"When one door closes another door opens" - how very true. But first
this one door needs to be tightly closed. Today we will close this door - the
GNS construction door.
The following theorem is an adaptation of Theorem 1.6.6. from William
Arveson, "An Invitation to C*-Algebras", Springer 1976, p. 30.
Originally I thought that I will follow Naimark (cf. Part 45),
but the proof in Naimark's monograph is, as I have realized today,
incomplete. So, here is the Theorem, as it is stated, in Arveson,
together with half of the proof (in this note we will follow Arveson's
notation, which is a good gymnastic for the mind). First the Theorem:
Then the first part of the proof
This first part of the proof should be clear. Since we are in finite
dimension, we can just skip the " π(A)ξ is dense" parts. We see that if
f is a pure state, then there are no non-trivial projections in the
commutant. But if there are no non-trivial projections, there are no
non-trivial operators at all. (Why?). That means, by Shur's lemma, that
the representation is irreducible.
If you have any questions - ask, and I will do my best to answer.
The second part, the converse, we can prove easier than it is done in Arveson, using the theorem from Part 45. Let us assume that f is a mixture of states, f=tf1 + (1-t)f2, where t is in (0,1). Then f1 is f-dominated (Why?). Therefore, using the theorem from Part 45, there exists a positive operator B in the commutant π(A)' such that
f1(a) = (ξ,Bπ(a)ξ).
Supposing that π is irreducible, B would have to be a scalar B=rI. From f1(1)=1, it would then follows that B=I. But then f1=tf, f2=(1-t)f, contrary to the assumption that f is a nontrivial mixture. Thus π must be reducible. QED.
And so we ready now to move to the next issue: the "baby" version of the Tomita-Takesaki theory of "modular automorphisms". Here I will make use of the paper by Roberto Longo "A simple proof of the existence of modular automorphisms ...".
P.S. 16-02-25 18:53
I am having math problem
P.S. 19-02-25 8:41 The next post will be about the baby version of KMS, Tomita-Takeski and Connes-Rovelli. It will be based on the following example taken from Emch's book "Algebraic Methods ..."
After that we will move into the universe of oriented 2-spheres in the 3-sphere (the universe of circles we have already tasted).
Looking for variations of the proof, stumbled upon the book (previously failed to understand there a single word):
ReplyDelete"Algebraic Methods in Statistical Mechanics and Quantum Field Theory" Emch, Gerard G.
https://ikfia.ysn.ru/wp-content/uploads/2018/01/Emh1976ru.pdf
Introduction begins with an ode to algebraic methods, which exert great influence on physics! In particular, they connect physics and mathematics by new, still tighter and deeper relations enriching them both.
The role of GNS-construction is emphasized as a powerful tool, overcoming the limitations of the Fock space. Especially, because it enables constructing a certain Hilbert space tailored for each particular system and problem (instead of using one and the same Fock space).
"Our" theorem stating that pure state corresponds to an irreducible representation is the Theorem 3, Chapter 2, Section 2.
There is much interesting around it. Distracting from the technical details, want to conceive its general meaning.
GNS is a 'good' - reliable and consistent construction giving a certain and one-to-one correspondence between states and representations.
Theorem 3 is the final justification of this fact, showing that pure state corresponds to irreducible representation, which intuitively seems to be the correct order of things.
Indeed, if a state does not dominate over any other state, then it is pure. Representation used to construct this state contains no subspaces. The represention matrix has purely diagonal, and not quasidiagonal, form. No complications, everything is ultimately simple-structured. So nice.
Ok, let us close the door with the sign "GNS-construction", but I would like to keep the keys for a while.
I like the book. It is on my desk - the Russian edition. It has an extended bibliography, 465 references, even my own two papers, and I like the author - we wrote a paper together when my office was next to his. I am happy that you found it. Today it would be somewhat outdated, but it covers more stuff than any other similar book.
ReplyDeleteIncredible. When i first found this book years ago, i could not imagine that will chat easily with the friend of the author )). Surely, he was a great scientist and an extraordinary person, it is seen in every phrase. And the Russian translation is excellent. But at that time the book was as if written in Chinease for me. Now the meaning starts to emerge, owing to you.
DeleteDo you really think that algebraization of quantum mechanics can reveal the secrets of quantum mechanics?
ReplyDeleteNo. But it is a very useful tool. The "secret" is somewhere else.
DeleteНекоторые авторы (Фейнман, Блохинцев и др.) искали его в броуновском блуждании, но спотыкались на комплексных амплитудах вероятности. Так не здесь ли главный секрет квантовой механики?
ReplyDeleteEther - the interface between information and manifestation.
DeleteТочно не здесь. Это общие слова, а нам нужна модель, в которой км появляется естественным образом.
Delete"нам нужна модель, в которой км появляется естественным образом."
DeleteWith this I fully agree. Geometric algebra and topology are other good tools.
А как же динамика, куда без неё? При этом, динамика должна быть хаотической.
DeleteDynamics of "what"? We have dynamics in quantum formalism. Can be chaotic. Can create quantum fractals. I think that "dynamics" should be derived, together with "time", as a secondary concept at the material level.
DeleteЯ согласен с тем, что динамика должна выводится. Но первоначально у вас должна быть такая модель, из которой следовала бы эта динамика. С другой стороны, существует модель случайного блуждания центра метафизического маятника по экваториальной плоскости, динамика которого удовлетворяет уравнению Шредингера.
DeleteBetter would be to get Pauli equation for a spinning particle.
DeleteТогда вам придётся приземлить вашу геометрическую алгебру, то есть рассматривать случайные блуждания шара в 4D.
DeleteЗаметьте, что поверхность шара трёхмерна.
