From Manjit Kumar, Quantum , Einstein, Bohr and the Great Debate About the Nature of Reality-Icon Books (2008):
Max Born, 38, a key figure in the future development of quantum physics, had arrived in the small university town from Frankfurt just six months before Pauli. Growing up in Breslau, capital of the then Prussian province of Silesia, it was mathematics, not physics, that attracted Born. His father, like Pauli’s, was a highly cultured medical man and academic. A professor of embryology, Gustav Born advised his son not to specialise too early once he enrolled at Breslau University. Dutifully, Born settled on astronomy and mathematics only after having attended courses in physics, chemistry, zoology, philosophy and logic. His studies, including time at the universities of Heidelberg and Zurich, ended in 1906 with a doctorate in mathematics from Göttingen.
I was walking the same stairs in the same building that Max Born was walking, when studying theoretical physics in Breslau ( after WWII renamed Wrocław ).
And then I was walking the same stairs during my year long stay in Gottingen.
So I feel some affinity with Max Born. And surely I feel lot of affinity with mathematics.
"The belief that there is only one truth, and that oneself is in possession of it, is the root of all evil in the world."
"I am now convinced that theoretical physics is actually philosophy."
"Intellect distinguishes between the possible and the impossible; reason distinguishes between the sensible and the senseless. Even the possible can be senseless."
Mathematics exercises the brain better than anything else. Yesterday I realized I need such exercises, since my unlawful use of a hammer (which may have been a kind of a self-punishment for past mistakes in my life) resulted in a psychic shock to my nervous system that I am still not quite able to completely overcome.
And so when the opportunity appeared, and it appeared, I decided to use it to its full. Here is my exercise, from N. Bourbaki - Elements de Mathematique. Algebre. Chapitre 9-Springer (2006) :
At the moment I have no idea how to solve these problems. But I will not cease until I solve them.
15-02-23 18:51 My guess is that Bourbaki defines here the Hodge star operator. Usually it is defined for bilinear forms, but in 9d Bourbaki is asking the reader to extend its properties to sesquilinear forms (so that it works also in quantum theoretical environment, for Fermionic systems), like those we are dealing with discussing twistors. It strikes me that these twistors do not twist. How to make them twist? Something to think about.
It occurred to me today, while reading Kumar, that energy is not in general quantized, contrary to misleading statements in some popular books. But angular momentum IS! It looks like angular momentum is more primitive than energy, perhaps more fundamental than space and time.... Godel rotating universe allows for time travel. Another hint?
Niels Bohr has no idea how to make his model of an atom until he earned from a mathematician John Nicholson that angular momentum may be quantized.
"Bohr had met Nicholson during his abortive stay in Cambridge, and had not been overly impressed. Only a few years older at 31, Nicholson had since been appointed professor of mathematics at King’s College, University of London. He had also been busy building an atomic model of his own. He believed that the different elements were actually made up of various combinations of four ‘primary atoms’. Each of these ‘primary atoms’ consisted of a nucleus surrounded by a different number of electrons that formed a rotating ring. Though, as Rutherford said, Nicholson had made an ‘awful hash’ of the atom, Bohr had found his second clue. It was the physical explanation of the stationary states, the reason why electrons could occupy only certain orbits around the nucleus."
Thus even mathematicians making "awful hash" may be sometimes useful....
16-02-23 17:43
P.A.M. Dirac, Heinrich Hora, J. R. Shepanski, "Directions in physics: lectures delivered during a visit to Australia and New Zealand August/September 1975", Wiley 1978
ISBN: 0471029971,9780471029977
17-02-23 8:37 While searching for articles related to Gryzinski and Eganowa accidentally found A. Guts, "Metaphysics of time and eventology": Александр Константинович Гуц, "Метафизика времени и эвентология" (2011) I love the term "eventology".
Книга, в которой собраны работы О.Ю. Воробьёва, основателя нового междисциплинарного научного направления - эвентологии, связывающего философию и математику с естественными, гуманитарными и социо-экономическими науками: статистической механикой и термодинамикой, теорией вероятностей, теорией информации, теорией решений, эвенто-менеджментом, психологией, экономикой, социологией и др. Эвентология - учение о событиях, возникшее из невыносимо лёгких наблюдений: "материя и разум - это просто удобный способ связывания событий воедино" (Бертран Рассел, 1946; О.Ю. Воробьёв, 2001) и "разум возникает там и тогда, где и когда возникает способность делать вероятностный выбор" (В.А. Лефевр, 2003). Математическая эвентология - основанный на колмогоровской аксиоматике новый раздел теории вероятностей, показавший свою эффективность в математическом описании и эвентологическом обосновании и развитии существующих теорий неопределённости (теории нечётких множеств (Лютфи Аскер Заде, 1965), теории возможностей (Лютфи Аскер Заде, 1978), теории свидетельств (Демпстер-Шафер, 1976)), а также теории перспектив (Канеман, Тверски, 1979, 1992), объединившей экономику и психологию, и теории спроса и предложения ("крест Маршалла"), краеугольного камня современной экономикс. Наряду с философскими и математическими вопросами "со-бытия" и бытия эвентология затрагивает экономические, социальные и психологические вопросы, над которыми каждый из нас задумывается и размышляет на протяжении жизни. В книгу включены материалы, рассчитанные на широкий круг читателей, интересующихся эвентологией и ее приложениями; особый интерес они представляют для специалистов, активно работающих в этой новой области, преподавателей, аспирантов и студентов старших курсов университетов, занимающихся теорией решений, эвенто-менеджментом, искусственным интеллектом, теорией вероятностей, математической статистикой и математическим моделированием гуманитарных, социо-экономических и естественных систем.
Содержание:
Пролог.
Общие эвентологические принципы.
Аксиоматика Колмогорова.
Математические эвентологические принципы.
Множество событий.
Обращение эвентологических распределений.
Зависимость событий.
Парус и ветер Фреше.
Эвентологические распределения.
Эвентологические случайные процессы.
Теория нечётких событий.
Эвентологический портфельный анализ.
Эвентологические обоснования экономикс.
Эвентологические сеточные методы.
Эвентологические модели в психологии.
Эвентологическая теория сет-предпочтений.
Эвентологическая теория принятия решений.
Эвентологическая теория игр.
Эпилог.
Эвентологические обозначения.
Литература.
Zeit ist nur dadurch, daß etwas geschieht und nur dort wo etwas geschiecht.
(E. Bloch)
More papers by the same author, actively searching for the truth, here. Guts gives a quasi-definite answer to the question: "What is Time?". Time is a form of realization of the consciousness.
11:08 It seems to me, but perhaps I am wrong, that Guts assumes, more or less, that Quantum Theory is the Ultimate Truth. That we are supposed to rely on its formalism even when looking for answers for our deepest questions of existence and meaning. Even if nobody understands it, the problem lies with our understanding, not with quantum theory. On this issue I would rather take sides with Gryzinski. But it's my subjective and certainly biased taste and choice.
13:39 Something to think about (apart of Bourbaki): A. Guts, Созидание Мира в Библии и науке Доклад на "Рождественских чтениях-2019" (Омск, ОмГУ, 9 декабря 2019)
P.S. All these links and comments above are exclusively for me. What I am writing is simply my personal journal. I am talking to myself. I am putting links, so that later on I will be able to find them easily. Right now I am stuck with Bourbaki. Can't prove a clear statement in an exercise. I am really disappointed with myself, with my mental/mathematical abilities. Shame!
18-02-23 19:40 Prepared pdf file with the proposition at the end. Will try to add my proof tomorrow. It took me several days. I hope finally today I got the idea how to do it.
I am aware of the fact that a bright student would solve this problem in five seconds instead of five days. On the other hand how can I think of ever solving a really hard problem if I can't solve a simple (as I believe it is) problem? Time needed for solving the problem really doesn't matter.
On the other hand a smart student has billions of problems that are waiting for solutions.... And there are problems of unknown difficulty (like Riemann Zeta) waiting for smart students.
20:30 Decomposable k-vector may be thought of as a non-entagled k-particle (fermions here) state. A generic (entangled) k-vector is a superposition (linear combination, or, simply, a sum) of decomposable ones.
