Long time, no see.
I have come to (from physical considerations) an unproved conclusion that GFR was right in a sense of multivaluedness of W function below \( -1/e \) :
Simultaneous considerations of branches W1 and W-1 of the Lambert function (and may be other symmetric) , and , related values of particular value of h( e^pi/2) = +-I are in fact , spinors or , as Cartan states it (free quote
):
Pair of 2 complex numbers that connect 3 complex coordinates so that it leads to an isotropic (Fundamental quadratic form=0) vector in some (my addition) coordinate space.
for example, values +I and -I satisfy these equations in 3D orthogonal coordinate system if x1=0:
x1^2+x2^2+x3^2 = 0
x1=0
x2= -2I
x3= -2
\( \eta0=+I \),
\( \eta1=-I \)
These coordinates can also be projective. In this case, the solution of an Absolute conic in plane ( 3 projective coordinates characterize point in plane, x1=0 means that point is somewhere at infinity (ideal point, vanishing point) ( usually it is x3=0, but that does not matter how we assign them) will be a spinor. In case +I, -I this Absolute conic will be a circle- in fact, all circles in that projective plane as they all cross in these imaginary points.
So the actual transformation \( h(I^{1/I}) \) performs is not from a real to complex number, but from real to spinor in projective space( "restricted complex number pair with certain properties" )
Ivars
I have come to (from physical considerations) an unproved conclusion that GFR was right in a sense of multivaluedness of W function below \( -1/e \) :
Simultaneous considerations of branches W1 and W-1 of the Lambert function (and may be other symmetric) , and , related values of particular value of h( e^pi/2) = +-I are in fact , spinors or , as Cartan states it (free quote
):Pair of 2 complex numbers that connect 3 complex coordinates so that it leads to an isotropic (Fundamental quadratic form=0) vector in some (my addition) coordinate space.
for example, values +I and -I satisfy these equations in 3D orthogonal coordinate system if x1=0:
x1^2+x2^2+x3^2 = 0
x1=0
x2= -2I
x3= -2
\( \eta0=+I \),
\( \eta1=-I \)
These coordinates can also be projective. In this case, the solution of an Absolute conic in plane ( 3 projective coordinates characterize point in plane, x1=0 means that point is somewhere at infinity (ideal point, vanishing point) ( usually it is x3=0, but that does not matter how we assign them) will be a spinor. In case +I, -I this Absolute conic will be a circle- in fact, all circles in that projective plane as they all cross in these imaginary points.
So the actual transformation \( h(I^{1/I}) \) performs is not from a real to complex number, but from real to spinor in projective space( "restricted complex number pair with certain properties" )
Ivars