DeleteThe Cartesian Product of a time disk and a 4D space ball is the home of the conformal group which is basically ether and is the symmetry of the Dirac equation and a 4-dim Feynman Checkerboard random walk.
DeleteThe Checkerboard vertices are home to a Clifford algebra the infinite tensor product of which can be used as a Heisenberg Hamiltonian Fock Space universe state. Isn't separable vs non-separable an entanglement thing where the non-separable entangled part has to go to a next universe state together?
These are two completely different concepts: nonseparable state and nonseparable Hilbert spaces The fact that they have the same adjective is unfortunate.
DeleteThanks. For fun I tried getting the Google AI to describe both concepts using similar words and just deleting "state" from "nonseparable Fock space state" goes from the one I know of to the one I didn't.
DeleteWhat is a "Feynman chessboard"? By chance, is this not wandering along rectangular broken lines? As for the “space ball”, this is also a concept that is incomprehensible to me. How much I don't know. However, I know that the “dance” must begin with the choice of metaphysical representation of geometric algebra.
DeleteArk, surprizingly, you answer the questions which i have not formulated yet but only trying to figure out - that is about nonseparable states and nonseparables spaces.
DeleteIn the paper of V.V. we discussed yesterday, the nonseparable states are considered to be states in nonseparable spaces, as far as i can see, pp. 16-17:
"A particle is a superposition of state vectors in a nonseparable Hilbert space H_2s+1⊗H_∞".
and below:
"The particle itself is a nonseparable state in this space".
Nonseparability means that an arbitrary elements cannot be represented as a superposition of basis elements, doesn't it?
While in the same paper V.V. writes (describing analogy between an electron and a qubit):
"The basis vectors can be represented by binary strings of the form |01110010 · · · 1001>. The general normalized vector can be expressed in this basis as a sum
Σ a_i |x>,
where 'a_i' are complex numbers, satisfying Σ|a_i|^2 = 1".
This is definitely a superposition of basic elements, and i do not understand why the space is nonseparable then.
Perhaps, we should ask Vadim?
This last question I can answer. Quoting from
Deletehttps://math.stackexchange.com/questions/323388/the-set-of-all-infinite-binary-sequences
" The set S of all binary sequences (which is a perfectly well-defined object) is uncountable."
Somehow I missed this sentence in V.V. paper. Thanks.
"Ether - the interface between information and manifestation."
DeleteAt the same time you discussed consciousness and information, you also made a remark that "current theory of electromagnetism is really bad because it has nothing to do with transdimensionality".
Roughly 5 years prior to that exchange, Aug 15th 1998, a remark was made that "gravity and magnetism are born of the same source" in context of 'multiple realities' or transdimensionality.
In the PS to "The Science series" note in July 2023, link to YT video of Rogan-Weinstein discussion was posted (not available to non-members; would be extremely grateful if you have an alternative workable link to share), from which a quote was posted, "gravity is the observer through something called a pull-back operation". At the approximately same time, the argument was made that perhaps more than observer, gravity would be the act of observation. Even if that's not completely true, it could be claimed that the source of gravity is observation which would imply, taking remark from Aug 1998 exchange into account, that observation is also the source of magnetism.
In the context of action guν(g) presented in Part 17 "When The Field appear", and comment made to Part 43 about possible role of cosmic EM fields as aether, would the scenario described bellow be mathematically or theoretically feasible?
Information "resides" in the scalar dimension, or the complex scalar plane if we include also the imaginary part in it. Observation as an exchange or flow or stream or current of information gives rise to magnetic field, or in general as seen in Part 17, the EM field, which then propagates also to the "vector" dimensions, and in such a manner informs or creates the 3-dim space as we know it. Through a sort of a Schwinger effect, the EM field populates this newly born 3-dim space with particle-antiparticle pairs and so makes the material world, complementary to the information flow giving birth to magnetic field, manifest.
Asking to know if this idea would be "worthy", from mathematical and theoretical point of view, investing some time and energy into it and developing more into details?
" The set S of all binary sequences (which is a perfectly well-defined object) is uncountable."
DeleteI've never actually tried to think about this and it's somewhat surprising since I've thought about related things. An infinite tensor product of small Clifford algebras would be countable for the number of small Clifford algebras and I've thought that branching between an infinite number of these infinite tensor product states would be uncountable. The small Clifford algebras kind of are responsible for the branches so the elements of the Cl(n) with n as infinity being uncountable kind of makes sense. It's kind of the power set for integers being the reals cardinality-wise.
"In the PS to "The Science series" note in July 2023, link to YT video of Rogan-Weinstein discussion was posted (not available to non-members; would be extremely grateful if you have an alternative workable link to share)"
DeleteGot the alternative link from friend, so you don't have to search for it.
https://youtu.be/h7CJoGKvx3U?t=6594
In general the initial observation with the initial entropy could give rise to Schwinger source particle-antiparticle pairs and the radiation era.
Deletehttps://vixra.org/abs/1807.0372
Each Schwinger Source particle-antiparticle pair should see (with Bohm Potential) the rest of our Universe in the perspective of 8 x 10^53 Monster Symmetry so a Schwinger
Source acting as a Jewel of Indra’s Net of Schwinger Source Bohm Quantum Blockchain Physics (viXra 1801.0086 )
can see / reflect 10^27 x 8 x 10^53 = 8 x 10^80 Other Schwinger Source Jewels of Indra’s Net.
How many Schwinger Sources are in the Indra’s Net of Our Universe ?
