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Physical model of (infinite) tetration=(NON-isotropic) turbulence
I was little wrong about spinors ( though they have a relation , obviously, to this).

At least in pure imaginary case , the values obtained by infinite tetration

are propagators ( of imaginary time) and their further research (further research of analytical properties of tetration) has to be done via Green function, a 2 point function!Yep!

I think Green function can be applied sucessfully also to real and complex , may be even quaternion etc . arguments z of h(z), but I am not sure.'s_fun...dy_theory)

But there is a hitch that these things might involve 2D complex projective space (and not just complex projective plane, plane in projective space is a very misleading name.... Argand plane will not be enough. The "curvature" of complex projective 2D space can be either 1 when its analoque to Argand plane, a real or imaginary constant , or a function of a special kind.)

This "curvature" appears in expression for distances /angles for complex projective 2D space, given by F.Klein

The "area'" "curvature" of complex projective 2D space is then:

Actually, one has to take multiplication of "distance" and angle constants since they both together characterize the space, c^2 =c1*c2 . We can also define another parameter, helicity of complex projective 2D space:

Which is a division of " distance" constant c1 by angle constant c2. From this it is obvious that in general, c1/c2 is a ratio of curvature /torsion of some other space, space of all Complex Projective 2D spaces with different "distance"' and angle constants c1 and c2.

Now, if c=1 we get Argand plane. In this case c^2 = 1 , I guess, so it is either c1, c2=1, or c1, c2=-1 ( corresponding to Argand halfplanes)

If c=const we get various complex projective 2D spaces with curvature

Since in Quantum physics we have operator which is a multiplication of propagators , it seems that in more general case this operator is a function of "curvature" of corresponding Complex projective 2D space:

However, if c= function of the curvature becomes quite intricate, but it seems that that has been already solved, since in definition of projective "distance" is not unique, as crossratio can have 24 different values in 6 groups by 4 , or, alternatively 4 groups by 6.

So in any arbitrary complex projective 2D space, (not just Argand conjugate halfplanes which is just a degenerate case of general complex projective 2D space) the "distance " between any 2 points has 24 values which may not be equal depending on "curvature" of complex projective 2D space.

These 24 different values seem to be adressed by Dedekind Eta function and involved invariants and which is homogenous of order -4 and -6 respectively.

So it not so simple, but it seems that most things have been already adressed. Except few ( like turbulence of time) .

Excuse me for little stretching of topic and not being exact ( yet) , but if You can offer any help, please do. The appearance of modular functions in relation to tetration is not at all a surprise to me.

This model of curved complex projective space does not involve any other metrical issues as those arising in transformations of various invariants (crossratio is a projective invariant (invariant to Mobius transfromations) , Klein invariant is invariant to Unimodular Mobius transformations etc ( g2, g3 and corresponding Discriminant and eta function) - that means that to work with these, ordering present on real number line is not needed explicitly, at least not as a starting point. Only in case c1=1 and c2=-1 (Argand half Planes) the ordering of numbers becomes the same as on the real number line, but that is so so so specific subcase that to focus on it too much is just misleading.


Messages In This Thread
RE: Physical model of (infinite) tetration=(NON-isotropic) turbulence - by Ivars - 02/20/2009, 10:31 PM

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