X-Message-Number: 20759
From: 
Date: Tue, 31 Dec 2002 10:01:12 EST
Subject: Non-local SUGRA , hell and time travel

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sugra
 
For the reccord, I have shown for the first time I think, how looks the 
translation formula between the simplest non-local and local force fields. It 
is something as E=mc^2, the translation equation between the mass and energy 
units. Here, I come back to the vector part of that equation, that subject 
being rather dry I suggest for a first reading to jump to the conclusion 
paragraph. It gives a new look at what is time, multi-time "dimensions" and 
supergravity, with bonus on heaven and hell :-).
 
The left hand side of the formula was the non-local one, it had four 
elements: A left derivative, a right one, a five indice local derivative and 
a 13 indice connection form or Christofeld symbol. On the right side was a 
single index ordinary-looking derivative and a "classical" 3-indice 
Christofeld:
 
Dl + Dr + Dabcde + Gfghijklmnopqr = Dw + Gxyz
 
Dabcde was the product of 3 derivatives: A left one (index a), a right one 
(index b) and the left and right derivative of the ordinary (index c) one, 
giving indice d and e. Each index can be written down or up (index or 
exposant position) depending on its transformation law in a coordinate shift. 
Clearly, a, b and c are of the same kind for example of the lower species. 
Usualy, a derived index jump to the other state: If c is written in the lower 
position, d and e must be in the upper. In the Einstein convention, if an 
index is repeated one time up and one time down, it is summed over: Dabcae = 
Dbce. So in fact, Dabcde = Dw stands for: Dabcab + Dabcba + Dabcbc + Dabccb + 
Dabcac + Dabcca. 
 
So, Dw is the sum of 6 quantities: Da + Da + Db + Db + Dc +Dc. That sum is a 
direct sum, a sum of objects with no common parts. Da, Db and Dc can be seen 
as 3 independent directions or dimensions. So, the indice a, b, c run on an 
arbitrary number of dimensions, but D is too an index limited to 3 
dimensions. 
 
In fact this is a narrow way to see the things, it assume that Da + Da = 2Da 
and so on for Db and Dc. When dealing with dimensions, one unit along x and 
one unit along x is different from one unit along x and one unit along y. The 
common way to combine dimensions is not the addition, it is the 
multiplication and there is the possibility of non-abelian (non-commutative) 
multiplication where the product A x B is different of B x A. In that case 
the way Da, Db, Dc are built matter. 
 
So, in the non-commutative product, there are two different Da ( Db, Dc). The 
number of dimension is 6. Now, that can be reduced: An object as Dabcde may 
be broken into a finite sum of objects: D1a x D2b x D3c x D4d x D5e. Each D1, 
D2,... is a vector in n dimensions. It can be seen as a variable or a unknown 
in a vector equation. The four summed over indice are then four unknown. We 
have six equations ( summing process) that is two more than necessary to 
solve the system. The six dimensional non-commutative space may then be 
reduced to four dimensions without information loss.
 
I move now to the G Christofeld object. There are 13 indice on the left and 3 
on the right, so 10 must be summed over: 5 down and 5 up.(The real object is 
somewhat more subtle, but that is of no concern here). One Down index can be 
summed with any of the 5 up indice, so there is 5 solutions. The second index 
can be summed with any of the four remaining up, so there is four 
possibilities for each one at the initial level. In all, there are factorial 
five combinations: 5x4x3x2x1 = 120 = 5! in short notation.
 
On the right side there is one up index and two down ones.  The up "survivor" 
may be any of the 6 ups, so there is 120 x 6 = 720 solutions. The same can be 
said for the first down index. The second down, being symetric with the 
first, must not count the same process over and over. The well known formula 
is: N!/(K! x (N-K)!) where N is the number of permuted indice (6) and K is 
the number of symetrical indice, here :2. So the result is: 6!/(2x4!) = 
720/48 = 15. So the compression process produces: 720 x 720 x 15 dimensions, 
that is, somewhat more than 7.5 millions. Because only 10 indice are summed 
over, there are only 10 independent equations or non-commutative dimensions. 
The number of commutative dimensions is larger: 6 solutions for the up index 
and 16 for the two down, so the total is: 16 x 6 = 96.
 
************ Conclusion*********
 
The translation formula between non-local and local fields displays some 
strange symetries. Working in any number of differentiable dimensions, the 
local derivative has always 3 commutative dimensions. It is tempting to see 
the 3-d euclidean space as that differential operator space.
 
The non-commutative version has 6 dimensions, the same number as the most 
general phase space built on the euclidean mechanics. That value reduces to 4 
when the information content is conserved in a minimum system. It seems that 
is space-time.
 
Moving the derivative from place to place, needs the Christofeld symbol. The 
number of euclidean dimensions is here 96. Modern physics seems not to know 
about that. The maximum phase space is enormous, more than 7.5 millions of 
dimensions. Recall that Special Relativity thermodynamics implies an infinite 
dimensional phase space. On experimental ground, it would not do much 
difference if that space was cut at the 7.5 million level. The 
pseudo-euclidean space-time is only ten dimensional, less than the mere 
euclidean system with 96 dimension. Ten dimensions is the world of 
Super-Gravity theory (SUGRA). SUGRA works only in 10 dimensions, but don't 
explain why that or where does they come from. Here we know: They are the 
minimum information preserving non-commutative version of quadratic 
derivative term. This is the smallest world able to see the effect of a 
moving non-local force.
 
Many papers have been produced on how to compress the 10 SUGRA dimensions 
into the 4 relativistic ones. That has not been very successful. Here is the 
answer: The relativistic 4 dimensions come from the local derivative; when 
SUGRA world comes from the moving object Christofeld. Both must be taken side 
by side, not converted one into the other. SUGRA is not the over-world, it is 
a side world.
 
What is time? Here, time is a dimensional parameter added to conserve locally 
the information of a static non-local force. General Relativity was built on 
a four dimensional continuum, with 4 equal dimensions. When moved to the 
information preserving space-time it must be no surprise to find some odd 
behavior as time travel in some geometries: It is simply the sign that GR is 
taken here outside its natural field of application. GR is not the theory of 
space-time, it is the geometrical theory of the four dimensional continuum.
 
I have said before that rest mass and disorder by unit of mass-energy 
(Boltzmann's constant) may be seen as time-like dimensions. These would 
complete the set of 6 pseudo-euclidean dimensions found in the local static 
derivative. Each dimensional parameter would be in fact an admixture of time, 
rest mass and randomness of energy. If we insist to single out a unique 
property so to have the minimum information preserving space, then we reduce 
a surface ( with mass and randomness coordinate axis) into a point. There is 
an infinite number of points in a surface, so there is an infinite number of 
time instants. Seen in time alone, the universe is infinite: no start, no 
end.
 
In a strictly mere time, particles would have no mass. This is conform to 
what predict gauge theories. Gauge fields are local fields so they have not 
to include something coming from information conservation in a non-local 
static field. A word for religious: They want a purely spiritual world, there 
they have. Adding the random bearing dimensional parameter, we get more and 
more randomness for a given energy: The temperature goes up even with low 
heat energy. This is hell :-). All of that without even looking at SUGRA 
side-world!
 
May I risk that: In Christofeld I trust.

Yvan Bozzonetti.

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