Based on gr-qc/0007006 by Paola Zizzi, the Inflation Era of Our Universe ended with Quantum Decoherence when its number of qubits reached 2^64 for Cl(64) = Cl(8)^8 self-reflexivity whereby each Cl(8) 8-Periodicity component corresponded to each basis element of the Cl(8) Vector Space. At the End of Inflation, each of the 2^64 qubits transforms into 2^64 elementary first-generation fermion particle-antiparticle pairs. The resulting 2^64 x 2^64 pairs constitute a Zizzi Quantum Register of order 2^64 x 2^64 = 2^128 . At Reheating time Tn = (n+1) TPlanck the Register has (n+1)^2 qubits so at Reheating Our Universe has (2^128)^2 = 2^256 = 10^77 qubits and since each qubit corresponds to fermion particle-antiparticle pairs that average about 0.66 GeV so the number of particles in our Universe at Reheating is about 10^77 nucleons which, being less than 10^80, can be reflected by Schwinger Source Indra Jewels. The Reheating process raises the energy/temperature at Reheating to Ereh = 10^14 GeV, the geometric mean of the Eplanck = 10^19 GeV and Edecoh = 10^10 GeV. After Reheating, our Universe enters the Radiation-Dominated Era, and, since there is no continuous creation, particle production stops, so the 10^77 nucleon Baryonic Mass of our Universe has been mostly constant since Reheating.
"Somehow I missed this sentence in V.V. paper"
Delete@Ark,
А эту статью вы видели?
Варламов В. В. О квантовании массы // Метафизика. 2023. No 1 (47). С. 115–134. https://www.researchgate.net/publication/369924062_O_KVANTOVANII_MASSY
Кстати, это хорошо ложится на мои представления об элементарных частицах как о пространстве (r,k)-торических узлов Клиффордова тора, на который действует алгебра бикватернионов.
@Igor, i am so glad that you see a topological analogy to the spin-tensor spaces of Varlamov. Probably, you could have a look at this paper https://arxiv.org/pdf/2311.16175, where his mass formula is verified by experimental data for particles, just to be sure that his theory is not only a speculative concept .
Delete@John, thank you for noting the Schwinger source theory. Rather unexpected for me, but this concept seems to be in line with the idea of universal interaction of everything formalized by Ernst Mach. This principle looks quite reasonable but extremely hard to mathematize (as evidenced by the cited paper https://vixra.org/pdf/1807.0372v3.pdf).
DeleteThe theory of views developed by Lee Smolin (i read about it in https://arxiv.org/abs/1805.12468, Section 5) is a less cumbersome construction based on the same principle that the local properties of an object are defined by what is sees around, i.e., by the environment. Ideas go round... )
Ideas do make the rounds. Zizzi's Machian idea in the Tony Smith cited paper was a development of a Freeman Dyson idea. Tony talked to Carlos Castro (Ark knew both in person) about it being a Machian dream and a Dirac-Eddington large number coincidence thing. Physics history can be just as fun as physics math. Smith and Castro were into the Wyler fine structure constant calculation and Wyler was invited to Princeton by Dyson and Smith got a couple Wyler papers from someone who got them from Dyson. I tend to think mass formulas relate to Wyler, will have to look at the Varlamov one.
Delete@Anna, спасибо! Очень интересно.
ReplyDeleteМоя топологические аналогии касаются вопроса о том, каким образом векторное поле замыкается на торе, то есть торические узлы (соответствующие фермионам) это линии тока соответствующего векторного поля.
DeleteIgor, where can i read about these topological aspects you mentioned in the most popular form, if possible?
DeleteАнна, это были мысли вслух. О гравитоподобном замыкании во введении "мат. заметок", а об моей интерпретации массовой формулы Барута в заключении "хаотической динамики". Ссылки на тексты легко найти на моей страничке в "воротах". С телефона неудобно вбивать.
DeleteАнна, я имел в виду стр 8, 45 по ссылке https://www.researchgate.net/publication/322369062_Matematiceskie_zametki_o_prirode_vesej
DeleteIgor, thank you for the reference, i will try to look at it. Although i'm afraid that 7-dim spheres are a bit tough for me...
DeleteВсё упрощается, поскольку на 7-сфере образуется вакуумное слоение с типичным слоем S^{3}\times S^{3}
DeleteИ в результате эволюции радиус одной 3-сферы значительно превосходит радиус второй 3-сферы.
DeleteQuestion to more math oriented and better versed in algebraic stuff;
ReplyDeleteis there an algebra or a math field that combines octonions O and quaternions Q? Any known algebraic structure that's made out of O and Q combined through let's say shared or joined or origin real dimension e0?
Split octonions that Alain mentioned are close, but not exactly there.
The background for asking this question is the exchange during April 22nd 2022, where it was said that O where "better" than Q for being "relevant to wave reading units" (meaning consciousness), and that there are infinite number of algebraic dimensions which are related to 3d space (and possibly time), where this infinite number stems from many iterations of "space and time".
So, the funny idea is that 7 "complex" dimensions of O might represent 7 densities of awareness, while the vector part of Q, which in original Hamilton version are 3 "complex" dimensions, represents 3d space (as it's usually used in known physics). The "shared" real scalar dimension e0 would be sort of information origin or let's say "pure existence", where consciousness would "arrange" itself (in sort of geometry) in O part, and also have a possibility to manifest itself in the material realm through vector part of Q as a kind of a "side space" (or bubble or maybe subspace) to the consciousness space of O.
I think Vadim Varlamov is discussing such things when he mentions Dyson's "threefold way" and develops his "algebraic QM".
DeleteOctonions are usually discussed in the context of Jordan algebras. John Baez wrote about it, and he is really good.
