“Though
boys throw stones at frogs in sport, the frogs do not die in sport,
but in earnest”,
so wrote Plutarch, the great Greek historian in
the first century A.D., in his treatise on Moralia.
When I was a boy, I was throwing stones at frogs, and I did it for sport. I did not think at that time that what was sport for me did not feel like that for the frogs. Didn’t I have a conscience? Or was it undeveloped? Or, perhaps it was simply my lack of knowledge? When I think about it today, it seems to me that nascent conscience was there; I knew, or I felt, that I was doing something wrong. I knew that this activity was something I had better not be bold about in front of my mother.
That's me, at the age 7 or 8. Innocent looking but irresponsible and dangerous. Gray hair grew much later.
Today I am sorry for what I was doing then. Today I know what I did not know then, or even if I knew, I did not know it deep enough. I had not suffered enough myself to understand the suffering of other beings; I did not know enough about the life of animals. Thus, I think, knowledge not only makes our lives richer, in the sense that it helps us in more efficient functioning in the world, it not only enriches our general culture, but also, if properly utilized, it can make us better in the moral sense.
How do we
acquire knowledge? We do it by experience, we do it by “osmosis”
– learning by watching what other successful people do and
assimilating their methods and habits - and we also acquire knowledge
through the use of scientific methods. Science is supposed to provide
us with knowledge that is well tested, true high quality knowledge.
This is in theory, but what about practice?
While I am a
scientist, I often talk to non-scientists. I realize that the image
of science from outside is not the same as that from inside.
Moreover, even when I discuss with other scientists, quite often we
disagree about the goals and about the methods used in science.
What is
better: knowledge or ignorance, hard truth or soft illusion?
Ignorance may be bliss, as some people say, but I think it is
blissful only in the same sense that addicts seek the bliss of their
drugs.
When a boy
throws stones at frogs, it can be merely ignorance, but when the boy
grows into a man and continues to behave in the same ignorant way,
there is something deeper wrong either with such individuals, or with
a world that permits – even encourages – that sort of anti-human
behavior.
P.S1. "I had not suffered enough myself to understand the suffering of other beings; " I think I did not have the necessary wirings in my brain. It took years of conscious self-observation to grow some of these necessary wirings, and then compensate their lack by unceasingly acquiring knowledge (software simulation).
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.
Göttingen and the World of Physics. An Evening with Gustav Born
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
18:53 I am also reading Langan and thinking about quantum theory.
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.
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.
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.
13:20 Yesterday there was a discussion "Jackson's "Measurement Problem" Analysis by James Keene"
"Interesting work by Jackson further takes us into the microscopic world
including some mysteries in quantum mechanics and illustrates some of
the strengths and weaknesses of the theory of continuous space and time
which has dominated thinking of physicists up to the present. The how
and why that mainstream theories fail is explored."
So, there seems to be a "measurement problem" in quantum mechanics. I was thinking: was there a measurement problem in classical mechanics? It seems that people did not have such a problem THERE. 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. This is what EEQT is about (nowadays its mutilated crawling baby version is being reinvented under the name "hybrid classical/quantum theory" or theory of "hybrid systems").
EEQT, by its very construction, reproduces ALL correct predictions of both classical and quantum mechanics, but correctly predicts also experimental data that have not been explained by these theories separately. At the same time EEQT provides the interface between classical and quantum - the sensitive boundary where the consciousness connects to. Still, the quantum part needs to be extended, since linearity is only an approximation. Thus, mathematically, we will have to deal with infinite dimensional grassmannians.
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).
"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.
"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.
And I've got a bell And I've got a song to sing All over this land
It is the song about the light and its mystery. And twistors and (later) also bi-twistors. And I am crazy about it all.
In fact I do have a hammer - the projective variety of isotropic lines in H2,2.
It occurred to me this morning that infinity is not a static concept. Like the horizon it moves when the observer, or rather the consciousness moves. It is a dynamical "thing". Yet it is objective, because consciousness is also objective. But that is for later.