19-02-23 8:37 John G. mentioned Raoul Bott. As I do not believe in coincidences, I checked Bott's "Lectures on K(X)". There is something that I will be needing very soon. Like this type of graphs:
12:15 It occurred to me last night that if I consider EEQT version of QED (I do not care for electro-week interactions for a while), we can shoot perhaps two birds with one shot. First of all we will get rid of the need of renormalization, and we will also have the answer to the question "what are the fundamental primitive events?". Of course we will have to use the conformal structure for that.
19:30 Have finished the proof, updated pdf. A smart student would certainly do it in five lines in five seconds. In fact I asked a mathematician, Marian Fecko, how to prove this property some ten years ago. I have a copy of my email to him, but his answer has been scrambled by some software bug. I do not remember today what was in his answer. What I do know is that, after that, I simply wrote in one of my papers that decomposability of $*_e x$ follows from decomposability of $x$, like it was instantaneous and easy. Can move now on, step by step. Yeah, theoretical physics is actually philosophy. In one of my old papers I was playing a bit with Hodge-star operator. Commenting on it I wrote: "The question is important because light-cone structure determines nine of the ten components of the metric tensor. Thus, in other words, we were asking to what extent is gravity intertwined with electromagnetism?"
20-02-23 21:30 A Reader asked me an apparently unrelated question. He asked me if proving theorems in a particular branch of mathematics can be considered as a way to salvation? I am not an expert in this domain, but as far as I know it is more or less clearly stated in the Bible: The way to salvation is by asking for forgiveness all person that have been hurt by our actions, and do it while they are still alive. And then repent, and repent aloud. But as I said: I am not an expert. Experts should be searched for among theologians
26-02-23 11:20 From my correspondence.
Received while ago:
Good day, Arkadiusz!
In the article sent to you (through Levichev) "Atomic Physics by Gryzinski..." you can immediately see my attitude to his outstanding works: he revived true physics, figuratively speaking, he returned to physics (experimental science!) experiment. Quantum mechanics, in principle, is unnecessary. He demonstrated this convincingly in his numerous writings. The fact that few people understand this is not surprising (most people do not think independently, only imitate scientific activity, without any serious education, both physical and philosophical).
All the best, I.E.
26-02-23 11:23
Part I ( Peter Keating )
Howard Roark laughed.
Chapter I, p. 9 ; opening sentence
"Do you mean to tell me that you're thinking seriously of building that way, when and if you are an architect?"
"Yes."
"My dear fellow, who will let you?"
"That's not the point. The point is, who will stop me?"
"What is mathematics about? Is there a mathematical universe glimpsed by a mathematical intuition? Or is mathematics an arbitrary game of symbols, with no inherent meaning, that somehow finds application to life on earth? Robert Knapp holds, on the contrary, that mathematics is about the world. His book develops and applies its alternative viewpoint, first, to elementary geometry and the number system and, then, to more advanced topics, such as topology and group representations. Its theme is that mathematics, however abstract, arises from and is shaped by requirements of indirect measurement. Eratosthenes, in 200 BC, demonstrated the power of indirect measurement when he estimated the circumference of the earth by measuring a shadow at noon, in Alexandria, on the day of the summer solstice. Establishing geometric relationships, solving equations, finding approximations, and, generally, discovering quantitative relationships are tools of indirect measurement: They are the core of mathematics, the drivers of its development, and the heart of its power to enhance our lives."
15:02 Some Readers of this blog may feel a strong urgency and a compulsive need to express their own views, ideas, and opinions, that are radically different than those I am presenting here - they just wish to go into a completely opposite direction - these readers are strongly encouraged to use their own websites or blogs for a public presentation of their private views, and taking responsibility for these presentations. It is a refreshing, useful, and educating experience. Worth the effort. On their own sites they can open their wings wide and fly as high as they wish feeling completely free.
15:24 Quoted from "Human Front:
"Ethical egoism is a view that one can be self-serving and moral. Writer and philosopher Ayn Rand (pictured) constructed her own egoist theory called ‘Objectivism’. Remarkably, she did this through her works of fiction.
Rand’s philosophical outlook was based on a trichotomy: phenomena inside the mind (subjective), phenomena outside the mind (intrinsic), and phenomena between the two (objective). She then argued that it is ethical to rationally promote one’s own best interests to form objective knowledge on what they ought to do in each situation: what’s best for themselves.
Rand’s view is teleological because it brings purpose into morality. "
I consider such a view, we may call it STS, as in fact existing, and followed consciously or not, by many, if not by most of the people. The opposite view may be called STO or, following Gurdjieff, "external considering.". It is my understanding that for some people external considering may be simply physically impossible. These people simply do not have appropriate wirings. But even without a proper wiring a simulation is possible. We can, for instance, simulate quantum computer on a classical computer. Of course such a simulation needs remarkably more effort and time.
"P.A.M. Dirac, Heinrich Hora, J. R. Shepanski, "Directions in physics: lectures delivered during a visit to Australia and New Zealand August/September 1975", Wiley 1978
ReplyDeleteISBN: 0471029971,9780471029977"
I agree with this! Beautiful!
I like this Gryzinski model. However, I'm not sure I'm not motivating this statement purely by a sense of mathematical aesthetics.
ReplyDeleteOn the other hand, it would be interesting to consider to what extent mathematical aesthetics are relevant and whether in fact more aesthetically pleasing models and more symmetrical equations are closer to the truth.
The final formula should be very simple. But it could be dissected in infinitely many qualitative ways. However, this is just such a shallow intuition, but a great deal has come from intuition.... However, it can sometimes let us down. Especially with very complex problems.
Well... One thing bothers me about all these articles you post. If such fundamental mistakes have already been made in such fundamentals as the atom then what are physicists supposed to do now?
ReplyDeleteWhat is a modern theoretical physicist or mathematician to do? Is he supposed to go back to the very basics and do a revision of all physics? And if not what is left for him? The quantization of gravity?!
"Even if nobody understands it, the problem lies with our understanding, not with quantum theory. On this issue I would rather take sides with Gryzinski. But it's my subjective and certainly biased taste and choice.".
ReplyDeleteI agree with you, however, you are now still philosophising. And the most important thing for me at the moment is what mathematical concretes can be extracted from all this philosophy. It's a bit like composing music. Some motifs appear, but there is still no whole.
And also in Engelking's book on topology there is an error every few pages. It is tiresome. These are minor errors, usually some typos. But nevertheless they get in the way a lot, they deprive the reader of confidence. Afterwards it is difficult to concentrate.
"A. Guts, "Metaphysics of time and eventology": Александр Константинович Гуц, "Метафизика времени и эвентология" (2011) I love the term "eventology".".
ReplyDeleteHe has very broad interests. He is relatively interdisciplinary. However, this seems to be what the entities he studies require.
You provide a lot of links and I can't keep up with reading it. Do I have to read it all today? Have you read it all in full? Do I have to address and expand on absolutely every point made there? That would take me an eternity...
ReplyDeleteWhat kind of reaction from the reader do you expect from the literature you provide?
"Can't prove a clear statement in an exercise. I am really disappointed with myself, with my mental/mathematical abilities. Shame!".
ReplyDeleteI suggest you look at it from a different angle. You already know so much that you don't think through established patterns and solutions don't show themselves to you immediately. You consider too many different possibilities in your head.
This doesn't have to be a shame. It could be a chance for you to see beyond that. Something more important than solving a single task. Look at it this way.
For me philosophizing is sometimes needed for the concrete math you already think about especially when you are as math limited as I am. If you have odd grade related to anticommutation relations spin components then what makes this mostly classical math also a quantum one?
ReplyDeleteWikipedia says "In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermions respectively. They play a prominent role in quantum statistical mechanics[1] and quantum field theory." However other than some doubling of some degrees of freedom (creation/annihilation), it seems really classical?
Degrees of freedom-wise spin seems very fundamental for fermions; spacetime seems fundamental in a metric with G-structure sense and energy seems maybe useful as values but not for its own degree of freedom maybe. This twistor however seems like conformal/dark energy bosons derived from an interface between dual light cones (4+4=8-dim). It's spin angular momentum/polarization/twisting derived from timelike/longitudinal/transverse degress of freedom in the even subalgebra.