Deletehttps://www.researchgate.net/publication/48198604_Division_Algebras_and_Quantum_Theory
DeleteSurely, Varlamov uses the idea of the Dyson 'threefold way' essentially in his algebraic particle theory. I am citing "Group Theory and Mass Quantization" https://arxiv.org/pdf/2311.16175:
Delete"The physical K-Hilbert space (K = R, C, H) is defined, where the canonical correspondence ω (state) ↔ πω (representation) realizes the Dyson “threefold way” as the symmetry between quaternion, complex, and real representations of group G_f.
With the respective representations we identify the charged (C), neutral (H), and purely neutral (R) states of the spectrum of matter, while the Dyson symmetry provides the dynamic relations between spin, charge, and mass in terms of tensor product".
The particles are classified accordingly (see the mass tables therein): 1) K = C – charged states; 2) K = H – neutral states; 3) K = R – truly neutral states.
"Baez is a very smart and wise man, it is pleasure to read him", Vadim told me (although i know it myself :)). The paper 'Division Algebras and Quantum Theory' is in my near store, i read it but understood not much of it. Have a strong intention to delve deeper.
DeleteThank you for references and supportive comments.
DeleteNow waiting for "time" to enter the stage in T-T theory if understood it correctly and see if the statement that "space and time are interchangeable" can find its place there.
Baez's Octonions paper is also nice.
Deletehttps://arxiv.org/abs/math/0105155
Geoffrey Dixon also likes a unified division algebra structure.
https://arxiv.org/search/hep-th?query=Dixon%2C+G+&searchtype=author&abstracts=show&order=-announced_date_first&size=50
Suddenly i thought that all these cyclic things like the threefold way, Brauer-Wall group, Budenich-Trautman spinorial clock https://www.fuw.edu.pl/~amt/amt2.pdf (and the way Varlamov builds his chessboard-like Weyl diagram of particles https://www.researchgate.net/publication/283072998_SPINORNAA_STRUKTURA_I_PERIODICNOST_ALGEBR_KLIFFORDA) resemble something in the homology theory. Perhaps, the construction of a long sequence with homomorphism connecting a pair of chain complexes of dimensions m and m+1. A short exact sequence of three groups and the mutual relations of three spaces R, C, H -- is there anything in common?
ReplyDeleteIt is all quite exciting, but I am constraining myself to more elementary problems: I want to understand just electromagnetism, gravitation and quantum theory. And, of course, spin. I don't understand spin yet. I know how we deal with it in quantum theory, but I do not understand its true nature. And I do not understand the true nature of the quantum theory. Mathematicians, as rule, and unfortunately, do not understand what it is that needs understanding!
Delete"I want to understand just electromagnetism, gravitation and quantum theory."
DeleteMe too.
BTW the more I learn and the more I know, the less I understand.
Cannot agree that understanding electromagnetism is more elementary problem, not to mention gravitation and QM. My weird analogy concerns quite an abstract and narrow subject. But it has an immediate relation to spinors since they connect spaces of neighboring dimensions - as was vividly shown in that video we discussed some days ago. What seemed to be disappearance and emergence in 2 dimensions appeared to be just circular motion of a cylinder when seen from the perspective of the 3rd dimension. This space of pairs connected via infinity is what we call 'spinor space', as far as i'm beginning to realize.
DeleteWell, I'm kinda less ambitious than you, I'd just like to understand consciousness in the context of multidimensionality or hyperdimensional reality.
DeleteIt somehow seems that EM, QM, spin, gravitation and other mathematical models as products of human mind would naturally come to the surface when we know what that mind is really capable of and where is its real place in the overall structure of things or reality.
Few years ago, used an empty set comparing it to individual consciousness, where the boundary played a crucial role as a place from where the observation happens. It seemed like there's nothing palpable inside and also nothing outside, in the sense of something to put your hands on, but that everything is permeated with the same kind of animating force where only this boundary or envelope or membrane made a distinction between one and another. You might guess that the reception of this comparison was not so great, to put it mildly.
DeleteThen in Ark's post about the prime factories saw that in math the product of elements in an empty set is defined to be equal to 1, which was nice to see that maybe that comparison was not so crazy and out of touch after all.
On another occasion, while discussing consciousness, completely freaked out my ex with a question where did she perceive the "seat" of her consciousness to be. It didn't help at all to tell her where I perceived mine was "seated".
So, the general impression has been that people are not very comfortable talking and thinking about these things. FWIW.
@Anna
Delete"This space of pairs connected via infinity is what we call 'spinor space', as far as i'm beginning to realize."
Where did you see "spinor space"? Was it real or complex? How many dimensions? With sigma or gamma matrices? Can you elaborate on the details?
@Saša "Consciousness" is too abstract a concept for me. I prefer particular rather than general. So, for instance, to understand how telepathy works - that I would also like to know, for sure.
DeleteArk, if i could only answer these questions ... Alas, it is only intuitive wandering. Lounesto explained me and that video confirmed that spinors are somehow the doubling of rotations, which is natural - you can occur at the same position on a circle by going clockwise or counter-clockwise. It sounds simple but there may be the key to deeper understanding. Spinors might be operators of doubling (and annihilating) in some sense.
DeleteAs regards the 'spinor space of pairs', i meant space whose elements are pairs of rotations, like drawing figure of eight, but moving to the left and right simultaneously; to do this, the object has to 'double itself' and 'recover into one' at the same place in the same position, so the visible result of such transform is zero, while some inner changes could occur.
Saša '...mathematical models as products of human mind would naturally come to the surface when we know what that mind is really capable of...'
DeleteYou should definitely meet my good friend, neuroscientist Anna Sverdlik. She is deeply involved in such things, few years ago she wrote a book with a long title: "How our emotions and bodies are vital for abstract thought: perfect mathematics for imperfect minds" https://philpapers.org/rec/SVEHOE
Now she is working on Lee Smolin's concept of views and Karl Friston's ideas. She also follows this Blog closely and is interested in most topics we discuss here.