For now let's start with this (one hour long) quite an interesting discussion: Einstein and the Quantum: Entanglement and Emergence
Einstein-Rosen bridge is the keyword. The bridge between the universe and its dual anti-universe. The double cover and the dynamical infinity layer between the two. Made of trapped light.
Once upon a time I even wrote a paper featuring this bridge. But as my many other projects it has not been finished, waiting for "better times". I realized yesterday that I do not even know where the degeneracy of the metric on the bridge comes from? Does it come from breaking the light-cone (conformal) structure, or from the scale factor? Will have to look at it again.
By the way: around 40 minutes into the discussion it is said that it follows from quantum theory that the amount of information in the universe is "conserved". I have no idea where this idea came from? Apparently these physicists think that they know what they are talking about. I don't.
H2,2 deserves a poem. It deserves a symphony. I am going to use my algebraic hammer and my complex projective bell to do just that. No more postponing for "later". Increasing the number of dimensions up to countable infinity and even further than that.
From the abstract of my paper: "Hanson-Regge gravitational instanton and Einstein-Rose bridge.We argue that a
region of space-time with vanishing vierbein but smooth principal connection
can be, in principle, detected by scattering experiments."
The paper features just one bridge. But if there is one, there may be infinitely many of them. And there are. Unstable gravity waves that can be read and utilized by consciousness units. If anything is objective (and there are more than one of such "things"), consciousness certainly is.
"Though we will see many similarities between the two psychoanalysts, ultimately I will argue that the quadrilaterals of Lacan operate in the opposite direction from Jung’s quaternities. They work to break open heuristic structures that, when closed in on themselves, result in dualisms and even monisms (in the case of Jungian quaternities, into a kind of monism of consciousness). "
Or better: Jung subconsciously knew about twistors? Collective unconscious?
Elements of this space are sometimes called "twistors" (when we double the dimension we get bi- twistors, containing twistors and anti-twistors, like in the Dirac equation). Twistors are spinors for the conformal group (rather than for the Lorentz group as it is for spinors).
From the "Einstein and the Quantum: Entanglement and Emergence" video:
00:43:40,880 --> 00:43:46,240
again that the information in a system
00:43:43,358 --> 00:43:48,639
any system is stored far away
...
00:45:30,480 --> 00:45:36,240
wherever the boundaries of space are
....
00:45:33,358 --> 00:45:39,358
now the space have a boundary who knows?
At the dynamical conformal infinity! On the "bridge".
Back to H2,2
We denote by (u,v) the Hermitian scalar
product, assumed to be linear in, antilinear in v. A basis e1,
e2, e3, e4 in is said to be
orthonormal if (ei,ej) = Gij, where
G is the diagonal matrix (-1,-1,+1,+1).
We denote by Q the set of all non-zero
isotropic vectors:
Q = {u ∈ H2,2: (u,u) = 0, u ≠ 0}
Q is a real manifold of (real) dimension 7. Why seven? 4 complex = 8 real. The condition (u,u) = 0 is one real condition, since (u,u) is real. 8-1 = 7.
Let C be the field of complex numbers, and let C* be the multiplicative group of complex numbers different from zero. Le R+ be the multiplicative group of positive real numbers, let U(1) be the circle group - the group of complex numbers of modulus 1. Then we have C* = R+ x U(1).
We define quotient manifolds:
Q' = Q/R+
Q'' = Q/C*
Q'' is 5-dimensional (because orbits of C* in Q are two-dimensional). Q' is 6-dimensional (because orbits of R+ in Q are one-dimensional). Q'' will be the manifold of all (compact) light rays. What is Q' ? At the moment I have no idea. Extra U(1) gauge freedom? Photon's polarization? It would be weird. Mathematically all is clear: Q' is a U(1) principal fiber bundle over Q''. But what would be the use of it in physics? If any ....