Not sure how to think of where quantum is coming from or how going from 8 to 6 to 5-dim structure should be thought of in the overall even/odd grade math structure but for me there is kind of always this math structure and I just want to know more how to philosophically think about it so it's more I need a philosophy for the concrete math rather than concrete math for a philosophy.
@John G
Delete"I just want to know more how to philosophically think about it so it's more I need a philosophy for the concrete math rather than concrete math for a philosophy.".
I have needed a philosophy for a long time and still do, but it is only now beginning to manifest itself more mathematically. Mainly through category theory. This is the discipline that allows me to model my mental constructs. I can't do this with linear algebra or complex analysis, for example.
I feel something like category theory is a discipline of mathematics that is isomorphic to my mind. I used to speak to myself in this language a lot, but I didn't know it because at the time I knew very little about the field, and even before that I didn't even know that such a discipline existed.
Of course, I think there is still a great deal that I need to learn and create in order to actually pour my mind into mathematics, but category theory is a revolution for me.
@John G
ReplyDeleteThe question is still what the meaning is at a deeper level and what we actually need this philosophy for. To explain something rationally?
Sometimes it is the case that something cannot be put into words, not even into music. Then only mathematical abstraction remains.
However, in order to explain to someone what I mean specifically, I would recommend Plotinus' "Enneads". It's like a kind of lecture on category theory, but without the formulas and systems of axioms. It's like such an outline of it and an attempt to give meaning to the abstraction. I am similarly influenced by Hegel. Possibly still Plato, Husserl, Kant. In the sense that I need these philosophers, they allow me to drift off into the realm of abstraction when I start to lower my flights. But the deepest abstraction is the edge between mysticism and very abstract mathematics. This is a very dangerous place to be. Apparently some people haven't returned from there.... Plotinus writes about it. People tend to run away from it, and I'm heading there....
Some relate Plotinus and his divine emanations structure to Gurdjieff's Ennneagram though maybe it's easier to go from Plotinus to Llull to Gurdjieff with Jung and digital/pluralistic idealist Leibniz in there somewhere too. Laura and her plotting of densities to the Tree of Life relates too I think.
DeleteFor category theory I'd go to the graded category and then the differential graded category quite quickly. So could "graded" apply to something that isn't algebra like cellular automata. Densities, personality, sacred geometry, and root systems seem rooted in the same categories.
As for graded category, here are publications of my PhD student Władysław Marcinek (1952-2003).
Deletehttps://ivv5hpp.uni-muenster.de/u/douplii/marcinek/
Always on wheelchair.:
https://inspirehep.net/authors/998845
http://ivv5hpp.uni-muenster.de/u/douplii/marcinek/marcinek-pub.htm
His PhD thesis:(at that time I was working, for a couple of years) with graded category)
https://www.sciencedirect.com/science/article/abs/pii/0034487789900207
But after his PhD Marcinek went much further than I ever dared.
Physics has had interesting people working in interesting areas even if the mainstream gobbles up a lot of people into uninteresting areas. You can probably say that about research in lots of disciplines.
Delete"But after his PhD Marcinek went much further than I ever dared.".
DeleteLack of courage may be the main reason why there is no meta-theory yet...
@John G
ReplyDelete"Some relate Plotinus and his divine emanations structure to Gurdjieff's Ennneagram though maybe it's easier to go from Plotinus to Llull to Gurdjieff with Jung and digital/pluralistic idealist Leibniz in there somewhere too. Laura and her plotting of densities to the Tree of Life relates too I think.".
It's all one and the same, but expressed in different languages. Someone has to dedicate themselves, try to know them all and express it on a meta-level. This requires a very long life or a miracle....
And one has to completely divest oneself of humanity in order to do so. This is more than religious order and celibacy. It requires looking completely outside the interests of this world.
A very hard thing to do. I would like to do only that in my life. And I don't need anything else.
And by the way - I saw your comment exactly when I was writing to myself in my diary about something very similar!
I wonder if it is legitimate to say that physics is the science that describes reality. If consciousness is objective reality and time is the means of realising it, it would be legitimate to say that 'real physics' is very abstract mathematics combined with mystical experience. Mathematical abstraction and mystical experience are the only ones that touch the deepest nature of our mind. At the deepest level we think very abstractly. Only the unlimited world of mathematics approaches this abstraction. Physics is very far away.
ReplyDeletePhysics tries to adapt the description to the phenomenon... Mathematics aims at the noumena from which all possible phenomena arise. At the highest level of abstraction, there is only one necessary primary concept - a kind of category of all categories. Physics does not touch this level. No physical theory is as beautiful as a pure mathematical structure - the latter is, despite appearances, closer to 'objective reality'.
I don't see the point of being a physicist. I see more sense in being a mathematician and a mystic.
I really like the Mackey-Arens theorem! It establishes a deep connection between the representation theory of a locally compact group and the study of its dual group, which is a topological space consisting of all continuous homomorphisms from the group to the unit circle in the complex plane.
ReplyDeleteThe theorem states that every irreducible unitary representation of a locally compact group G is equivalent to a subrepresentation of the left-regular representation of G on L^2(G), the space of square-integrable functions on G. In other words, any unitary representation of G can be realized as a collection of linear operators acting on the functions in L^2(G) in such a way that the group action is preserved.
Furthermore, the theorem provides a duality between the representation theory of a locally compact group and the study of its dual group. Specifically, it states that the dual group of G is isomorphic to the group of equivalence classes of irreducible unitary representations of G!!!
Do you think it is possible to carry out a proof of Mackey-Arens theorem more or less as follows?:
ReplyDelete1. The first step is to show that every unitary representation of a locally compact group G can be realized as a subrepresentation of the left-regular representation of G on L^2(G). This is achieved by constructing a family of intertwining operators that map the original representation into the left-regular representation.
2. Next, it is shown that the left-regular representation of G on L^2(G) is decomposable into a direct integral of irreducible subrepresentations. This involves using the spectral theorem for self-adjoint operators and some basic results from functional analysis.
3. Using the decomposition of the left-regular representation, it is possible to define a Fourier transform between functions on G and functions on the dual group of G. This is a key step in establishing the duality between the representation theory of G and the study of its dual group.
4. Finally, it is shown that the dual group of G is isomorphic to the group of equivalence classes of irreducible unitary representations of G. This is achieved by constructing a family of canonical representations of the dual group and showing that they are all irreducible and unitary, and that any two such representations are equivalent if and only if the corresponding irreducible subrepresentations of the left-regular representation are equivalent.
I cannot find proof of this theorem anywhere.
@John G, @Ark
ReplyDeleteAnother crazy concept. I'm also interested in what you think about it.
Information can curve the state space in quantum mechanics. This idea is closely related to the concept of quantum entanglement and the idea that the state space of a quantum system can be geometrically curved in a way that depends on the entanglement structure of the system.
One way to think about this is through the concept of the quantum state space. In quantum mechanics, the state of a system is represented by a vector in a complex vector space. This vector space is known as the state space, and its geometry is intimately connected with the dynamics of the system.
When two or more quantum systems become entangled, their state space becomes curved in a way that depends on the entanglement structure of the system. This curvature can be measured using mathematical tools such as the Fisher information metric, which quantifies the distance between quantum states.
In recent years, researchers have explored the idea that the curvature of the state space can be influenced by the amount of information that is available about the system. For example, the amount of entanglement between two systems can be increased by adding more information about their state, and this can lead to a change in the curvature of the state space.
Overall, while the idea that information can curve the state space in quantum mechanics is still a topic of ongoing research, it is a promising area for exploring the fundamental connections between quantum mechanics, information theory, and geometry.
Ark once mentioned this, but I don't know the mathematical description. Did you create that description? What would time be in such a theory?
Well since it would include entanglement in time, each decoherence you would be choosing a complete worldline through time in a complete universe state throughout time but it won't be what you or the universe live through since a new state gets chosen every decoherence.
DeleteThe future of your current state (your worldline and the universe) however would represent what you are being subtly curved towards. There are also real things happening in that future of your current state just not exactly what you will experience when you get there but some version of you in the future is currently experiencing that. There's also some version of you in the past of your current state doing things different than you.
