Superstrings, Knots, and Noncommutative Geometry in \(E^{{\text{(}}\infty {\text{)}}} \) Space

摘要: Within a general theory, probabilisticjustification for compactification which reduces aninfinite-dimensional spacetime $$E^{{\text{(}}\infty {\text{)}}} (n = \infty )$$ to afour-dimensional one (DT n 4) isproposed. The effective Hausdorff dimension of this spaceis is given by $$\langle \dim _{\text{H}} E^{{\text{(}}\infty \rangle d_{\text{H}} 4 + \Phi ^3 ,{\text{ where }}\Phi 1/[4 ]$$ PV number and φ (√5– 1)/2 the golden mean. derivation makes use various results from knot theory,four-manifolds, noncommutative geometry, quasiperiodictiling, Fredholm operators. In addition somerelevant analogies between $$ , statistical mechanics, Jones polynomials are drawn.This allows better insight into nature theproposed compactification, associated space, thePisot–Vijayvaraghavan 1/φ3= 4.236067977 representing its dimension. This dimensionis in turn shown be capable naturalinterpretation terms invariant andthe signature four-manifolds. brings work near context Witten andDonaldson topological quantum field theory.