I am not 100% sure if all of the above is correct. Perhaps I am wrong. Then I will retract and fix it. In old good times internet user Bjab was kindly catching and fixing almost all my mistakes. But that is a history now - now I am responsible for all, and have to catch and fix it all by myself. Normality in life. Interests come and go.
Why is the space of all light rays 5-dimensional? Cut spacetime by the hyperplane t=0. Every light ray intersects this hyperplane in exactly one point. This gives us three coordinates. But then we have to specify which direction on the heavenly sphere the light goes to. This adds additional two (angular) coordinates. Together 3+2=5.
Kaironic fields are, in fact, fields on Q'', unlike other fields used in physics. I didn't know that when I was creating the theory of Kairons (it began there: Anticausal Currents for Massless Particles, A. Jadczyk, Preprint Wroclaw-491, Dec 1979. 11p, never published as a paper). Kaironic fields on Q' would describe charged kairons. Massless charged "particles"? These are not yet experimentally discovered (at least officially, in mainstream science). Little is also known theoretically about such strange fairy-tales creatures.
From "our" perspective light is something that interacts with our retina, the quanta. From a different perspective, that of a block universe compactified, higher density perspective, light ray, the whole ray, with its all "history" is just a point in a 5-dimensional "space". The whole history is a point. Another history - another point. Choosing between different histories is just selecting different points. Our life is just a point in the manifold of all possible lives. Applied mathematics - that's what it is. Consciousness is light trapped by gravity. Approximately.
We denote by P the canonical projection P":Q→Q''. The projection P" can be implemented in two steps: first taking the quotient with respect to R+ to obtain Q', then quotienting Q' by U(1) to obtain Q''. We denote the corresponding projections by P' and π respectively. Thus we have
P" = π∘P'
Q'' = Q'/U(1)
Q' and Q'' are compact.
In order to discuss the topology we will need a "split'. Choosing a split in , Q' itself will split. It will split into the product S3xS3. Choosing another "split" - it will split into S3xS3 differently. But what is this split? In fact we will need to operators: ":split" and "flip".
Notation: For the ease of notation in the following we will use the letter V to denote H2,2. If A is a linear operator on V, we denote by A* its adjoint with respect to the indefinite scalar product (u,v) in V.
Definition: A split is a direct sum de3composition V- ⊕ V+, where V- are two two-dimensional subspaces of V, with the scalar product (u,v) positively defined on V+ and negatively defined on V-.
If e1, e2, e3, e4 is an orthonormal basis in V, then it defines a split, where V- is the subspace spanned by e1, e2, and V+ is spanned by e3, e4. Let S(V) be the set of all splits. Every split determines a unique operator S defined by Su=-u on V- and Su=u on V+. Then S=S* and S2=1. Moreover the scalar product (u,v)S defined as
(u,v)S = (u,Sv) = (Su,v) .
is positive definite on V. Thus the set S(V) of all splits can be equivalently defined as the set of all such operators.
Definition: Give a split S, we define a flip as a linear operator F on V such that F=F*, F2=1, and anticommuting with S: SF+FS=0. Evidently such an F exchanges positive and negative eigensubspaces of S. If e1, e2, e3, e4 is an orthonormal basis and if S is the canonical split defined by this bases, then, written in a bloc matrix form, with 2x2 block matrices we have
S =
-1 0
0 1
and the canonical flip can be defined by the block matrix
F =
0 1
1 0.
Notation: From now on I will us TeX notation "^" and "_" for superscripts and subscripts respectively.
The topology of Q' =P'(Q) and Q'' = P"(Q)
It will be convenient to introduce an orthonormal basis e_i (i=1,2,3,4). Le u be in Q. Then the condition u is Q implies
|u_1|^2 + |u_2|^2-|u_3|^2-|u_4|^2 = 0
or
|u_1|^2 + |u_2|^2 = |u_3|^2-|u_4|^2 >0
The last inequality follows from the fact that we have excluded the zero vector from Q. Thus let
r^2 = |u_1|^2 + |u_2|^2 = |u_3|^2-|u_4|^2 , r>0
We can then replace u by u/r, and we have P'(u)=P'(u/r) - the same point of Q'. After that replacement we have
|u_1|^2 + |u_2|^2 = |u_3|^2-|u_4|^2 = 1.