This reminds me of when Doctor Who's time machine became humanoid and was trying unsuccessfully to talk about the past, future and present and finally just stated tenses are difficult aren't they.
"18-02-23 19:40 Prepared pdf file with the proposition at the end.".
ReplyDeleteDo I understand this correctly?
A k-vector x is decomposable if it can be written as the exterior product (wedge product) of k linearly independent vectors, i.e.,
x = x1 ∧ x2 ∧ ... ∧ xk,
where each xi is a vector in the underlying vector space E. Here, the symbol ∧ denotes the exterior product, which is a way of combining vectors to form a new multivector.
In other words, a k-vector x is decomposable if it can be expressed as the wedge product of k linearly independent vectors, meaning that x can be written as the determinant of a k x k matrix whose columns are the vectors x1, x2, ..., xk.
If a k-vector x is not decomposable, it is called indecomposable or non-decomposable. This means that it cannot be expressed as the exterior product of k linearly independent vectors, and hence cannot be written as the determinant of any k x k matrix.
I analysed the proof of Gödel's theorem today. I will describe it briefly.
ReplyDelete1. Gödel first constructs a formal system, called G, that is powerful enough to represent arithmetic and that can express statements about itself. The system includes a set of axioms and a set of rules for deriving new theorems from the axioms.
2. Gödel then constructs a statement, called G, that asserts its own unprovability within the system G. That is, G says "I cannot be proved within the system G".
3. Gödel shows that if G is provable within G, then G is false. This is done by constructing a self-referential statement that contradicts the assumption that G is provable.
4. Gödel then shows that if G is not provable within G, then G is true. This is done by constructing a different self-referential statement that asserts its own unprovability within G.
5. Finally, Gödel shows that the system G cannot prove or disprove G, and hence that G is true but unprovable within G.
The proof of Gödel's second incompleteness theorem builds on the first incompleteness theorem and uses similar techniques to show that any consistent formal system that is powerful enough to represent arithmetic cannot prove its own consistency.
Now look at this proof through the prism of ephesis. And I also have a thought about EEQT. Maybe I'll write about it later when it has clarified a bit...
"https://www.sciencedirect.com/science/article/abs/pii/0034487789900207".
ReplyDeleteI wrote a note about it today:
A generalized Lie-Cartan pair is a pair of Lie algebras (g, h) equipped with a bilinear form B: g × h → R, which satisfies certain compatibility conditions. This structure is important in the study of symmetric spaces and other related topics in mathematics.
The first condition that must be satisfied is the Cartan-Killing form condition, which requires that the bilinear form B be nondegenerate on h and that it be invariant under the adjoint representation of h on itself. That is, for all x, y ∈ h and z ∈ g, we have:
B([x, y], z) = B(x, [y, z])
The second condition is the Lie-Cartan condition, which requires that the bracket operation [·, ·] on h be induced by the bilinear form B via an isomorphism between h and its dual h*. That is, for all x, y ∈ h, we have:
B(x, [y, z]) = [B(x, y), z] + [y, B(x, z)]
where z ∈ h.
The third condition is the generalized Jacobi identity, which requires that for all x, y, z ∈ h and u, v ∈ g, we have:
[B(x, y), B(z, u)v] + [B(y, z), B(u, v)x] + [B(z, x), B(v, u)y] = 0
The structure of a generalized Lie-Cartan pair is important in the study of symmetric spaces, which are spaces that possess a certain kind of symmetry. In particular, the bilinear form B plays an important role in defining the geometry of a symmetric space, and the Lie algebra h is closely related to the isotropy group of the space, which is the group of symmetries that leave a given point invariant.
Generalized Lie-Cartan pairs also have applications in other areas of mathematics, such as the theory of Lie algebras and Lie groups, differential geometry, and representation theory.
I have also previously listed potential EEQT problems. For now, this is a draft version. What do you think?:
ReplyDelete1. Lack of experimental evidence: EEQT is a purely theoretical framework, and it has not yet been tested experimentally. While there are some proposals for how EEQT could be tested using existing experimental setups, none of these proposals have been verified by experiment.
2. Non-locality: EEQT proposes a new type of non-local interaction that violates the principles of locality and causality. This raises serious questions about the consistency of EEQT with other well-established physical theories, such as relativity.
3. Lack of mathematical formalism: EEQT is still in the early stages of development, and there is currently no well-established mathematical formalism for the theory. This makes it difficult to make precise predictions or to compare EEQT with other theoretical frameworks.
4. Philosophical assumptions: EEQT makes several philosophical assumptions about the nature of reality, such as the existence of irreducible events and the rejection of reductionism. These assumptions are controversial and not widely accepted by the scientific community.
5. Compatibility with other theories: EEQT proposes a new type of interaction that is not present in standard quantum mechanics, and it is not yet clear how EEQT can be made compatible with other well-established physical theories, such as quantum field theory or general relativity.
"And there are problems of unknown difficulty (like Riemann Zeta) waiting for smart students.".
ReplyDeleteAnd then there is the problem of meta-theory. Requiring 16 hours of mathematics a day. The mind can go mad from something like that.
"On the other hand how can I think of ever solving a really hard problem if I can't solve a simple (as I belive it is) problem?".
ReplyDeleteYou really don't believe in yourself very much.... I've never seen it as deeply as I did today. Those clinical psychology studies are even of some use to me!
„A generic (entangled) k-vector is a superposition (linear combination, or, simply, a sum) of decomposable ones.”.
ReplyDeleteYes, that is correct.
I thought to myself that a generic k-vector can be expressed as a linear combination of decomposable k-vectors. In other words, it can be written as a sum of wedge products of k linearly independent vectors, with coefficients that are complex numbers.
The coefficients represent the degree of entanglement between the different decomposable k-vectors in the superposition...
I still had this problem today: Mathematical framework for UFT.
ReplyDeleteOf course, I think it is category theory. Let me explain why.
One of the main advantages of category theory is its ability to capture the relationships between mathematical structures in a highly abstract and general way. This can be useful when studying the relationships between different physical theories, including those related to the fundamental forces.
By using category theory to capture the relationships between theories, we can gain insight into how these theories might be unified into a single framework.
However, one thing is very important - this union must manifest itself at the level of mathematical structures. It is the physics that must adapt to the mathematical structures, not the other way around.
Mathematics is more fundamental, an aspect of spirit that physics lacks...
Here is my today’s proof of the density theorem:
ReplyDeleteFirst, assume that the image of F is dense in D. This means that for any object d in D, there exists an object c in C and a natural isomorphism η: F(c) → d. We want to show that d is a colimit of objects in the image of F.
To do this, consider the diagram D(F) of all diagrams of the form F(d') → F(d), where d' ranges over all objects in the image of F. This diagram is essentially the image of F, viewed as a diagram in D.
Since the image of F is dense in D, we know that any object in D can be written as a colimit of objects in the image of F. In particular, d can be written as a colimit of the diagram D(F). Let (d', f') be a cocone for the diagram D(F) with colimit d. By the definition of a cocone, this means that for any object d'' in the image of F, there is a unique morphism g'': d'' → d' in D(F) such that f' o F(g'') = η'' o η'^{-1}, where η'' : F(d'') → d and η' : F(d') → d are the natural isomorphisms provided by the density assumption.
Now, let c' be the coequalizer of the pair of morphisms F(g''), where d'' ranges over all objects in the image of F. Since F is full and faithful, this means that c' is also the coequalizer of the pair of morphisms g'': F(c'') → F(c') for any objects c'' in C. Moreover, we have a natural isomorphism η': F(c') → d by taking the cocone (d', f') to (c', f), where f is the unique morphism F(c') → d in D(F) such that f' o F(g'') = f o g'' for all d'' in the image of F.
It remains to show that c' is a colimit of objects in the image of F. To do this, let (c'', f'') be a cocone for the diagram F(F(d')) → F(c') → d, where d' is an object in the image of F. Then, there exists a unique morphism g''': c'' → c' in C such that F(g''') o f'' = f o F(F(d')) for all d' in the image of F. Since F is full and faithful, this means that g''' is also the unique morphism F(c'') → F(c') such that g'' o F(F(d')) = F(g''') o f''. Therefore, we have a cocone (c'', g''') for the diagram F(d') → F(c') in C.
