Eine Kleine Al Gebra - Romanze: Andante (C major)

 Xmas 2022

Romanze: Andante

The 2nd movement is also a short one when compared to other contemporary pieces of the “Romantic” era. It has a gentle and slower tempo, with a recurring A section and a B section similar to the 1st movement. ...

Therefore We will go slowly and gently, in a romantic mood,  with the parametrization of the set of symmetries (or maximal positive subspaces of X) as defined in the previous post. But first of all, dear Reader:

Please do backups of your hard drives. Today. An email that I received today from a Friend contained this part:

"...everything I wrote, all the articles I had collected over the years, everything for myself, which is thousands of files, all of it crashed...

As if someone decided that it would not be enough for me to have Carcinoma, it is necessary that the Disk, on which all my Life is, is irreparably peeled off. Well, it has peeled off..."  

Backup!

Make your backup. Save your files. Save your life. Today! 

Lest us start with the star. 

Recall from the previous post:

Given an orthonormal basis X can be identified with C^m+n.

Let † denote the standard hermitian conjugate (conjugate transpose). Then, (z,z') can be written as

(z,z') = z^†J₀z'

where J₀ is the diagonal (m+n)x(m+n) matrix J₀= diag(-I_m, I_n) and I_m, I_n are the mxm and nxn unit matrices respectively. The group U(n,m) is the set of all matrices U satisfying

U^†J₀ U = J₀

Let A be an (m+n)x(m+n) matrix written in a block form as A ={{A,B},{C,D}}. We denote by A* its conjugate with respect to the indefinite scalar product (z,z'). Then A* is given by 

A* = J₀A^†J₀  

or, in block form 

A* ={{A*,-C*},{-B*,D*}}

where * on the right, applied to the blocks, denotes the ordinary hermitian conjugate. It's confusing, isn't it? Anyway...

Allelujah!

Another version here!

P.S.1 Today from Irina Eganova in Novosibirsk I received a copy of the paper by M. M. Lavrent'ev "Ad disputandum". At the end the author recommends the ideas of the late Polish nuclear physicist M. Gryzinski. Grzyzinski has his own, mostly classical,  ideas about explaining many quantum phenomena. Some time ago I have looked into Gryzinski's ideas and criticised them. I do like Lavrent'ev. He is certainly sincere. But is he right about Gryzinski?Now I will have to look at Gryzinkski's ideas again.

P.S.2 From my working desk. Here is a piece  from Lavrent'ev paper that concerns us, as we are interested in merging the two very different theories, General Relativity and Quantum Mechanics, into one new simple and clear conceptual, theoretical and mathematical framework. Lavrent'ev is quoting and discussing Ginzburg. (slightly corrected automatic translation from Russian to English below may not represent adequately all the subtleties of the text)

1. Monopolism in science: loss of logic

"The fact that I do not at all abuse the charge of pseudoscience, when it comes to ideas and constructions that I do not share, is clear, I think, from the discussion with Academician A.A. Logunov. I have a negative attitude to his criticism of general relativity and to his own relativistic theory of gravitation, I wrote about it."

Having read these weighty sounding lines of Academician A.L. Ginzburg any reader can draw a conclusion that the position of Academician A.A. Logunov is erroneous. A. A. Logunov's position is erroneous, because if it is so clearly treated negatively it means that it has been unqualified fallacy. It cannot be otherwise! So publicly to express one's negative attitude (exactly negative!) to a theory is admissible only in one case: if it is shown to be erroneous. It is admissible only in the one and only case that it is shown to be erroneous.

But then we go on to read: "At the same time, in the article [2] I specifically emphasize that since Logunov's views have not been rigorously refuted, to declare them "pseudoscience" would be unacceptable and, of course, I did not and I do not do it."

But where is the logic?! If "Logunov's views are not strictly refuted" (and there is no such thing as a non-strict refutation!), then what is the reason for being negative to them! In this case it is not only inadmissible to declare these views as pseudoscience, it is also inadmissible to propagate (which, by the way, lasts more than one year) one's own personal rejection of them. The great Plato condemned such "discussions" more than two thousand years ago. He called them "eristics” (from the Gr. eris - dispute, discord) and defined them as a purely sophistical dispute for the sake of argument, self-assertion. It is astonishing that such a gaffe was overlooked by such a representative editorial board.The editorial board of Vestnik RAN, or perhaps such a "logica" has already become the order of the day.

Translated with www.DeepL.com/Translator (free version)