This is a unique parametrization of the points of Q' (with respect to the chosen orthonormal basis). It is now clear that Q' is topologically (but also as a smooth manifold) homeomorphic (and even diffeomorphic) to the product of two 3-spheres:
Q' ≈ S^3 x S^3
Then Q'' = (S^3 x S^3)/U(1)_{diag}
A kind of a Hopf fibration. The group U(1) (multiplication of u by complex numbers of modulus 1) acts on both S^3 simultaneously, so we wrote it as U(1)_{diag}. We notice that this product representation depends, in fact, on the split, and not on the orthonormal basis itself. Choosing a flip allows us to identify the two copies of S^3. But it is easier to see it using an orthonormal basis adapted to the split/flip pair.
Back to the transcript from the World Science Festival video:
01:02:21,280 --> 01:02:25,119 people have been trying to put gravity
01:02:22,798 --> 01:02:27,440 and quantum mechanics together for a
01:02:25,119 --> 01:02:30,720 long time ...
01:02:40,559 --> 01:02:45,519 i very strongly believe what's going on
01:02:43,199 --> 01:02:48,639 is that quantum mechanics and gravity
01:02:45,519 --> 01:02:50,719 are simply so deeply connected so
01:02:53,280 --> 01:02:57,760 that trying to separate them into the
01:02:55,440 --> 01:02:59,280 classical theory of gravity and quantum
01:02:59,280 --> 01:03:03,839 will inevitably lead and then put them
01:03:01,679 --> 01:03:05,440 back together again separate them into
01:03:03,838 --> 01:03:06,880 these two things and then put them back
01:03:11,920 --> 01:03:16,240 except that quantum mechanics and
01:03:13,679 --> 01:03:18,159 gravity are almost the same thing
While quantization of gravity may be an absurd, physicists, who have nothing better to do do than to follow the mainstream, are organizing conferences and are all enthusiastic about it.
And what am I doing doing here, on this blog? These twistors... Is that quantum mechanics? Or is that gravity? Or is that just simple and pretty mathematics?
Perhaps an example first, and an image that should be kept in mind when happily playing with these abstract algebro-geometric structures.
Let e_i (i=1,2,3,4) be an orthonormal basis. Then u=e_1+e_3 is an isotropic vector. v=e_2+e_4 is another isotropic vector. Moreover u and v are orthogonal to each other - they span a two dimensional totally subspace. All vectors in this subspace are isotropic and mutually orthogonal. The vector u belongs to this subspace. Now consider the set of all totally isotropic subspaces to which u belongs. It is the set of all spacetime points on the light ray associated with u. Spacetime points are represented in our picture as totally isotropic subspaces of V. Light rays are represented by one-dimensional isotropic subspaces. Given light ray u, we can consider the "pencil" of all totally isotropic subspaces containing u.
These are "events" on the null geodesic (aka light ray). Given a spacetime point - totally geodesic subspace of V, we can consider the set of all isotropic lines of this subspace. ("Point" is the plane itself, not its origin - thus not the zero vector of the plane)
These are light rays crossing a given space time point - the apex of the light cone. This is a kind of "sacred" duality between points and lines of elementary Euclidean geometry. Point is an intersection of lines. Line is a collection of points.
P.S.1. 13-02-23 13:20 Perhaps I will investigate closer the problem of massless charged particles (write a paper on this subject?). If nothing in nature prevents them from existing - they should exist. But if so, what do they do?