Thanks for suggestion and socializing offer. Hopefully your friend is not repelled by all the crazy sounding stuff expressed here, some of which are also utterly incorrect in mathematical sense. If Ark agrees, we might start calling these discussions like "meetings at Ark's, in the Open System caffe just next door".
Delete"...is not repelled by all the crazy sounding stuff expressed here"
DeleteDon't worry about it, Anna is professional in crazy stuff, her speciality is psychiatry :)
"Meetings at Ark's" sounds cool, i like the idea!
We will supplement three spatial dimensions with three gauge dimensions. Locally, we will wind the spatial dimensions onto a 3-torus stretched onto a 3-sphere without poles. We will do the same with the gauge dimensions. As a result, the spatial dimensions at the macro level acquire the symmetry of the Euclidean space, and at the micro level, the symmetry of the SU(3) group. Similarly, after winding, the complete symmetry of the gauge space is SU(3). So, locally, our space is a product of two 3-spheres without poles, in which the diameter of one sphere increases and the diameter of the second sphere decreases as the evolution proceeds. Then fermions are tori lying on two spheres at once, and bosons are tori of the gauge sphere.
Delete"21-02-25 17:30 World Scientific publisher wants me to start working on the second edition of my "Quantum Fractals" book. So I started thinking: what possibly I would like to add or improve? Should I agree? I really don't know."
DeleteVery nice news, congrats.
If it's not too much of a distraction from the course you set off for yourself, why not? It might be fun...
"21-02-25 17:30 World Scientific publisher wants me to start working on the second edition of my "Quantum Fractals" book."
ReplyDeleteThis is another proof of your talent to explain complex things in simple words and descriptive pictures.
As regards including some stuff from the Blog into the Book, of course, it is entirely the choice of the author, i would only like to note that the Book is quite voluminous even in the first edition, its further expansion may scare off potential readers. In my opinion, the Blog is worth publishing as a separate book, certainly with color illustrations.
@Saša, Anna
DeleteThanks you for the feedback. In the meantime I am in a state of a phase transition, regrouping, collecting/organizing my thoughts. It takes longer than I intended. I have lost the view of the forest for the trees, and it needed to be fixed.
Dear Ark, phase transition is the most interesting state, expecially the transition between topological phases. This is another wide area of modern study, but it is somehow connected to our talks here. For me, the main magic is in the interplay between continuous and discrete, topology and algebra. The Atiyah-Singer theorem was a great insight of how algebra manages to organize the variety of elusively flowing forms into the rigid order of discrete classes.
DeletePerhaps this chapter of our adventure can end at Part 46? The door is closed, and it's a good place to stop for a rest.
The door is closed, but I can hear the child's voice behind the closed door: "Daddy, so what is spin????"
DeleteThis could become a kind of obsession... I'll tell my neuroscientist friend Yana about the symptoms :)
DeleteYou've done a great work and made significant progress towards the answer, isn't that enough for now for a traveler on the endless path of knowledge?
Of course, I'm not advocating packing one's luggage, but does the Tomita-Takesaki theory have anything to do with spinors?
So far Tomita-Takesaki has nothing of significance to do with spinors. But it deserves a mention for the following reason: Tomita-Takesaki has some vague relation to time, and spinors have some vague relation to time.. Though I am not yet able to draw a genealogy tree. Perhaps it is not even a tree, rather a net of relations.
DeleteWhat kind of problems are you discussing with Yana?
If the origin of spin is due to the internal rotation of the electron's trajectory (as a point on a circle), then why is the electron able to rotate both clockwise and counterclockwise? Maybe, in fact, there is not a circle, but a double circle, and on one circle the wind blows in one direction, and on the other in the other direction.
Delete"But it deserves a mention for the following reason: Tomita-Takesaki has some vague relation to time, and spinors have some vague relation to time.. Though I am not yet able to draw a genealogy tree. Perhaps it is not even a tree, rather a net of relations."
DeleteIf the suggestion to "replace time with consciousness" (February 10th 2018) is going in the right direction, then it seems to make more sense to have a net of relations instead of a tree-like structure. Kind of like live biological neural networks, mapping or reflecting the inter-connections like synapses among neurons in complex nervous systems.
And these relations or relationships, might perhaps be nicely represented by the algebra or algebraic structures.
And then time, as events or things happening, rises as a product or result of observation, that is "recording the information by consciousness equals time" (May 29th 2021).
That's another idea or pranalytical impression to share here; how does it sound to you?
I know it is not a Bourbaki style math (yet), but we'll get there in due time. First stop and sort of a starting point seems to be to reach a definition that then we can work with. If we by then already have a whole landscape opening up in front of us, that's even better, the voyage will be more fun and enjoyable.
Spinors have relation to projecting and to doubling, i.e., to disappearance and birth, expressed as transitions between spaces of adjacent dimensions. I suspect that generally any change can be considered in terms of adjacent dimensions because any change implies passing a boundary and boundary is always 1-dim less than the bulk. In order to have a flow, one needs a gradient. Can't it be a 'gradient of dimension'? Tomita-Takesaki theory deals with a flow, which seems to be something more general than just our time, because algebra and topology operate with spaces of arbitrary dimension. Probably, our time can be considered as a specific case of general changes, a flow taking 3d massive bodies along the 4th dimension.
ReplyDeleteOccasionally found bulky text "An Analysis of the ‘Thermal-Time Concept’ of Connes and Rovelli" https://www.theorie.physik.uni-goettingen.de/forschung2/qft/theses/dipl/Paetz.pdf
With Yana (Anna Sverdlik) we discuss a great number of various topics including the crazy ideas about the Tomita flow i mentioned above. She is interested in unusual mathematical ideas like Vekshenov's 'fundamental rotations' and explores the relations of mathematical concepts to our thinking patterns.
