As I moved through the thinking of those extra dimensions it became apparent to me that the conceptualization of that distance scale was a strange world indeed. How, if we had accept the move to non-euclidean views could we not of accepted the consequences of this move?
Dazzled with the amazing properties of this new mathematical realm, everything seemed a bit magical, as if, experiencing for the first time a taste that is strange indeed? How would I recognize this strange dynamical world, if I had not understood this move to include the geometry that Kaluza and Klien adopted, to gather together another reality of photon engagement with that of gravity?
Fig. 1. In quantum chromodynamics, a confining flux tube forms between distant static charges. This leads to quark confinement – the potential energy between (in this case) a quark and an antiquark increases linearly with the distance between them.
So at the same time you had this distant measure, how could we resolve what was happening between those two points?
Without some supersymmetrical reality(supergravity) how could any point emerge from the brane if it did not recognize the evolution of those dimensions?
So how does this point expand? This is a simple enough question?
A theorem which is valid for a geometry in this sequence is automatically valid for the ones that follow. The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.
In the above picture Michael Duff draws our attention too, I was drawn to the same principals that Klein demonstrated in his ideas of projective geometry, as the dimensions are revealed?
IN this effort and recognition of appropriate geometry, I had wondered, that if the same consistancy with which these two had demonstrated the principals, euclidean
postulates fell in line, as a basis of this method of applicabilty? Does one now see this thread that runs through the geometry?
Having accepted the road travelled to GR we have come to recognize the royal road has lead us to a strange world indeed. First it was Reimann with Gauss looking over his shoulder, and Maxwell joining Faraday in this celebration, with Einstein bringing all the happy go lucky, into a fine example of what has been implied by the harmonious nature, structure of strings in concert?
But I am not happy yet. If one could not see what was happening between those two points, what’s the use of talking any math, without the co-existance of the physics?
The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This, in fact, also gives rise to quantization of charge, as waves directed along a finite axis can only occupy discrete frequencies. (This occurs because electromagnetism is a U(1) symmetry theory and U(1) is simply the group of rotations around a circle).