Two papers on the subject that I am aware of:
Kurt Lechner, Electrodynamics of massless charged particles, JOURNAL OF MATHEMATICAL PHYSICS 56, 022901 (2015)
Ivan Morales, Bruno Neves, Zui Oporto, Olivier Piguet, Behaviour of Charged Spinning Massless Particles, https://arxiv.org/abs/1711.04127
P.S.2. Since C2⊕ C2
≈ C2 ⊗ C2, we may be as well plying with an entangled pair of qubits. The universe in a pair of qubits? As above so below? Or as below so above? Are those the glimpses of the physics of the future? Is that what quantum future is about?
Once in a while we have to learn something new. Once in a while we have to learn again what we have already forgotten. I need to learn again about G-structures. They are as exciting as G-spots, or even more. So, here is the piece from Kobyashi-Nomizu Vol. 1, p. 288
3. Chern [3] defined the notion of a
G-structure on a differentiable manifold M, where G is a certain
Lie subgroup of GL(n; R) with n = dim M. In our terminologies, a
G-structure on M is a reduction of the bundle of linear
frames L(M) to the subgroup G. For G = O(n), a G-structure is nothing
but a Riemannian metric given on M (see Example 5.7 of Chapter
I). For a general theory of G-structures, see Chern [3], Bernard
[1] and Fujimoto [1]. We mention some other special cases. Weyl [3] and E. Cartan [3] proved the
following. For a closed subgroup G of GL(n; R), n>=3,
the following two conditions are equivalent:
(1) G
is the group ofall matrices which preserve a certain non-degenerate quadratic form
of any signature:
(2) For every n-dimensional manifold
M andfor every reduced subbundle
P of L (M) with group G, there is a
unique torsion-free connection in P.
The implication (1) -> (2) is clear
from Theorem 2.2 of Chapter IV (in which g can be an indefinite
Riemannian metric); in fact, if G is such a group, any G-structure on
M corresponds to an indefinite Riemannian metric on M in a similar
way to Example 5.7 of Chapter I. The implication (2) -> (1) is
nontrivial. See also Klingenberg [1].
Let G be the subgroup of GL(n; R)
consisting of all matrices which leave the r-dimensional subspace Rr
of Rn invariant. A G-structure on an n-dimensional manifold M is
nothing but an r-dimensional distribution. Walker [3] proved that an
r-dimensional distribution is parallel with respect to a certain
torsion-free linear connection if and only if the distribution is
integrable. See also Willmore [1, 2].
Let G be GL(n; C) regarded as a
subgroup of GL(2n; R) in a natural manner. A G-structure on a
2n-dimensional manifold M is nothing but an almost complex structure
on M. This structure will be treated in Volume II.
This way I will recall for myself, as well as for any interested Reader, the necessary definitions.
Not every smooth manifold can carry a group structure. For instance the two-dimensional sphere. It can be acted upon by a group (for instance the 3D rotation group), but it cannot be a group itself. A good reason for it is that any n-parameter Lie group has n nowhere vanishing, nonsingular, vector fields. Such vector fields exist on the 3D sphere (that is impossible to imagine, because it needs four dimensions), but not on a two-dimensional spherical surface. The fact of life. Or "The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe)"
"Hairy ball". Any vector field on the 2D sphere must have at least one singular point. The graphics represents one of the two polarization vector fields of a photon, as seen by an observer moving with the speed of light, drawn on the unit sphere in the momentum space. Thanks are due to JS and AMS for their interest, help, and cooperation with this unfinished project.
Wikipedia has a smoother and prettier illustration of a one-pole vector field on the 2-sphere:
What we will need is a fundamental one-form on a Lie group. But to discuss this form we will need vector fields, the notion of left-invariance, and the construction of the Lie algebra of a Lie group
Not to be continued....
1. Good news this morning. After four years of work and several serious revisions (like fixing a false proof of a false theorem!) my paper "On the Bundle of Clifford Algebras Over the Space of Quadratic Forms" has finally been published today, 06/02/23. I started writing it as summary of all hat I know about Clifford algebras, but then it developed all by itself into an original publication. Its first original title was "Summa Cliffordiana". End of this saga.