Now, since F is full and faithful, we know that every object in D is a colimit of objects in the image of F if and only if every object in C is a colimit of objects of the form F(d') for d' in the image of F. Therefore, c' is a colimit of objects in the image of F, which completes the proof.
The density theorem in category theory has a number of interesting philosophical implications, especially in the area of ontology.
One way to interpret the density theorem is in terms of the relationship between objects and their properties. Specifically, the theorem suggests that any object in a category can be thought of as a colimit of objects that share certain properties. In other words, an object is defined by the collection of objects that are "close" to it, in the sense that they can be combined in a certain way to form the original object.
This view of objects and their properties has been applied to a number of different areas of philosophy, including metaphysics, epistemology, and even aesthetics. For example, the density theorem has been used to argue that the objects of our experience are not fixed, objective entities, but are instead defined by their relationships to other objects and to our own experiences.
The density theorem can also be seen as providing a foundation for the idea of emergence, which is the notion that complex systems and phenomena can arise from simple interactions between their constituent parts. The theorem suggests that the behavior of a complex system can be understood in terms of the behavior of its simpler components, which can be combined in a certain way to produce the emergent phenomenon.
You can probably see what I am getting at again... Ennead-like motifs.
Still, this is an important observation: While there is no direct relationship between Plotinus and category theory, there are some interesting connections between their ideas...
ReplyDeleteOne of the key ideas in Plotinus' philosophy is the concept of the One, which he saw as the ultimate source of all existence. The One was seen as a transcendent, ineffable reality that could not be fully understood or described. This concept has some similarities to the idea of a universal category in category theory, which is a category that contains all other categories as subcategories. Like the One, the universal category is an abstract entity that cannot be fully grasped by human understanding.
Another interesting connection between Plotinus and category theory is the idea of hierarchy. Plotinus saw reality as arranged in a hierarchical structure, with the One at the top and various levels of being descending from it (ephesis). Category theory also makes use of hierarchy, with categories and functors arranged in a nested structure.
Finally, both Plotinus and category theory are concerned with the nature of existence and the relationship between different levels of reality. Plotinus' philosophy emphasizes the idea of emanation, in which lower levels of reality emerge from higher levels. Category theory, in turn, provides a powerful tool for describing relationships between different levels of abstraction in mathematics.
Now think: EEQT and phase transitions. A concept about densities. Time separates these levels, penetrates them, in this lies its cruelty, which is why it has become a problem in my life.
We do not formulate our problems sufficiently abstractly. We limit the scope of applicability of our own concepts at our own request...
There are fewer problems in category theory than in other areas. The problems of CT are reducible to 2:
ReplyDelete1. The Size of Categories: One of the most fundamental problems in category theory is understanding the size of categories. Categories can be small (i.e., have a set of objects and a set of morphisms) or large (i.e., have a proper class of objects and/or morphisms). The study of large categories is still relatively new, and many of the techniques developed for small categories do not carry over to the large case.
2. Higher Category Theory: Category theory itself is a theory of categories, but it is also possible to study categories of categories, categories of categories of categories, and so on. These higher categories can be used to describe more complex structures and phenomena, but the theory of higher categories is still in its early stages, and many questions remain unanswered.
You wrote that you don't even think about difficult problems when you can't solve easy ones. It seems to me that sometimes we can't solve the easy problems, only the difficult ones. We are operating at some other level of abstraction. We look for a double bottom in easy problems. Difficult problems are more like our mind.
From your twistor point of view that K-vector and Hodge star dual sounds like you are relating spin components in our light cone to spin components in the other light cone. The boson view would be something different, course I'm thinking in terms of a Hodge star map for Clifford algebra. Don't worry too much about doing something more slowly, you'd still be in good company; from Woit's blog:
ReplyDeleteBoth Atiyah and Witten are extremely quick on their feet. I remember one time at MSRI talking to Raoul Bott, who had just walked away from Atiyah and Witten, shaking his head. He told me he found listening to the two of them “scary” since they were so much quicker than he was. Bott is a great mathematician also, but one who has to think everything through slowly and carefully to understand it, quite different than Atiyah or Witten.
Thanks John! In aprticular for mentioning of Bott. His monograph "Differential forms in algebraic topology" may be even of some use for me. He has Grassmannians (though only compact case) and tautological bundles. This is what I was thinking about last night. I even got up from the bed to write my thoughts, in particular I wrote :
Delete"... Then you have tautological bundle. Over each spacetime point you have null vector it represents".
What a "coincidence"!
"... Then you have tautological bundle. Over each spacetime point you have null vector it represents".
DeleteI thought to myself that the base space of tautological bundle is a complex projective space and the fibers are complex lines.
Each point in the base space corresponds to a complex line in the fiber, and the tautological bundle is defined to be the collection of all such lines.
More specifically, the tautological bundle is the subset of the product space consisting of all pairs (p,v), where p is a point in the complex projective space and v is a non-zero vector in the complex line corresponding to p.
The physical meaning of the tautological bundle is related to the concept of null directions in spacetime. In general relativity, null directions are defined as the directions in which light rays propagate. The tautological bundle can be used to describe the space of null directions at each point in spacetime, by associating a complex line with each point in spacetime. However... "As above so below"…
Everything that exists in physics must have some counterpart in a higher (objective) reality. This has already been discussed on this blog (consciousness as objective reality). Hence, I thought of something like this:
In category theory, the tautological bundle can be presented as a natural transformation between two functors.
Let $\mathbb{CP}^\infty$ denote the category whose objects are complex projective spaces $\mathbb{CP}^n$ for all non-negative integers $n$, and whose morphisms are given by complex projective maps. Let $\mathrm{Vect}_{\mathbb{C}}$ denote the category of complex vector spaces and linear maps.
The tautological bundle is defined as a functor $T:\mathbb{CP}^\infty\to \mathrm{Vect}_{\mathbb{C}}$ that assigns to each complex projective space $\mathbb{CP}^n$ the complex line bundle over $\mathbb{CP}^n$ whose fiber over a point $[v]\in \mathbb{CP}^n$ is the one-dimensional subspace of $\mathbb{C}^{n+1}$ spanned by $v$.
We can define another functor $I:\mathbb{CP}^\infty\to\mathbb{CP}^\infty$ called the identity functor, which simply maps each object and morphism to itself.
Now, the tautological bundle can be presented as a natural transformation $\tau:I\to T$ between these two functors. For each complex projective space $\mathbb{CP}^n$, the tautological bundle provides a linear map from the vector space $\mathbb{C}^{n+1}$ to the space of sections of the line bundle over $\mathbb{CP}^n$, which is a functorial construction.
This could be extended even more interestingly by introducing more general spaces etc.
And now my question: Why is no one doing this? Do we prefer to stay with theories that describe phenomena or move towards noumenas (functors)?
The concept of a tautological bundle can be extended to certain non-associative algebras, including the octonions, which are a type of normed division algebra and can also be described in terms of functors in a way similar to the complex tautological bundle. But to make sure they are not just 'words, words'…
ReplyDeleteThe octonionic multiplication is given by the octonionic multiplication table.
This multiplication table is used to define a left action of the exceptional Lie group $G_2$ on the octonions by automorphisms that preserve the norm.
Let $\mathbb{O}$ denote the space of octonions, and let $G_2$ act on $\mathbb{O}$ via the above left action. Then $S^6 = G_2/SO(4)$ is the space of unit octonions, where $SO(4)$ is the stabilizer of a chosen point in $\mathbb{O}$.
We can define a functor $T:S^6\to \mathrm{Vect}_{\mathbb{R}}$ that assigns to each point $p\in S^6$ the tangent space $T_pS^6$ to $S^6$ at $p$, which is an 8-dimensional real vector space. We can then associate to each point $p$ and each nonzero vector $v\in T_pS^6$ the corresponding pure imaginary unit octonion $j_v$ in the direction of $v$.
More precisely, for each $p\in S^6$, the functor $T$ assigns the vector space $T_pS^6$, and for each nonzero $v\in T_pS^6$, it assigns the pure imaginary unit octonion $j_v\in \mathrm{Im}(\mathbb{O})$. This gives a well-defined tautological bundle over $S^6$.
