2D surfaces (manifold). A 2x2 matrix act on a 2D vector.
I'm happy to work in my "algebra" only with split-numbers, because chirality (mirror symmetry) is very important. We are more or less on the surface of this "mirror"... ;=))
To finish, the "infinitesimals" of dihedrons is the simple matrix ((0 1) (0 0)) and (( 0 0)(1 0)) one right and one left.
Wildberger knows how it is important to construct algebraically all the notion of derivative, and so on...
With this, exp(x) is just equal to 1 + x x can be a number, or a multivector...
We are very close to the solution ;=))) Trust me ;=)