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.
We will have to talk a little bit about Lie groups ("Lie" for Sophus Lie, Norwegian mathematician, pronounced /liː/ LEE)) and Lie algebras.
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
- Nomizu, Katsumi (2. from left)
- Kobayashi, Shoshichi (2. from right)
- Klingenberg, Wilhelm P.A. (left)
Location: Oberwolfach
Author: Ferus, Dirk (photos provided by Ferus, Dirk)
Source: Dirk Ferus, Berlin
Year: 1974
Copyright: Dirk Ferus, Berlin
6. 11:35: Discipline, discipline, discipline
7. 12:00 Paying attention to reality left and right:
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.
10. 09-02-23 9:30 Fighting with the meaning of U(2) and its double cover I reminded myself about a very good resource that everybody should have: John Baez, "Introduction to Algebraic and Constructive Quantum Field Theory". It has a chapter on Clifford systems and also sections about group representations and renormalization. See also this by the same author (His uncle Albert Baez was a physicist, a co-inventor of the X-ray microscope, and father of singer and progressive activist Joan Baez. Albert interested him in physics as a child.).
Bjab -> Ark
ReplyDeleteObserwator nie może poruszać się z szybkością światła. (Tym bardziej nie może kosić z taką szybkością)
Observer can't move with light velocity? I think that non-material observers not only CAN do it, but actually they DO IT, and without special problems.
DeleteBjab -> Ark
DeleteNieistniejące byty mogą wszystko, mogą doskonale kosić nieistniejącą trawę ("observer mowing")
A dlaczego nie istnieją niematerialni obserwatorzy? Otóż fraza: "non-material observer" ma w googlu mniej niż 150 znalezisk - więc nie istnieją.
It is moving now. Thank you.
ReplyDeleteNon-material observers exist. Watch the Steven Spielberg's movie "Just like heaven (2005)", or read the book by Marc Levy "Et Si C'était Vrai" ( Polish translation: "Jak w niebie" lub "A gdybyby to była prawda".
The movie is based on the book, not that deep, but perhaps more entertaining.
Google is biased. Shows different things to different people. We watched the movie last night. It's true! Ghosts should be helped rather than just dismissed as non-existent.
I checked. It is not, in fact, Stephen Spielberg's film. He has just obtained rights. My mistake.
Deletehttps://en.wikipedia.org/wiki/Just_like_Heaven_(film)
"There IS an objective reality (though not necessarily a purely material one), contrary to what the recent fashion is telling us.".
ReplyDeleteThe question that immediately arises: How do you define objective reality?
https://corporatecoachgroup.com/blog/the-difference-between-objective-and-subjective-reality
ReplyDeleteThis definition is quite primitive and shallow. Some very important issues are not covered. It seems a bit like naive realism.
But it is good. I will soon write about objective reality from the perspective of the philosophy of Plotinus and Hegel.
https://www.youtube.com/watch?v=YYgOT2DZ-J8&ab_channel=TheAtlasSociety%2CLtd
ReplyDeleteI already like this video better than the previous link, but it's still being told from the perspective of an objective realist.
First of all, I don't agree with the statement that mysticism is a cognitive dead-end.
For my part, I recommend a different perspective on similar issues: https://en.wikipedia.org/wiki/Enneads
I am fascinated by some aspects of his philosophy.
And look here:
ReplyDeletehttps://www.researchgate.net/publication/289972304_Time_and_eternity_from_plotinus_and_boethius_to_einstein
And there:
"This article seeks to show that the views on time and eternity of Plotinus and Boethius are analogous to those implied by the block-time perspective in contemporary philosophy of time, as implied by the mathematical physics of Einstein and Minkowski. Both Einstein and Boethius utilized their theories of time and eternity with the practical goal of providing consolation to persons in distress; this practice of consolatio is compared to Pierre Hadot's studies of the "Look from Above", of the importance of concentrating on the present moment, and his emphasis on ancient philosophy as providing therapy for the soul, instead of mere abstract speculation for its own sake. In the first part of the article, Einstein's views are compared with those of Plotinus, and with the elucidation of Plotinus' views provided in the Arabic Theology of Aristotle.".
And maybe this:
ReplyDeletehttps://www.geniuses.club/genius/plotinus
I have a short portion of questions. But first some hypothetical facts.