The octonionic tautological bundle can also be described in terms of a natural transformation. Let $I:S^6\to S^6$ be the identity functor, and let $i:T_pS^6\to \mathbb{O}$ be the inclusion of the tangent space into the octonions. Then the tautological bundle is a natural transformation $\tau:I\to T$ that assigns to each point $p\in S^6$ the map $\tau_p:T_pS^6\to \mathrm{Im}(\mathbb{O})$ given by $\tau_p(v)=j_v$ for each nonzero $v\in T_pS^6$.
In this way, the octonionic tautological bundle can be understood as a natural transformation that relates the geometric structure of $S^6$ to the algebraic structure of the octonions.
What is most important to me, the octonionic tautological bundle has interesting connections to both quotient algebras and to the concept of time in physics.
Firstly, the octonions have a natural quaternionic subalgebra, obtained by taking linear combinations of the unit octonions $e_i$ with indices $i\in{1,2,3}$. This quaternionic subalgebra is isomorphic to the quaternions, and it can be shown that the quotient algebra $\mathbb{O}/\mathbb{R}$ of the octonions by their real subalgebra is isomorphic to the algebra of pure quaternions. This quotient algebra can be identified with the space of tangent directions to $S^6$, and the octonionic tautological bundle can be thought of as assigning to each tangent direction the corresponding pure imaginary quaternion, which is a unit quaternion.
Let's look at it from an M-theory perspective that unifies the various string theories in 11-dimensional spacetime. In this theory, the spacetime manifold is taken to be a product of 10-dimensional spacetime and an extra dimension that is compactified on a 7-dimensional manifold known as a G2 manifold. This G2 manifold can be constructed as a quotient of the unit octonions $\mathbb{O}^*$ by a discrete group of automorphisms, and the octonionic tautological bundle over $S^6$ is related to the geometry of this G2 manifold.
To be continued…
Continuation of the previous comment:
ReplyDeleteFurthermore, the octonionic tautological bundle has a deep connection to the concept of time in physics. In particular, it has been conjectured that the space of pure imaginary octonions, which is the fiber of the tautological bundle over each point of $S^6$, can be identified with the space of internal time in M-theory. This identification is based on the fact that the pure imaginary octonions form a 7-dimensional subalgebra of the octonions, which can be thought of as a „time-like” subspace (this can be important from a philosophical perspective, even on issues such as free will considerations - but that's for a long treatise).
The challenge of M-theory is the difficulty in making testable predictions or experimental verifications. One reason for this is the high energy scales involved in the theory, which are currently beyond the reach of current experimental technology.
Here, again, category theory can help provide a more abstract framework for studying the relationships between different physical systems and their properties, which could potentially lead to new insights into the behaviour of M-theory in extreme conditions and more. Much more!
There are no paradoxes in our consciousness at the highest level (the first hypostasis, the mind in Plotinus' philosophical system). Mathematically, the closest to this level (we mean something like sixth density) is precisely category theory. Category theory does not touch sixth density, but when extended appropriately...
Paradoxes are in physics. The higher level resolves these paradoxes.
Category theory is more than physics. It is the language of a higher reality, the one closer to "consciousness".... And its realisation through time. Husserl once wrote about this philosophically (he wrote about feelings without mathematical description). In physics no one can do it. Mathematically I think it can be done.
Is this the 'new physics' we are all waiting for? Physics that is capable of describing the spirit.... This is, after all, what we are looking for...
"Another thought from last night: K-K theory usually produces gravity and EM in 4D from metric in 5D. Yet if we take two-form in 6D we have 15 parameters and 15=6+9. Six for EM and 9 for the light cone structure of gravity. So gravity from EM rather than EM from gravity.".
ReplyDeleteIt seems to me that the relationship between gravity and electromagnetism in KK theory is more subtle than simply deriving one from the other. You simplify the beauty. I do not like it.
In KK theory, the extra dimension is typically assumed to be compactified in a particular way that produces a specific internal structure, often in the form of a fiber bundle. The metric in the higher-dimensional spacetime is assumed to have a certain symmetry that allows it to be written as a direct sum of the four-dimensional metric and the metric on the internal space. The gauge fields for electromagnetism are then identified with the components of the metric on the internal space that are orthogonal to the four-dimensional metric, while the gravitational field is identified with the entire metric in the higher-dimensional spacetime.
In this picture, both gravity and electromagnetism arise from the same underlying geometry, but they are not necessarily derived from each other. Instead, they are both seen as different aspects of the geometry of the higher-dimensional spacetime.
Regarding your comment about the two-form in six dimensions, it is true that there are 15 parameters in a general two-form in six dimensions. However, it is not accurate to say that these correspond directly to the parameters of electromagnetism and gravity in four dimensions. The parameters of the two-form are related to the geometry of the internal space, and it is not clear how they would correspond to the physical properties of the four-dimensional spacetime.
The geometric interpretation of electromagnetism can be seen as a natural transformation between two functors, one that takes geometries to electromagnetism and another that takes geometries to pure gravity.
What do you think about it?
"It occurred to me last night that if I consider EEQT version of QED (I do not care for electro-week interactions for a while), we can shoot perhaps two birds with one shot.".
ReplyDeleteThe EEQT version of QED ought to be an attempt to incorporate the electromagnetic force into the EEQT framework.
In the EEQT version of QED, the electromagnetic field would be described by a quantum field that interacts with the curvature of spacetime in a manner similar to the gravitational field.
This interaction between the electromagnetic field and the curvature of spacetime would be described by a set of field equations that are similar to those of general relativity…
This is where a new road opens up... Information, gravity, consciousness, time... I like this direction.
„The geometric interpretation of electromagnetism can be seen as a natural transformation between two functors, one that takes geometries to electromagnetism and another that takes geometries to pure gravity.”.
ReplyDeleteTo describe the functors in more detail, we can consider a category of geometric spaces. The objects in this category are geometric spaces, such as manifolds, and the morphisms are smooth maps between them.
The first functor, which takes geometries to electromagnetism, is given by associating to each geometric space a gauge field that lives in the extra dimensions of spacetime. This functor can be thought of as a kind of "dimensional reduction" that collapses the extra dimensions down to the familiar four-dimensional spacetime of special relativity. The action of this functor on morphisms is given by pulling back the gauge field along the morphism.
The second functor, which takes geometries to pure gravity, is given by associating to each geometric space a Riemannian metric, which describes the curvature of spacetime. This functor can be thought of as the "usual" way of describing gravity in terms of the geometry of spacetime. The action of this functor on morphisms is given by pulling back the metric along the morphism.
These morphisms correspond to smooth maps between Riemannian manifolds, which preserve the metric structure. These maps can be thought of as diffeomorphisms that preserve the geometry of spacetime.
This is going to sound crazy, but... These diffeomorphisms... This is what consciousness perceives approximately as time. But it is very difficult to describe mathematically. I’m working on it.
I believe this is possible because, for the moment, category theory was able to capture what I was not able to express in any other way. Music didn't say it, no philosophy said it, category theory said it...
In a way I am even happy. I believe that I have not thrown any words to the wind. Category theory came to me and told me that my dreams made sense after all... It's like coming back to life after a very serious illness. After an illness of body, soul and mind.
"So why not simply expand classical mechanics by ADDING to it also non-commuting operators, rather than removing all commuting, as it is done in the standard QM. There is no "measurement problem" then.".
ReplyDeleteBut is this approach sufficient to reproduce the full range of phenomena that are described by quantum mechanics? In particular, classical mechanics with non-commuting operators does not provide a natural way to describe phenomena such as entanglement, superposition, and the wave-particle duality of matter...
Additionally, while adding non-commuting operators may avoid some of the difficulties associated with the measurement problem in standard quantum mechanics, it does not necessarily provide a complete solution. The measurement problem arises not just from the fact that the operators do not commute, but from the fact that the act of measurement fundamentally changes the state of the system being measured. This is a consequence of the non-locality and non-determinism that are inherent in quantum mechanics.
Non-locality (in the sense of the absence of time and space) is the very essence. The essence of the One. The category of all categories.