ReplyDeleteCurrently she is deeply engaged into studying correlation between Karl Friston's ideas and Smolin theory of views. I think Yana will be glad if you communicate her directly, i am not a good 'interfacer' (it is her word) in this case.
She is interested in unusual mathematical ideas like Vekshenov's 'fundamental rotations' and explores the relations of mathematical concepts to our thinking patterns.
ReplyDeleteУважаемые Анны, по ссылке
https://mega.rudn.ru/file/22_1241_Метафизика%204%2046-666.pdf
на стр. 37 есть рисунок 3. Это как раз о том, что на восьмёрке поток направлен в одну сторону, а на составляющих окружностях по отдельности в противоположную сторону. В целом, развиваемый автором переход от дискретного к непрерывному интересен. Немного напоминает переход от понятия суммы ряда к аналитическим функциям. Если у вас есть связь с автором, то предложите ему посмотреть мою конструкцию метафизического маятника.
Igor, if you mentioned the 'Metafizika' journal, i guess, you may be interested in the Thursday seminars of Prof. Vladimirov (Chief editor of Metafizika), discussing fundamentals of physics. If you wish, i could send you a link to tomorrows' session.
DeleteSorry for not inviting everyone else, but the seminar is held always in Russian. Articles in Metafizika https://journals.rudn.ru/metaphysics/index/index have English abstracts and author email addresses, so interested readers can contact the authors directly.
@Ark, please let me know if you regard the information above as an inappropriate advertising, i will delete it immediately.
Анна, спасибо за приглашение к участию в семинаре, но я пока воздержусь - сначала надо помочь Арку понять спин.
DeleteReading Vekshenov S.A. “Non-standard” formalism of quantum theory I. He writes:
Delete"We can translate the notion of spinor into a purely algebraic plane and and treat it as an element of the minimal left ideal of a complex Clifford algebra. This gives it a clear algebraic meaning, but takes it even further away from its intuitive content. "
The Author pushes forward the primitive idea of a "rotation" comes directly from our "consciousness". But he does try to explain where our consciousness and our thoughts are coming from. I don't buy it, though I understand that a mathematician may be satisfied with such an approach.
@Igor, "помочь Арку понять спин" is a goal that we all share here. Incidentally, Fig.4 on page 42 of that paper presents quite an original illustration of spinor, belonging to Alexander Efremov, one more researcher of fundamentals.
DeleteI had the same idea of interlocked gears, shown on the image here:
Deletehttps://ark-jadczyk.blogspot.com/2024/10/the-spin-chronicles-part-3-spin-frames.html
The idea is clear, but the devil hides in the details. And these are still muddy.
"The idea is clear, but the devil hides in the details. And these are still muddy."
DeleteА идея с "гироскопом" вам подходит? "Гироскоп" необычный, внешне он похож на сферу, а на самом деле это тор, натянутый на сферу без полярных шапок. Но самая главная его деталь в том, что группа движений этого гироскопа является унитарной группой.
Without math it may be good as a sleeping pill - to put you to sleep. Words, words, words. What is needed is a Bourbaki-style math: definition, lemma, proposition, theorem.
DeleteНадеюсь вас вдохновит тот факт, что произведение OTO', где O,O' элементы группы SO(2), а T - диагональная матрица diag[e^{i\varphi},e^{i(2\pi-\varphi)}] , принадлежит группе SU(2), то есть, специальная унитарная группа порождается движением тора и его вращением на сфере.
DeleteIgor, you are good at creating bed-time stories. Why don't you write a book for children with fairy tales and color pictures? Bourbaki-style mathematics is needed. Not too formal, but nevertheless precise enough for other people to use use it for their needs.
Delete@Ark, i was amazed even at that time when first saw yours interlocked gears in Part 3. But please note that, in contrast to your gears, Efremov's gears are tilted by 90 grad angle to each other. This may be a kink emphasizing their weird nature.
DeleteBy the way, Efremov also prefers rigorous mathematical manner - less words, more deals. Don't remember whether i showed you his paper THE FRACTAL STRUCTURE OF SPACE ENTAILS ORIGINE OF PAULI EQUATION
https://www.researchgate.net/publication/334573118_PAULI_EQUATION_AS_AN_EXCLUSIVE_ALGEBRA_SAVING_CONDITION
It is directly related to spinors and algebras.
@Anna
DeleteThank you! I am so excited! Will start reading right away!
@Anna Started reading Yefremov, and instantly I see that I will need to read his previous papers in order to understand where he is taking his assumptions from? At first sight they look to me somewhat arbitrary and artificial. But I am sure it will be very useful for me. So thanks once more!
DeleteGood evening, everyone!
DeleteI would like to clear up Sergey Vekshenov's idea of rotation. Here, we primarily encounter a semantic issue. This type of rotation has nothing to do with consciousness; rather, Sergey refers to it as mental—that is, purely abstracted fundamental rotation, abstracted also from any environment. As far as I understand, he does not attribute any additional meaning to the term "mental" in this context.
PS.
DeleteAnonymous is me, Anna Sverdlik
"purely abstracted fundamental rotation, abstracted also from any environment."
DeleteHi! Welcome!
If there's no environment, i.e. no reference points or lines, what makes this "abstracted fundamental rotation" really the rotation as usually understood the meaning of that term?
How to distinguish this "rotation" from for example translation or boost or any other type of 'mental motion' for that matter?
If it's just a mental picture for the math not a physically real thing then reference points as well as the rotation could be part of the picture. The spin vector components might be the closest thing to physically real since it gives you a real rotational direction in normal space. The direction is used for probabilities and you then have to think about where in reality probabilities come from but I think of probabilities as related to pre-existing connections between pre-existing universe states kind of path integral-like.