ReplyDelete1) From the results so far, it appears that under certain assumptions, new models of relativistic quantum mechanics can be obtained. They boil down to finding a self-adjoint representation of a certain quadratic algebra with 10-generators in the inner p. Hilbert space of the described system, called HRA (Hamiltonian relativistic algebra). The assumption that this space is finite dimensional reproduces the well-known Dirac theory for particles with spin 1/2.
2) Photons do not have a "good", or exactly universally accepted, quantum mechanical description (except perhaps the BB approach). The standard model is a kind of quantum field theory, whose results (which work perfectly well at the LHC!) are obtained perturbatively without a clear mathematical structure...
3) We also managed to find a certain class of HRA representations related to irreducible representations of Lie algebras of the Poincare group with fixed mass and spin, which lead to new, previously unconsidered models of relativistic quantum mechanics.
Q1) Also, can representations with zero mass and set helicity (e.g., photon representations) be used to generate new representations?
Q2) Can such a free "Dirac" equation be solved? How to deal with the problem of spectrum outside the Hilbert space?
Q3) The renormalization problem. Mathematically, many things are very wrong with it. And this is a serious problem. Very serious...
in the inner Hilbert space* (little groups etc.).
ReplyDeleteFor quantum mechanics interpretations, objective is kind of Objective Collapse interpretations vs. Copenhagen's you can't say what is actually happening. Objective collapse does seem better. I first came across Plotinus when reading Augustine's book. That block B-series of time "nothing changes" philosophy seems correct in the sense of all possible states pre-existing eternally but then you need an A-series of time "everything changes" philosophy for continuously going from one pre-existing state to another. Besides Plotinus, it's also a Parmenides vs. Heraclitus thing.
ReplyDeleteProbably need definitions for material vs. immaterial too. All information/consciousness at the highest unbroken symmetry level would seem immaterial at least in the sense of anything known but would massless particles fall under immaterial or not? I don't even know how to use normal adjectives for things like a degenerate metric or conformal inversion.
@John G
DeleteLike Plotinus, Parmenides taught that as soon as one ceases to think about all the illusory objects that dazzle the senses, one's thoughts immediately merge with that which truly exists - the perfect, one and only true being.
However, I prefer Plotinus' philosophy. It is more sublime, and reminds me of mathematical category theory, only that it is unformalized. In addition, I see mystical insights in his writings.
Besides, I also see some similarities between Plotinus' philosophy and EEQT, this time, these changes.... They are like phase transitions. Between ways of "being". It's a bit like time itself, not just its manifestations, it brings it closer to the nature of time, to the noumena behind the manifestations called time.
Plotinus, however, continues to be beautiful. So is Dirac's equation implying the existence of antimatter (Getting to know the mind within yourself). Only, there is still a problem - renormalization.
ReplyDeleteAnd now the question: how to apply Plotinus' peculiar philosophy to the concept of renormalization? Well. Going down the ladder of entities, something must be given up. Physical theories would like to be perfect on every scale. But scale is isomorphic to the hierarchy of entities. This will not be possible. An approach from the other side is needed, because mathematically it just doesn't work out....
Just how to explain it to others....
It is necessary to sacrifice the Hetman... General Relativity has to be sacrificed. The question is: how? But it is rather this direction that physics is unfortunately about. "Unfortunately", because General Relativity is beautiful. And one more thing puzzles me... Space and anti-space. The manifold dense...
But first... Mackey's theorem...
Unitary representations of compact groups as the direct sum of irreducible representations. And the same happens in topological structures. Now it is necessary to put this together. How? This is already fairly straightforward. Category theory.
This is beautiful. This regularity seems to occur at every possible level!!! However, writing about it requires further analysis.
Does anyone know of any literature where I could find answers to the questions from yesterday's comment at 7:15 PM?
ReplyDeleteI have looked for them in Varadarajan's book, but the answers are not there.
"G is a certain Lie subgroup of GL(n; R) with n = dim M.".
ReplyDeleteA representation of a group G on a vector space V over a field K is a group homomorphism from G to GL(V), the general linear group on V and dimension of V is called the dimension of the representation.
In the case where V is of finite dimension n it is common to choose a basis for V and identify GL(V) with GL(n, K), the group of n-by-n invertible matrices on the field K. In our case K=R and V=M.
Can it be understood in this way?
Can we write like this?:
ReplyDelete\rho: G -> GL(n,R)
And then the kernel of a representation \rho of a group G is the normal subgroup of G whose image under \rho is the identity transformation, i.e.
ker(\rho): {g \in G, \rho(g) = I}.