"Thus, mathematically, we will have to deal with infinite dimensional grassmannians.”.
ReplyDeleteI will translate everything you write into the language that expresses my thoughts best among the languages I know. We'll see what comes out of it....
Thus, in category theory, a Grassmannian could be a special kind of object called a "Grassmannian object," which is defined in terms of a sequence of vector spaces and linear maps between them. In the case of infinite-dimensional Grassmannians, these vector spaces are infinite-dimensional Hilbert spaces, and the linear maps are unitary operators.
A Grassmannian object would be, therefore, a way of encoding the idea of a "space of subspaces" of a given vector space. More precisely, a Grassmannian object is a sequence of vector spaces, V_0, V_1, V_2, ..., together with a sequence of linear maps, f_01: V_0 -> V_1, f_12: V_1 -> V_2, f_23: V_2 -> V_3, and so on.
These linear maps satisfy certain axioms that ensure that they encode the properties of subspaces of the vector space.
However, the axioms that define a Grassmannian object are purely linear, which means that they do not include any non-linear operations. This is because the Grassmannian object is a purely algebraic object that encodes the linear properties of subspaces of a given vector space.
Where is there a place for non-linearity here? I see two ways.
One approach is to introduce non-linear operators that act on the Grassmannian object, such as superselection operators, which select certain subspaces of the Hilbert space and ignore others (But what is the criterion?).
Another approach is to introduce non-linear functionals that act on the Grassmannian object, such as the functionals used to define the path integral in quantum field theory. See, for example: https://en.wikipedia.org/wiki/Path_integral_formulation
These functionals are non-linear because they involve integrals over all possible paths between two points in space-time, which is a non-linear operation.
Question, however: How deep is it and how much is necessary at this point? I don't know myself if I "feel" it well enough...
Try to argue more clearly what you are writing. Prove mathematically that this must be the case.
„In one of my old papers I was playing a bit with Hodge-star operator.”.
ReplyDeleteStudying psychology, reading Plotinus, serious study of mathematics and other circumstances have changed something in me. Currently, I am primarily searching for the deep identity of each being. Mathematically I do this at the level of category theory (the closest to the highest hypostasis of all mathematics), and so I looked at this operator in the context of your question about gravity and electromagnetism.
In the language of my mind (category theory), the Hodge duality operator can be thought of as a natural transformation between two functors. The first functor takes as input a differential form and returns its Hodge dual, which is a form of complementary dimension. The second functor takes a differential form as input and returns a corresponding vector field.
The Hodge duality operator thus provides a way to relate different types of geometric objects, such as forms and vector fields, in a way that is consistent with the geometry of the underlying space (There will be other fields. There will be fields of consciousness, but more general than all the fields we have come to know so far. This is where there will be a unification of interactions - it's not just a unification of these four interactions!!!). This can be seen as a kind of "categorification" of the Hodge duality operator, where we are replacing specific objects with more general, abstract objects in a categorical framework (Oh this is the answer to many dilemmas.... What isn't a morphism or an object but isn't a class either - I'll elaborate on that soon, right now I'm writing my thoughts as they fly).
I actually had to read what you wrote. But I think I found something even better! The arrival time operator as a morphism of EEQT phase spaces, but it has to be considered with a broader notion of field.
And one more thing...
Neutrinos. I think I have the answer to why it exists for neutrinos!!!!!!!!!!!!!!!!!!!
Oh, it wasn't pointless! I'll try to explain it later.
It also occurs to me that there is a curiosity that I was not aware of. Classes of solutions to mathematical problems of a certain type. But that's also to be considered later. I'm thinking of a thousand problems at once...
These thousand problems and thousand perspectives that I think about are actually one problem.... This is how this problem becomes apparent to a person. People around me think I'm doing a thousand things at the same time. I myself have started to believe them. And in fact I have never done anything else - except one thing. Except what I've been thinking about every day for as long as I can remember. Every day and every second of my life.
ReplyDeleteThe problem is so powerful that it required considering many perspectives and breaking down very many mirrors along the way. So that I could see the Dawn again.
How is it that we think our lives are like music? We keep noticing leaps, intervals, sequences.... It's all so uneven... What is it about this object (morphism) that so strenuously disturbs our perception in this 3D world?
I find it difficult to write. But my point is that I think I have found all the motifs. Now I just need to compose a piece. I'm learning to play to play the piece. But at least I have the piano - the category theory. So I can record and write notes.
A 4-dim Feynman Checkerboard has conformal symmetry (via satisfying the Dirac equation) and path integrals. It kind of is very classical with only the random walk paths making it nonlinear. Even the propagator fits in the overall symmetry at the vertex. What would EEQT do to a Feynman Checkerboard? Technically the conformal group is linear in 6-dim and some kind of nonlinear as seen in 4-dim. I think conformal gravitons could handle things like entanglement/tunneling.
ReplyDeleteAs for that 6+9=15 EM plus gravity thing. The idea of a graviweak sector is not unheard of. I don't know if you really could symmetry break down to it but they do form a block in Hodge star maps. Einstein had a 4x4 block and a 6-dim "antisymmetric" corner was thrown out for relativity and I think Einstein did try to relate it to EM unsuccessfully and that could be related to the whole conformal group being needed for gravity but Einstein wasn't actually working with Lie groups.
The diagonal of the 4x4 block could be thought of as the flat 4-dim metric which I think fits Einstein and a Hodge Star map. The 6-dim corner for EM could be 4-dim electroweak plus a dilation and the Feynman propagator/affine one-form. It's basically everything you could make using only the other light cone's 4-dim metric (from the twistor point of view). The 6-dim "symmetric" corner that Einstein kept could simply be thought of as the rotations/boosts you can make from the XYZT vectors. There's another 4x4 block in the Hodge Star map for translations, special conformal transformations and gluons which all involve one vector from XYZT and one from the other light cone.
Bott also found that 8-dim Bott periodicity that puts a spotlight on octonions that you can break into quaternions for dual light cones.
@John G
ReplyDeleteContinuation of the previous comment.
Conformal Field Theories as Conformal Categories: One way to combine the conformal group and category theory is to view conformal field theories as examples of conformal categories. In this framework, the conformal group acts as a group of automorphisms on the category, and the objects and morphisms in the category correspond to physical states and operators in the theory.
Categorical Formulations of Conformal Symmetry: Another approach is to use category theory to develop a more abstract formulation of conformal symmetry. This involves defining a category whose objects and morphisms correspond to the physical quantities and symmetries of the theory, respectively. By studying the properties of this category, I could gain insights into the behavior of conformal field theories - it's quite similar to mystical revelation.
Homotopy Theory and Conformal Field Theories: Another way to combine the conformal group and category theory is to use tools from homotopy theory to study the properties of conformal field theories. In particular, the homotopy type of a space is related to its conformal properties, and this connection can be used to develop new insights into the structure of conformal field theories.
To all of this, however, I have one very serious comment. The creator must feel and love what he does. And it must come from the depths of the mind, otherwise it cannot succeed.... There were many rebellious physicists who decided to seek their fortune in mathematics. However, I have my doubts that they have ever asked themselves for what reason they fall in love with a particular theory. Is it because they are hoping for empirical confirmation or some kind of reward? I think the reward in itself is to gain insight. I am referring to the circumstances in which all work becomes a passion.
To be honest, I had to understand a great deal about my own psyche and my emotional problems in order to delve into category theory. At the moment I am really comfortable with this framework and want to understand it thoroughly.
@John G
ReplyDelete"Bott also found that 8-dim Bott periodicity that puts a spotlight on octonions that you can break into quaternions for dual light cones.".
I see and understand this as follows (correct me if I am wrong):
In the case of 8-dimensional spaces, Bott periodicity tells us that the homotopy groups of certain spaces repeat periodically every 8 dimensions, and that the algebraic objects that appear in these periodicities are octonions.
This interests me. There was also a session once with Cassiopaeans that talked about octonions and their non-associativity. I remember when I read it, it was something that was in line with my own insight. But now I see it a little differently.
