Linear fractional action of U(n,m)

 First we will review of definitions and results from previous posts.

We have discussed indefinite metric complex vector space X, endowed with a scalar product (z,z') of signature (n,m), where m,n ≥ 1. Let ei (i=1,2,...,n) be a basis in X. Then each z in X can be decomposed into the basis vectors

z = z1 e1+...+zm+n em+n

We call the basis orthonormal if the scalar product, when written in this basis, takes the form

(z,z') = - z1*z'1 -...- zm*z'm -.+ zm+1*z'm+1 +...+ zn*z'n ,         (*)

where zi* denotes the complex conjugate to zi.

Selecting an orthonormal basis in X, we identify X with Cm+n, = Cm⊕Cn. The vectors in X we then write as columns z={w,v}, with w ∈ Cm and v ∈ Cn. The scalar product (z,z') takes then the form

(z,z') = -w*w' + v*v'

where * applied to vectors in Cm and Cn denotes the hermitian conjugate.

We denote by J the set of all maximal positive subspaces of X. A subspace V of X is called positive if the scalar product (z,z') restricted to V is positive definite. By Sylvester's Law of Inertia 





each such V is necessarily n-dimensional. We are interested in the set J of all such subspaces. In a convenient parametrization by mxn complex matrices Z with Z*Z<I (that we will introduce below), the pseudo-unitary group U(n,m) will act on J by (generalized) linear fractional transformations. In mathematics J is an example of a bounded symmetric domain of type I(one).

Note: There are also infinite-dimensional generalizations - see "Bounded Symmetric Domains in Banach spaces" by Cho-Ho Chu. We will restrict ourselves to the finite-dimensional case.

Strictly speaking we are interested in the group U of all isometries of X endowed with the scalar product (z,z'). But once we have selected an orthonormal basis of X, then U becomes identified with U(n,m) - the group of all complex (m+n)x(m+n) matrices preserving the scalar product (*). Introducing the diagonal  block matrix  J₀

 J₀= diag(-I_m, I_n) ,

we have written the condition on matrices U from U(n,m) as

U^†J₀ U = J₀

_{where the dagger} † denotes the hermitian conjugate. We notice that J₀ itself  is in U(n,m).

Note: We are using bold letters to denote (m+n)x(m+n)  matrices.

We have defined  J is as the set of all maximal positive subspaces of X. Equivalently we could have defined J as the set of all linear operators J acting on X satisfying the three conditions:

1) J=J*, 

2) J²=I, 

3) the sesquilinear form (z,z')_J defined by  

(z,z')_J =(z,Jz')

 is positive definite. 

Note: Notice that J₀ ∈ J.

If V is a maximal positive subspace of X, and if W is its orthogonal complement, then J corresponding to V is defined as the unique linear operator defined as the identity on V and as minus identity on V. Conversely, if J satisfies the conditions 1),2),3), then its eigenspace belonging to the eigenvalue +1 is a maximal positive subspace of V.  This follows by an elementary linear algebra.

We have shown that every J ∈ J is of the form :

Where Z is an mxn matrix satisfying Z*Z<I (which is equivalent to ZZ*<I) uniquely determined by J. Moreover the maximal positive subspace determined by J (that is the eigensubspace belonging to the eigenvalue +1) consists of all vectors z of the form

Let now U be an isometry of X (equipped with the indefinite scalar product (z,z')). It is elementary to show that if J is in J, i.e. J satisfies the conditions 1)-3), the J'=UJU* also satisfies these conditions. It is also elementary to show that if V is the eigensubspace of J belonging to the eigenvalue +1, and if V' is the eigensubspace of J' belonging to the eigenvalue +1, then

V' = UV.

Therefore vectors of V' are again necessarily of the form

Here Z' is another mxn matrix satisfying Z'*Z"<I, determined uniquely by Z and by U. We will now find an explicit form of Z'.

To this end we write U,z and z'  in a block  form and calculate the result:

Now, if z is nonzero, then z' must be also nonzero, since U is invertible. It follows then that the nxn matrix CZ+D must be invertible. Indeed, if z is nonzero, then also v is nonzero. If there existed nonzero v such that (CZ+D)=0, then we would have (z',z')≤0, while we should have (z',z')>0. Therefore, setting v'=(CZ+D)v , we get

Since this should hold now for any v, comparing with the prvious expression for z' we get

And this is our final formula - a generalized linear fractional transformation. It automatically follows that if Z*Z<I, then Z'*Z'<I.

In the next post we will discuss what happens to this formula when we leave the safe ground and  try to do something "forbidden", namely extend the above transformation formula to Z such that Z*Z=I. For m=n=2 such Z parametrize points of the "Shilov boundary" - the compactified Minkowski spacetime of events equipped with the flat conformal causal (light-cone) structure.

P.S.1. Everything presented in this note requires only elementary linear algebra. In particular I did not use any computer algebra software, like for instance Mathematica, or Reduce, which I love to use when it helps. 

Well, I used one line of code (which I am not particularly proud about) to get the formula (8) from The Sound of Silence:

Reduce[2 x + x y + 2 x Sqrt[1 + y] == y, y]

The function graph above this formula comes with the code.

P.S.2 (31-12-22) Started reading Christopher Langan's "Introduction to Quantum Metamechanics". Observations from the first page: Langan rightly complains about the state of quantum mechanics. Mentions the need for "post-quantum mechanics" (post-QM). (I think he borrowed this term from Jack Sarfatti?) But then Langan writes about it: "Because this theory is necessarily a metatheory (or theoretical metalanguage) of QM, it is called Quantum Metamechanics or QMM)."

I do not see any necessity for post-QM to be a metatheory. What I see is the necessity of having a better theory than the standard QM.

P.S.3 31-12-22 13:00 Encouraged by Irina Eganova I have started reading "World as Space and Time" by <a href="https://en.wikipedia.org/wiki/Friedmann_equations">A.A. Friedman</a>. Beautifully written! The book (in Russian) is accessible for reading online <a href="https://reallib.org/reader?file=583994&pg=7">here</a>

P.S.4 Searching the net for Friedman and "space-time boundary" I have stumbled upon "Category Theory in Physics, Mathematics and Philosphy", Ed. Marek Kuś and Bartłomiej Skowron, Springer 2019, and there the paper by Michael Heller and Jerzy Król "Beyond the Space-Time Boundary". Interesting reading though only superficially related to my own projects. The authors claim that " The standard geometric tools on M do not allow one “to cross the boundary”. Well it all depends on what they call "standard".

Happy new Year! May your dreams come true! But while chasing your dreams pay close attention to reality left and  right!