X-Message-Number: 21984
Date: Sun, 15 Jun 2003 10:05:00 EDT
Subject: Science theories from infinities

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The infinite, defined as the next quantity beyond the last one in a given 

frame, is a powerful tool to invent new theories. I don't bother here about what
is beyond the infinite,  Cantor has done that with the theory of transfinite 

In the integer domain, if the next integer after N is defined as N+1, that 
is, the new value of N is  the old one and one: N:=N+1 then I have pointed out 
that N+ 1/2 (or any fraction not giving an integer) is a definition of the 

infinite. For real numbers, i from the complex field is a representation of the

Now, quantum mechanics uses indeed complex numbers and more than one century 
ago, Hamilton build the next number field, the quaternions. So, some have 
jumped to a quaternionic form of QM. 

Assume someone make an experiment giving a proof that quaternionic quantum 
mechanics is real, what can we say? A complex number can be written as a 2X2 
matrix, in the same way, a quaternion can be put as a 4X4 matrix, there are no 

complex-like numbers in the 3X3 matrix domain. Are all 2X2 matrix the picture of
a complexe? the answer is no, there are the so called zero divisors. The 

square of a ZD is zero, even if ZD is not 0. there is a simple example, with 
any real number:

0 a
0 0

Such a matrix is a definition of an infinite for the complexes. If there is 
some physics with the larger quaternions, then there will be with the simpler 
infinite. You can then use all the power of Cantor set theory to extend 
classical QM in the transfinite domain.

Another similar possibility as pointed out before is to build the transfinite 
classical mechanics. Why would it work? Simply, because classical mechanics 

is build to work in 3 dimensions and that we know of an extension working in 4,
namely Special Relativity. Any SR element taken in isolation and thrown in 

classical mechanics will form an infinite for it. The first consequence would be
to introduce an endless set of different spaces: Beyond the "basic" space E, 
there will be the function space E*, the operator space E**, ... There is a 
space e' whose the function space e'* is E, this is the fiber bundle space. 

There is a space e" whose the operator space is E, this is the Green's functions
space, and so on.

A system X can be described at a time t by a so-called "state vector" V. At 
the next instant it will be described by another state vector W. You can think 
of V and W as vectors in the phase space ( space of coordinates and impulsion 
here). A matrix U links V and W, so that: W = UV. here, U is defined on the 

E** operator space. Now a weird question: What about a matrix T such that: WT =
V ? Beware: in most cases the matrix product don't commute: WU is different of 
UW for example. Here, T acts "on the other side", in fact it is the inverse 
of an operator, that is a green function.

What I have done here?

Using the fact that classical physics can be extended into Special 

Relativity, I have used that fact to predict an infinite version of classical 
the first effect being the creation of an endless set of spaces. I have then 
used the operator space to define the flowing of time. Then using the opposite 
space, the one of Green's functions, I have built a time reversal 
"anti-operator". Ready to banking on anti-Green operators? :-)

Yvan Bozzonetti.


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