Bott periodicity can be understood in terms of the K-theory of complex vector bundles, which is a functor from the category of topological spaces to the category of abelian groups. The periodicity arises because the K-theory of a space X is related to the K-theory of its suspension SX, which is defined by attaching an extra copy of X to a cone over X. Specifically, Bott periodicity states that:
K^0(X) \cong K^8(SX)
K^1(X) \cong K^7(SX)
K^2(X) \cong K^6(SX)
K^3(X) \cong K^5(SX)
K^4(X) \cong K^4(SX)
This periodicity is related to the fact that the octonions form a division algebra over the real numbers, which is related to the exceptional Lie group G2. In fact, the periodicity can be understood in terms of the Clifford algebra associated with G2, which is an algebra that encodes the structure of spinors in 8 dimensions.
One way to think about this in category theory is to view the K-theory functor as a cohomology theory, which assigns a sequence of abelian groups to each space X, and satisfies certain axioms like homotopy invariance and the Mayer-Vietoris sequence. The periodicity then arises from the fact that the K-theory of the suspension of X is related to the K-theory of X itself, but shifted by 8 dimensions. This can be thought of as a kind of self-duality of K-theory, which is related to the duality symmetry of the octonions.
Overall, Bott periodicity provides a striking example of the deep connections between topology, algebra, and geometry, and shows how seemingly unrelated areas of mathematics can be brought together through a common theme - category theory.
I already know what I want to try! Clifford algebras and Category Theory! Today I am going to deal with this topic. Thank you for the inspiration, John!
@John G
ReplyDeleteThis morning I read about it: https://en.wikipedia.org/wiki/Mayer%E2%80%93Vietoris_sequence
This sequence arises from a technique called "excision," which allows us to remove a subset from a space without changing its homology. This is possible if the subset is "nice" in a certain sense, meaning that it satisfies certain conditions related to the topology of the space (the closure of the subset should be contained in an open subset of the space and the complement of the subset should be compact - I already understand why, but this is not the essence of the matter at the moment).
In category theory, the Mayer-Vietoris sequence can be understood as a special case of a more general concept called a "short exact sequence" in an abelian category. This is a sequence of objects and morphisms in the category that satisfies certain conditions related to homomorphisms and kernels. The Mayer-Vietoris sequence is a specific example of a short exact sequence that arises from the excision technique in algebraic topology.
I am currently still moving towards the most general hypostases (Neoplatonism). This perspective allows us to see this world not in its temporality, but as solutions to a more general equation.
I have always been bothered by time. Now I am looking for what is beyond it. I feel that with category theory I will finally be able to break free from the power of time. Just thinking about it makes time go realistically slower. Minutes pass more slowly. It's some kind of union of mystical experience and mathematical abstraction. This is no compromise. It is a total sinking in.
When the νοῦς begins to emerge and make itself visible it is able to see the world again. It is like being born again/resurrected. Physics is waiting for the νοῦς... Physics waits for love from the level of νοῦς. For someone willing to look at it in its entirety without looking at contemporary models - just some low hypostases of ψυχή...
I just 'accidentally' found this: https://www.salon24.pl/u/arkadiusz-jadczyk/1066172,jedynosc-i-istnienie-na-przykladzie-algebr-clifforda
ReplyDeleteAnd there:
„To be able to talk about Clifford algebra, one must additionally have a symmetric bilinear form on V. Something we denote by the symbol B(x,y). When x and y are columns, of numbers, then our bilinear form is of the form:
B(x,y)= x^T B y”.
I think that in terms of category theory, one could view the bilinear form as a natural transformation between two functors, which take objects in a category to matrices and elements of the base field, respectively. The bilinear form is then a natural transformation between the matrix functor and the Hom functor, which assigns to each pair of objects in the category a set of morphisms between them.
The use of the matrix representation of the bilinear form allows for an efficient and concrete representation of the algebraic structure of the Clifford algebra, but it is not necessarily essential to its definition or interpretation in terms of category theory.
What is essential seems to be the construction of the free associative algebra over the underlying vector space with a quadratic form. The Clifford algebra is then defined as a quotient of this free algebra by an ideal generated by certain relations. This construction can be formulated in a purely abstract and algebraic way without reference to matrices or bilinear forms, which highlights the intrinsic categorical structure of the Clifford algebra. The key idea is to encode the bilinear form as a degree-two symmetric multilinear map on the free algebra, which is used to define a certain quadratic relation. This relation encodes the "squares" of the vectors in the original vector space, and leads to the Clifford algebra, which is a graded associative algebra with a canonical Z_2-graded structure.
In Plotinus' philosophy, the One is the highest hypostasis or principle of reality, from which all other principles are derived. In a mathematical context, one way to view the One is as the most general algebraic structure, or the structure that has the fewest axioms or constraints.
From this perspective, we can view other algebraic structures as being derived from the One by imposing additional axioms or constraints. For example, a commutative ring can be viewed as a specialization of the One in which multiplication is commutative, while a field can be viewed as a further specialization in which every nonzero element has an inverse. Similarly, a Clifford algebra can be viewed as a specialization of the One in which a certain set of relations between elements is imposed, while other algebras, such as Lie algebras or associative algebras, impose different sets of relations.
In this way, we can view the One as a unifying principle that underlies all algebraic structures, while the other structures can be seen as further hypostases that arise from the One by imposing additional constraints. This perspective is in line with Plotinus' view that all reality is ultimately derived from the One, and that the One is the source of all unity and diversity in the Universe.
We can define a quadratic form on an octonion algebra, which is a more general type of algebra than a Clifford algebra.
I no longer question the physical sense of these considerations. If the abstraction were as high as possible, other hypostases should reveal themselves. I think it is necessary to seek the deepest abstraction we are capable of creating. This is the most exciting experience of my life!
But as a philosopher, which somehow I am also (although I don't admit it lately) I still have to think about 'eventology'. What is 'eventology'? What is the 'metaphysics of time'?
The One is decomposable. This is simply an assumption for now. Consciousness is not something we can describe mathematically. Consciousness is what causes us to be able to describe anything. This Neoplatonic „One” is precisely pure consciousness.
ReplyDeleteIn this context, I need to rethink 'eventology' and 'metaphysics of time' and I need to find the right categories for entities. So, I need to start with some initial system of axioms, with revision and with classification.
However, I cannot be afraid to surrender completely to this abstraction. Currently, I happen to be afraid that it will consume me and then reject me. Nevertheless, people are afraid of the world all their lives and run away. I don't want to run away. Such a life would be lost...
@Ark
ReplyDeleteRecommend me some new article to read. I would like something that is as abstract as possible. I would like it to be related to 'eventology' and 'metaphysics of time'. The other requirement is that it should be an article that you feel and that it should be mathematically rigorous.
Do you have any more mathematical problems that are not very difficult but are interesting and abstract?
ReplyDeleteI wonder if these mathematical structures I am writing about are not too primitive and temporal. What are the most abstract mathematical structures you know of?
ReplyDeleteWell that Bott periodicity octonion with its dual transverse-longitudinal-timelike structure is quite abstract. You can see it in personality where its a flexible-decisive-public interactionish structure. You can see it in cellular automata where it's easier to directly relate to specific multivectors in physics. Primitive idempotents can be seen on diagonals for a Hodge star map for physics or a cellular automata rule space partitioning.
ReplyDeleteThe timelike vector of the octonion is used for lots of multivectors and is maybe most temporal for the metric. The proper time/proper distance/affine one-form might kind of order your worldline events but it's not always timelike (doesn't even use the timelike vector). Maybe just counting state changes is temporal but I don't think a degree of freedom keeps track of that.
Getting to know interesting things in physics seems more like the universe somehow being nice to us even if we don't overly deserve it for real salvation reasons.
This blog compared to the earlier ones is becoming more Cass-like in multiple ways; that's probably a good thing given things like the state of the planet. On the Cass forum though there would be a network to go over this in a Gurdjieff-like way and a forum structure to take the discussion to a different topic.
ReplyDeleteEven math/physics discussions could be taken to a different topic. Topics kind of have to be focused because even if an idea is wrong, you have to stay focused to get to that realization. Maybe the blog posts here could be mirrored on the Cass forum with some place for more divergent discussion?
What I am doing is purely technical and, in fact, quite boring. Just trying to tie up some loose ends. Nothing to be excited about and rather peripheral to anybody's interest, including C's.
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