Ivancevic_Applied-Diff-Geom

Introduction 45the Casimir effect, namely from quantum fluctuations in the string. Thesize of this contribution depends on the number of dimensions since for alarger number of dimensions, there are more possible fluctuations in thestring position. Therefore, the photon will be massless – and the theoryconsistent – only for a particular number of dimensions.The only problem is that when the calculation is done, the universe’sdimensionality is not four as one may expect (three axes of space and oneof time), but 26. More precisely, bosonic string theories are 26D, whilesuperstring and M–theories turn out to involve 10 and 11 dimensions, respectively.In bosonic string theories, the 26 dimensions come from thePolyakov equation. However, these results appear to contradict the observedfour dimensional space–time.Two different ways have been proposed to solve this apparent contradiction.The first is to compactify the extra dimensions; i.e., the 6 or 7 extradimensions are so small as to be undetectable in our phenomenal experience.The 6D model’s resolution is achieved with the so–called Calabi–Yaumanifolds (see Figure 1.5). In 7D, they are termed G 2 −manifolds. Essentiallythese extra dimensions are compactified by causing them to loopback upon themselves. A standard analogy for this is to consider multidimensionalspace as a garden hose. If the hose is viewed from a sufficientdistance, it appears to have only one dimension, its length. Indeed, thinkof a ball small enough to enter the hose but not too small. Throwing sucha ball inside the hose, the ball would move more or less in one dimension;in any experiment we make by throwing such balls in the hose, the onlyimportant movement will be one-dimensional, that is, along the hose. However,as one approaches the hose, one discovers that it contains a seconddimension, its circumference. Thus, a ant crawling inside it would move intwo dimensions (and a fly flying in it would move in three dimensions). This‘extra dimension’ is only visible within a relatively close range to the hose,or if one ‘throws in’ small enough objects. Similarly, the extra compactdimensions are only visible at extremely small distances, or by experimentingwith particles with extremely small wave lengths (of the order of thecompact dimension’s radius), which in quantum mechanics means very highenergies. Another possibility is that we are stuck in a 3+1 dimensional (i.e.,three spatial dimensions plus one time dimension) subspace of the full universe.This subspace is supposed to be a D–brane, hence this is known as abrane–world theory. In either case, gravity acting in the hidden dimensionsaffects other non–gravitational forces such as electromagnetism. In principle,therefore, it is possible to deduce the nature of those extra dimensions