What is impossible takes a little longer

 To each mxn matrix Z such that

Z*Z < I

wehave associated a "symmetry" operator J defined in block matrix form as

The pseudo-unitary group U(m,n) acts on the set of symmetries by

U: J ↦ UJU*

Which translates into linear fractional action on Z, written in a block matrix form

U: Z ↦ Z'

But what if Z*Z = I? It looks as then we encounter a catastrophe. The expression for J has then 0 in the denominator! J explodes. J is in such a case impossible! What to do?

What to do in general when we are facing a catastrophe? Fasten sit belts and keep cold blood. Get smart! That is what we will do now. We adjust and continue as if nothing has happened, except that we avoid explosions. Explosions never do any good. Their effects are, unfortunately long lasting. We have to learn how to control explosions. And that is what we will do now.

We take the bull by the horns. We skip the intermediate steps and dive into the center of the cyclone. Do we really need J? Yes, it is useful, but do we need it NOW? How did we get the formula for transformations of Z? We were considering vectors of the form

When Z*Z < I, we have (z,z) ≥ 0, and = 0 if and only if v=0. Thus, if Z is fixed and v runs over Cn, vectors z generate an n-dimensional positive subspace of X.

What if Z*Z=I? Then ||Zv||2 = ||v||2 and vectors z generate a two-dimensional subspace V of X consisting of isotropic vectors: (z,z)=0. If n≤ m – it is a maximal isotropic subspace.

From now on we we will assume that n≤ m, otherwise we would have to make some adjustements. Anyway, for applications in physics we usually take m=n.

Now, if V is maximal isotropic, and if U is in U(n,m), then V'=UV is also maximal isotropic, since U is an isometry

Warning: We use bold letter U to denote isometries of the indefinite metric space X. We will use normal U to denote isometries (i.e. satisfying U*U=I) from Cn to Cm. U is represented by an (m+n)x(m+n) A,B,C,D block matrix, while U is nxm matrix. Context IS important! 

Let us find a general form of a maximal isotropic subspace, say V. V is necessarily nxn dimensional (recall that we assume n≤ m). Let z be a non-zero vector of V. Written as a column vector {w,v}, it is evident that v is a non-zero vector, otherwise we would have (z,z)<0. If z,z' are two vectors in V, then if v=v' we must have w=w', otherwise z-z' would be negative and it should be isotropic.  Thus, for z in V w is uniquely determined by v. By linearity is is easy to see that w must depend linearly on v, moreover we must have ||w||=||v||. Therefore z is of the form z={Uv,v}, where U is an isometry, i.e U*U=I. Conversely any such U detrmines a maximal isotropic subspace consisting of vectors {Uv,v}.

Now, let U be an element of U(n,m). Then Uv consists of vectors {U'v',v'}, for some other isometry U'. Writing U in a block matrix form, we get

U'v'=(AU+B)v

v'=(CU+D)v

exactly the same way as we did it before for Z.

Exactly the same way as before, for Z, we deduce that CU+D must be invertible, since if there would exist a nonzero v' such that (CU+D)v'=0, then z' would have to be of negative (z',z'), which is impossible, since it is isotropic, or zero, which is impossible since U is invertible.  Therefore the same linear fractional transformation formula holds

The maximal isotropic subspaces of X form what it is called the Shilov boundary of the domain of maximal positive subspaces considered before.

In the coming posts we will discuss its relation to our four-dimensional spacetime for m=n=2.

P.S.1. In physics we start with m=n=2. The elements of X are called twistors. They are bi-spinors for six-dimensional extended spacetime with metric of signature (4,2) - the space of conformal relativity. Our division into blocks is nothing but a representation of a bi-spinor as a pair of spinors. Our U, elements of U(2) group,  will correspond to the events of our four-dimensional space time. What is perceived as a simple point in our space time is, as we will see, a Plato projection of a null geodesic (light ray) in six dimensions. Transition from 6 to 4 (or from 4 to 6)  is like a phase transition. Then we can go from 6 to 8, from 8 to 10, and from 10 to 12 (Burkhard Heim's world  and seventh density) by adding each time two dimensions, always of signature (1,1). BTW: Distance between 4 and 12 is 8, so there is a place for octonions ....

P.S.2. And another by the way:

"Contraryto the majority of studies that have focused on health effect of redmeat, this study argues that total meat consumption, in general,benefits people health, which leads to greater longevity. This hypothesis is supported by a study conducted by Campbell advocatingthat total meat consumption may offset the detrimental effect of redmeat on people’s health. "

P.S.3. From an email I have sent to my Friend this morning:

Here are my thoughts on the subject. The graph taken from Irina's "Riemann" paper:

Graphs taken from my Kairons:

Elementary solution of the Kairon wave equations have support on hyperplanes tangent to the light cone. Therefore, in particular, the source can produce wave packets that allow for propagation of information so that  we can learn about the "true" («Истинный») state of the source.

By the way, I think that my Hilbert space carries also a natural unitary representation of the conformal group, and that the whole construction can be extended to U(2) or its double cover. But this I left for the future. For convenience I am attaching the Kairons paper again.

Still working on the explicit form of the action of SU(2,2) (or rather SO(4,2)) on the double cover. Step by step.

Best,

ark

P.S.4 

The double cover of U(2) - A+L

P.S.5 Pretty soon we will  have to ask a help from the PC, as there will be a need to do some tiresome calculations. Much of these calculations can be done with Mathematica as it can crunch both symbols and numbers, and also produce graphics. Unfortunately it is quite expensive - unless you are some kind of a student -  then it gets cheaper. For a review of Mathematica see here. Symbolic noncommutative calculations are often simpler to deal wisth using FREE Reduce computer algebra software.

P.S.6. Good news! Physicists are now rediscovering EEQT!. Except that they rename it into a commercial name: "Hybrid Quantum-Classical Master Equations" (2014), and pretend they have discovered it all by themselves, like Lajos Diosi, who knows all my papers but will avoid quoting me.
Some of them, surprisingly, still remember that before "Hybrid Systems" there was EEQT. Example: "The constraints of post-quantum classical gravity" , by Jonathan Oppenheim and Zachary Weller-Davies 2022.
But their understanding of the subject is still rather superficial. Hopefully, with time, they will take all the goodies from my papers, re-own it, and sell to the wide public.
I fully understand why physicists in my Alma Mater town of Wroclaw will never quote me - according to them I have made "wrong choices", I am a "black sheep" - they are all "white wolves", following main stream and a "proper" religion and a "proper" politics. They think that one day I will feel sorry for my choices, perhaps after my death, so they hope.

P.S.7. It so happened that I am forced to visit the forest of homotopy groups. I have zero knowledge of this forest. Beasts are hiding behind every tree. I am completely dumb. Right now I will be looking at Trautman's "Double covers of pseudo-orthogonal groups". It's all new for me! Except, perhaps, of Clifford algebras, where my knowledge is slightly more than just simple zero. Adventures, adventures. Life is certainly not boring.

P.S.7. By "chance" (of course, as always) I have just received a message from Igor Bayak on Linkedin.com, announcing his paper "Chaotic dynamics of an electron". And what do we see in the Abstract?

Abstract

First, we construct the image of the torus on the two-layer shell of the sphere and note that the isometries of the image of the torus on the sphere generate the unitary group U(2), and then we establish that, as a result of the action of the modular group on the sphere, it is factorized in such a way that the minimal (one-element) equivalence classes are given by the set of primes. 

The group U(2)!  When I am just looking for the action of SU(2,2) (in fact O(4,2)) on its double covering space!