Antisymmetric bilinear forms define there morphisms of Clifford algebras within Grassmann algebra. They may have something to do with the action of consciousness which does not act using what we call "force", like other forces of nature. Matter fields do not necessarily care which "gauge" we are using. Consciousness, on the other hand, does care. Thus "All is hidden in plain sight". But that is for the future.
2. 06-02-23 16:20 I decided to change my mind. I will not write about trivial matters. Makes no sense. This blog is more like a personal journal. Writing thoughts down helps in making them more clear to oneself. I have a lot of fuzzy thoughts, often contradictory, and not in an evident way. Writing these thoughts down enforces clarity and shows explicitly hidden contradictions.
3. Concerning Lie group, Lie algebras, and invariant metrics I have just discovered very nice posts from the past on another blog
Studying them now with a true interest, even though I do not necessarily agree with the way it is written there. I do not understand everything there; as it is written, it is not sufficiently clear .
4. 07-02-23 7:30 :
"In life never do as others do.”
“Either do nothing—just go to school—or do something nobody
else does."
Right now I am going to school. To teach myself about geodesics of left invariant metrics on Lie groups. I thought there was a time when I understood this subject pretty well, but now I am not so sure.
Whenever you meet an obstacle - use it to your advantage, as an opportunity to become stronger and to grow.
A.J.
5. 07-02-23 10:35 Reading now:
Katsumi Nomizu, Invariant Affine Connections on Homogeneous Spaces
7. 12:00 Paying attention to reality left and right:
Steady decline (of the readership of this blog)
8. 12:30 Freedom and free will. It is true that we can't do everything. But the theory that we do no have free will is only a theory. We are free to accept it or not. Even if we can't do everything, a every moment, there is infinitely many different choices that we are free to make. We can use this freedom or not. It is our free will. Choices that we do not make are being made, randomly or not, by "fate", these are our lost opportunities.
This post is an English rendering of an old post (May, 2019) from my now abandoned Polish blog.
I have been interested in gravity since
childhood. Then, as a student, I puzzled over Einstein's field
equations. I shared my thoughts with my colleagues - one of them
even immortalized it in a photo.
Einstein's equations of gravity
Then my interests became a bit more
serious. I received a German Humboldt scholarship, during the
scholarship I stayed in a hotel for scholarship holders, it was
Europa Kolleg,
Europa-Kolleg Hamburg
near the DESY (Deutsches
Elektronen-Synchrotron) research center in Hamburg, where I worked.
I was working on advanced gravity. In
the hotel, in the same corridor, a couple of doors away, lived
another grantee - a Russian. Since I knew Russian, every now and then
he would invite me to his room for a discussion. On the table he
would put a bottle of vodka, a loaf of bread, and we would start the
discussion. The end of the discussion I usually did not remember, as
well as the return to my room. I only remember that I was returning,
as it were, against the laws of gravity and certainly not along the
shortest (geodesic) line - as Einstein commanded.
Then I dealt
with various things. For example, the basics of quantum mechanics. I
wrote one book on theories of gravity and one on quantum mechanics and quantum fractals. Both books are apparently ahead of the era - no
one reads them. Maybe in a hundred years.
Recently, somehow, I
went from Clifford's algebras to gravity. I started thinking about
antigravity and that's how I came across "bimetric theories."
I began to correspond with authors of papers writing about these
theories and negative masses. I wrote a short note criticizing the
errors I noticed in their works. The result is that one of these
authors stopped talking to me because I did not defend him in a
dispute over priority with the other. However, I still have good
contact with the other author. This one drew my attention to yet
another theory of gravity - which seems to be becoming more and more
fashionable lately . It is called unimodular gravity. This phenomenon
escaped my attention, so I began to study what it was about. And I
found out that it goes to the problem of the cosmological
constant.
Nobel Prize-winning physicist Steven Weinberg wrote
a nice review of "The cosmological constant problem."
Weinberg's paper is from 1988, but is still cited as a classic. You
can download it from here.