Delete@Anna Sverdlik
DeleteWelsome and thank you very much. So, do I understand it correctly, thoughts are made of rotations, and material objects are made of rotations. What would be the main difference between "thoughts" and "material objects"? Some "density of rotations"? And how light is made of rotations? Are these questions tentatively answered somewhere?
John G: but I think of probabilities as related to pre-existing connections between pre-existing universe states kind of path integral-like.
DeleteIf the probability of a particle's state is the product of the sums of probabilities, where the product of probabilities runs through the time index, and the summation is based on spatial variables, then the probability of a particle's state includes all possible variants of the particle's position in the past. For quantum mechanics, the main thing is that this probability should be complex, and this is achieved solely through additional compactified dimension.
Surely, FR (fundamental rotations) is an abstraction for mathematics, like the notion of group. Group is not a physical object. If you apply group action to something, i.e. represent it, you get a concrete representation. If you apply FR to a wheel, you get a rotating wheel.
DeleteMoreover, FRs are conceived as a basis for new mathematics intended especially to treat the strangeness of quantum mechanics. This approach tries to go beyond the set theory and to deal naturally with entangled and nonseparable entities, which are principally NOT "made of parts".
Returning to the topic of fundamental rotations. Sergey would explain it better himself, but I'll give it a try. This is not about "thoughts," "material objects," "made of," or "density"—whether in quotes or not. It is about a fundamental mathematical abstraction that emphasizes processuality and movement, as opposed to the set-theoretic approach. This also relates to Saša's question:"If there's no environment, i.e., no reference points or lines, what makes this 'abstracted fundamental rotation' truly a rotation, as the term is usually understood?"
DeleteIn response, Sergey offers an analogy with the concept of a number. Numbers, too, exist nowhere, are abstracted from objects and from any environment, and are an absolutely mental construct.
The difference—and here I’m adding my own perspective—is that the neocortex, and consequently explicit human thought, have been evolutionarily fixated on objects (and later on their quantity) because survival depended on grasping and holding onto objects—otherwise, one would starve. However, object should be reached. The very concept of an object emerged from movement and is literally rooted in it—physically, in the brain. Before the neocortex evolved, there were no objects.
No one questions the legitimacy of the concept of number, because in our brain, crippled as it is by evolution, movement must always be tied to something (i.e., to an object—whether a banana, a point, a line, or a number). And that is a problem.
@Anna and Anna
DeleteThank you. For natural numbers we have nine Peano-Dedekind axioms. Do we also have an analogous set of clearly stated axioms for "rotations"?
I am still having problem with "сложность представления вращения, которое нигде не существует кроме сознания."
That means before human beings evolved "rotations" did not exist? If so, how could anything got evolved? Are such questions not allowed for some reason?
The problem is not betting on movement, but rejecting that bet. To generate new things (like the number of revolutions), rotation doesn't have to be an abstract concept. Algebra of complex numbers (quaternions, biquaternions, etc.) It is also generated by rotation, but not in an abstract form, but in the form of linear vector fields describing the rotational motion of matter.
Delete@Anna S
Delete"because survival depended on grasping and holding onto objects—otherwise, one would starve."
So objects and neocortex should exist (even starving?) before?
This resembles the problem of what was the first egg or hen :)))
The vast majority of things in physics are spinors or bivectors so rotations as fundamental works a lot I guess. Scalars and volume forms would be exceptions.
Delete@Annas and Anonymous
DeleteWell, leaving aside that I see the concept of a number rather differently that what's stated in the comment, just a reflection on the statements about movement and object.
First, any kind of movement, no matter how abstract, presuposes some kind of a reference, without it there is no movement, by definition and by experience. Just try to define it or picture it mentally or in an abstract manner to really be a movement, like rotation or reflection or whatever kind of motion.
Second, a reference, mental or abstract or physical, means in fact that there is an object or a subject, same thing at the end of the day, so an object as a reference precedes movement, not the other way around as stated in the comment. The object as a reference can be even something completely abstract, as mind or consciousness, but it is still an object or a subject of action.
FWIW.
P.S. Without a reference, be it inside or outside, there is no way to tell the difference between the movement and the stationary state of being.
Delete@Sasa
Delete"there is no way to tell the difference between the movement and the stationary state of being."
Maybe because they (differences) do not actually exist.Optical illusions are an example of this.
In linear space, rotations are the basic form - so the properties of the screen/wall determine the properties of the shadows - speaking in Plato :))))
3 years ago I bothered Ark excessively in an attempt to show that change is fundamental if not the fundamental concept of all. Quite similar to what's stated about the movement in previous comments here, and in fact movement in its most broadest sense of meaning can be viewed as change.
DeleteSince then I realized that change of itself only does not possess sort of ontological existence, that is it does not come into being in the sense of what change really is, without being applied onto something other than itself. If applied to itself it basically gives "no change", that is sort of its negation. So to really be the actual change, no matter how abstract concept we might talk about, it necessarily needs an object of application, again no matter how abstract that object might be like for example a pure potential.
And quite similar is with the concept of movement, it only becomes actualized when there is a reference of some sort, be it an actual mental object or even just space or time in which this movement is being performed. It simply needs some sort of being to move or in general change.
@Anonimous
DeleteWithout some difference or reference of changing the state of being, how can we tell that rotation or movement is really happening?
From where I stand it seems that no difference directly implies no change, that is no movement as movement, be it rotation or whatever, is by definition some kind of change.
@Sasa
Delete"From where I stand it seems that no difference directly implies no change,"
Then You are excused :)))
Thank you.
