By Frederick S. And Frederick H. Bailey Woods
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Extra info for A Course in Mathematics Volume II
Some branches of pure mathematics have the pleasant characteristic that what seems plausible at first sight is generally true. In this theory anyone can make plausible conjectures, and they are almost always false. Nothing short of absolute rigour counts; and it is for this reason that the Analytic Theory of Numbers, while hardly a subject for an amateur, provides the finest possible discipline in accurate reasoning for anyone who will make a real effort to understand its results. Godfrey H. Hardy, 1915 [Hard2, pp.
Lapidus relativity, continues to shrink to yet smaller lengths, string theory insists upon a radical reinterpretation of what actually happens. String theory claims that all physical processes in the Garden-hose universe in which the radius of the circular dimension is shorter than the Planck length and is decreasing are absolutely identical to physical processes in which the circular dimension is longer than the Planck length and increasing! This means that as the circular dimension tries to collapse through the Planck length and head toward ever smaller size, its attempts are made futile by string theory, which turns the table on geometry.
What is more, the resulting generalization is (especially in its supersymmetric forms) free of the ailments that plague quantum general relativity. If the logic has seemed a bit thin, it is at least in part because almost all we know in string theory is a trial and error construction of a perturbative expansion. [The Feynman-Polyakov path integrals over moduli spaces of Riemann surfaces] are probably the most beautiful formulas that we now know of in string theory, yet these formulas are merely a perturbative expansion ...