There is a small problem with this
cosmological constant. We have two great theories, we (we
physicists, and we humanity) are very proud of them. These are the
General Theory of Relativity and Quantum Mechanics. We like to
highlight their great successes. Time and again we read that quantum
mechanics may be strange, may not really explain anything, but if we
take the pragmatic attitude of "shut your mouth and get down to
the calculus" - then quantum mechanics is never but never wrong.
Similarly, the General Theory of Relativity. It predicted, as it
should, the strange motion of Mercury, it predicted, as it should,
the deflection of light rays by the Sun, it predicted various lensings and such, it predicted black holes, and recently a
photograph of such one was even in the paper. So what is this little
problem? This is what Weinberg wrote about.
When we combine
the powers of these two theories together, we can calculate from here
the value of the cosmological constant responsible for the observed
course of expansion of our Universe. And we get the number as needed.
This number obtained by using our best theories, of which we are so
proud, we compared with observations. It turns out that it
doesn't quite agree.
Perhaps it is for want of other crises to
worry about that interest is increasingly centered on one veritable
crisis: the theoretical expectations for the cosmological constant
exceed observational limits by some 120 orders of magnitude.
The
theory comes out with a number 10 to the power 120 times too large. How many times
too big? By 10 times? 120 times? No, 1 with 120 zeros times too big.
Weinberg writes that we have something to worry about. We have a
CRISIS.
That's what the newspapers say, that
it's supposedly like in the picture, but what is it really
like?
From this paper we learn, among other things, that from
combining the forces of ordinary quantum theory and Einstein's theory
of gravity, it follows that our Universe should have a diameter of 31
km:
"... one can easily compute that R = 31 km [17]. So
this value drastically affects the solar system, since there would be
no solar system.
It seems prudent to look for a way out."
I'll just mention that the authors seem to show a bit of
black humor here.
To lift your spirits, let me briefly say
what the idea of unimodular gravity is all about. Normally, gravity
is described by a 4x4 symmetric matrix of gravitational potentials -
the space-time metric. These are the dynamical variables of the
ordinary theory of gravity. Unimodular gravity arises when we impose
constraints, i.e., when we find that not all these ten functions are
independent functions. Constraints are the imposition of a condition on the
determinant of a matrix, that it should have a predetermined value and
that's it. For this "predetermined value" can be taken (in
the chosen system of units) the value of 1 (variations then have
trace zero). One , odin, uno - hence also the name "unimodular".
When this is done, the cosmological constant appears only as an
integration constant, not predicted by the theory, and then we can
take its value as we are comfortable. In boxing, this is "evading
the blow". It has its own value. Even if we lose the fight, it's
only a matter of time and not right now.
It is not easy to be
a physicist. In order to survive one needs black humor and
faddishness. One such fad in our house is daily gymnastics, a
combination of Chinese, Tibetan and Western exercises. We have our
own repertoire. The picture below shows how we
exercise. I myself am lying down and hiding my face in the grass on
the top left. It's because of this cosmological constant.
And here is one of my own old attempts at proposing a new theory of gravity based on a conformal structure rather than on Riemannian metric: A
Note on Conformal Field Equations. Int. J. Theor. Phys.
18, (1979) p. 107-112.
I am going to return to this subject again and again.
P.S.1. Thursday Feb. 2. 2023, 12:40: As I wrote above, it is not easy to be a physicist. This very morning I was subjected to the direct attack - the smash by an unstable gravity wave (materialized as a variable g heavy hammer). Here is the result:
This event cause a number of secondary effects. The end result being: we won the war. The office chair has been fixed. And, at the end of the day, results is all that counts.
P.S.2. My present task: t will understand every line in the paper W. Kopczyński and L.S. Woronowicz, A geometrical approach to the twistor formalism, Rep. Math. Phys. Vol 2, pp. 35-51 (1971). It is a paper of an indescribable beauty!!!
And I will understand it at least as deep as the authors of this true pearl understood it!