DeleteUnfortunately, that does not change the fact that if there is no difference between movement and stationary state of being, as you suggest, then the very meaning of movement, and rotation, is completely lost. Then we might call it "point" or "object" as well, or whatever word we fancy, as the real meaning behind the concept and the word used serms to be rather irrelevant.
@Sasa
Delete"then the very meaning of movement, and rotation, is completely lost"
It's true. But that's what physicists do - by treating entire functions (from t=-00 to +00 ) as constant objects/vectors of a linear Hilbert function space.
For "Shut up and calculate” meaning is not important :)))
That's true, but luckily for us readers at Ark's, we are here because of the meaning and consequently understanding what's behind the Feynman's proclaimed approach.
DeleteInventing new ways to mindlessly do what's already done rather successfully with the ways already in use seems like a rather wasteful way to exert our energy and time, and frankly smells quite a lot like hubris, egotism and narcissism.
I think I speak for most of us here by saying that we are not here for the fame and laurels, but for better and deeper understanding of our reality and closer and connection to truth.
@Sasa
DeleteOK - however, playing with linear objects can sometimes be useful (computationally) but does not lead to an understanding of our non-linear reality even at a mathematical level :)))
@Anonymous
Delete"OK - however, playing with linear objects can sometimes be useful (computationally) "
Not only computationally. Think about the wave equation and it use for our understanding what waves are. On the converse, nonlinear wave equations are useful computationally, but they do not help us much in understanding the waviness.
Both approaches can be useful for our understanding, and one complements the other. And nonlinear objects are often constructed out of linear bricks. To define nonlinearity you need to define and to understand really well linearity first.
In addition, linearity in theoretical constructions is important, since in a nonlinear system it can manifest itself as local linearity. It's not for nothing that a Riemannian manifold is locally Euclidean. The same thing happens with algebras on manifolds - see Section 3.2 (page 49) at the link
Deletehttps://www.researchgate.net/publication/329252706_MATHEMATICAL_NOTES_ON_THE_NATURE_OF_THINGS
@Ark
Delete"To define nonlinearity you need to define and to understand really well linearity first."
And presumably that's why physics hasn't progressed in 100 years? :)))
@Igor Bayak
Delete" nonlinear system it can manifest itself as local linearity."
E.g., at the point of bifurcation or area of chaos? :)))
E.g., at the point of bifurcation or area of chaos? :)))
DeleteI do not dispute that linearity is of little use in global matters.
@Igor Bayak
Delete"I do not dispute that linearity is of little use in global matters."
OK. But in terms of understanding a holistic/global view is probably essential? Unless we are looking for devils (in the details) ? :)))
"But in terms of understanding a holistic/global view is probably essential? Unless we are looking for devils (in the details) ? :)))"
DeleteNaturally, from the point of view of understanding, it is necessary to see the whole picture of the universe. At the same time, we must take the materialistic principle from philosophy, the principle of least action from physics, and mathematicians must choose a suitable space in which moving matter generates all the laws of physics.
@Anonymous
Delete"Unless we are looking for devils (in the details) ?"
Yes, we are indeed. Non-abelian Yang-Mills theory is non-linear, but it is based on the concept of linear connection. The same with general relativity. There is place for nonlinearity, and there is the need of linear basic structures. The devil is in the details, and if you are not paying attention to details, you can easily create chaos out of order.
@Igor Bayak
Delete"At the same time, we must take the materialistic principle from philosophy,...."
Why _must_ ? If we have free will ... :)))
@Ark
Delete"you can easily create chaos out of order"
Yes - and there is no other way to create something new whether in theory or reality :)))
Of our own free will, we must take this from philosophy.
Delete@Igor Bayak
Delete"we must take this from philosophy"
And where did philosophy get this from? :)))
Probably, the primacy of matter in relation to consciousness is inherent in consciousness.
Delete@Igor Bayak
Delete"Probably, the primacy of matter in relation to consciousness"
How high is this probability?
So far it's not really known what consciousness is and there are some who talk about consciousness beyond body/matter?
How high is this probability?
DeleteLet it be a hypothesis.
@Igor Bayak
Delete"Let it be a hypothesis."
OK. We have free will when it comes to hypotheses too - and everyone can put them up as they see fit.
@Saša 'not here for the fame and laurels'
DeleteQuite true. Sergei Vekshenov clearly realizes that his ideas bring him nothing but headache and a lot of misunderstanding; nevertheless he persistently moves in the direction he considers right, as any of us.
Another his basic idea is that we use numbers only to aracterize quantity, though they can also characterize ORDER. Like 'five' and 'the fifth'. This would capture dynamics, and Sergey suggests using this second aspect of numbers on the equal basis with the quantitative one.
To my mind, in traditional mathematics, this double role of numbers is already performed by such objects as spinors, matrices, algebra elements, which can be either an operator or an operand with equal success.
Sergey tries to grasp this duality and formalize it at the fundamental level.
@Anna
DeleteDo you share his view and belief that "numbers exist nowhere"?
First of all, wish you a happy transitioning and that the new phase is more to your liking than the one left behind.
ReplyDeleteIf by chance you find out in the process what a consciousness is and how telepathy works, give all of us here a call, if you will.
And a small word of caution about voices behind tightly closed doors; they do not necessarily belong to whom they appear to represent. FWIW.
Hugs.
Yesterday occasionally recalled about the Dirac’s belt trick, another simple but striking demonstration of strange spinor nature:
ReplyDeletehttps://yandex.ru/video/preview/16172307603127052892
It visualizes the fact that the fundamel group of SO(n) is not trivial and, speaking topologically, there is a non-contractible loop in SO(n), contractible if you go around it twice.
Delete