Fractional dimensions in superstring duality

[Loren Booda]

Instead of superstrings having 10 unitary dimensions (6 of compactified space and 3+1 of ordinary spacetime), imagine these 6 compactified spatial dimensions being of fractal value (3/6=1/2) relative to the 3 apparent dimensions of space.

This corresponds as a dimensional duality to the 6 dimensional phase space of quantum mechanics, where each of three orthogonal dimensions is assigned both a spatial and a momentum phase (6/3=2).

Is there any merit to this pattern being a legitimate duality?

 

[sol2]

Why 10 Dimensions?

When strings vibrate in space-time, they are described by a mathematical function called the Ramanujan modular function.26 This term appears in the equation:27

[1-(D - 2)/24]
where D is the dimensionality of the space in which the strings vibrate. In order to obey special relativity 9and manifest co-variance), this term must equal 0, which forces D to be 26. This is the origin of the 26 dimensions in the original string theory.

In the more general Ramanujan modular function, which is used in current superstring theories, the twenty-four is replaced by the number eight, making D equal to 10.28

In other words, the mathematics require space-time to have 10 dimensions in order for the string theory to be self-consistent, but physicists still don’t know why these particular numbers have been selected.

http://www.ecf.utoronto.ca/~quanv/String/string9.html

 

[selfAdjoint]

Yes, and the numbers arise for physical reasons (cancelling anomalies). You can't build a consistent string theory in other than 26 dimensions or a consistent superstring theory in other than 10.

 

[Loren Booda]

I like the link - it's fairly understandable. Forget the "duality" I mentioned, but couldn't the "compactification" of the 10 minus 4 extra dimensions take place as with 6 fractal dimensional values that fit just as well our apparent 4-dimensional spacetime?

 

[selfAdjoint]

Yes, you can play tricks with compaction. See my post on the string computation of Bekenstein entropy. Here they take 10 dimensional branes and superstrings, compact five of the 9 spatial dimensions onto circles, and work in the resulting effective 5 dimensional Minkowsi spacetime. Perfectly reasonable, but note that the argument doesn't just ignore the compacted dimension; what happens in them is just as important as what happens in the "big" dimensions.

 

[Loren Booda]

Rather than imposing the familiar "compactified" dimensions on the Planck scale, I am proposing that extradimensions can be considered interstitial to integral-value spacetime by assigning them fractal values. Fractal space is just as effective as compactified space in representing the resonances underlying physics.

 

[sol2]

Rather than imposing the familiar "compactified" dimensions on the Planck scale, I am proposing that extradimensions can be considered interstitial to integral-value spacetime by assigning them fractal values. Fractal space is just as effective as compactified space in representing the resonances underlying physics.

That's a pretty tall order considering the predictability that is required from http://wc0.worldcrossing.com/WebX?14@213.PIgGcbusp83.0@.1dde4729/1 In terms of discrete states of existence, you should see more alliance with LQG. Look up quantum gravity in the Monte Carlo effect.

Yet we see where such meaning given to quantum gravity could have indeed given a complex issue to consider in the energy plot considered in this example.

it turns out that within string theory ... there is actually an identification, we believe, between the very tiny and the very huge. So it turns out that if you, for instance, take a dimension - imagine its in a circle, imagine its really huge - and then you make it smaller and smaller and smaller, the equations tell us that if you make it smaller than a certain length (its about 10-33 centimeters, the so called 'Planck Length') ... its exactly identical, from the point of view of physical properties, as making the circle larger. So you're trying to squeeze it smaller, but actually in reality your efforts are being turned around by the theory and you're actually making the dimension larger. So in some sense, if you try to squeeze it all the way down to zero size, it would be the same as making it infinitely big. ... (CSPAN Archives Videotape #125054)

what is being discribed makes sense in this as well.

The familiar extended dimensions, therefore, may very well also be in the shape of circles and hence subject to the R and 1/R physical identification of string theory. To put some rough numbers in, if the familiar dimensions are circular then their radii must be about as large as 15 billion light-years, which is about ten trillion trillion trillion trillion trillion (R= 1061) times the Planck length, and growing as the universe explands. If string theory is right, this is physically identical to the familiar dimensions being circular with incredibly tiny radii of about 1/R=1/1061=10-61 times the Planck length! There are our well-known familiar dimensions in an alternate description provided by string theory. [Greene's emphasis]. In fact, in the reciprocal language, these tiny circles are getting ever smaller as time goes by, since as R grows, 1/R shrinks. Now we seem to have really gone off the deep end. How can this possibly be true? How can a six-foot tall human being 'fit' inside such an unbelievably microscopic universe? How can a speck of a universe be physically identical to the great expanse we view in the heavens above? (Greene, The Elegant Universe, pages 248-249)

There are certain realizations that must be understood in D5 branes? The fifth dimension has been realized here?

words attributed to Hardy and Ramanujan( the issue about the taxi cab). And, just for the record:

1729 = 1 cubed + 12 cubed
or
1729 = 9 cubed + 10 cubed

Could someone explain?

 

[sol2]

There are problems with this picture.

http://www.nature.com/nature/journal/v411/n6841/images/411986af.0.jpg

Maybe if we look at displacement?

Physicists also measure the extra dimensions in terms of the energy needed to probe them. A particle accelerated to 1 trillion electron volts (TeV) has, according to standard arguments from quantum mechanics, a wave aspect with a wavelength of about 2 x 10–19 m. It can therefore explore facets of the subatomic world on that scale. Doubling the energy means seeing features half that size, and so on. So far, the smallest length scale observable with accelerators is a little greater than 10–19 m.

The idea of extra dimensions dates back to at least the 1920s. At that time, physicist Oskar Klein, building upon work by mathematician Theodor Kaluza, added a curled-up fifth dimension to the familiar universe in an ingenious but unsuccessful attempt to unite the forces of electromagnetism and gravity.

Physicists believe that the four forces—electromagnetic, weak, strong, and gravitational—were joined as a single superforce at the time of the Big Bang. In theory, they could merge only if the forces were about the same strength under conditions of high energy. However, gravity is much weaker than the others.

As some researchers today explore extra dimensions, they are on the lookout for implications regarding unification of the four forces. Other scientists striving for models that unify the forces have found extra dimensions a useful tool.

http://www.sciencenews.org/articles/20000219/bob1.asp

Brane World models only carry this unique topology solution a bit further returning in part, though often modified in present format, to Klien's solution to the unification of gravity to electromagnetism through an additional dimensional set. The beauty of this path as we should rightly call it is that it is a natural progression of Einstein's dream of a pure geometric explanation for everything we see around us in nature.

I had to tie posts together