Exotic Smoothness and Quantum Gravity II: exotic R^4, singularities and cosmology

TL;DR

This paper investigates how exotic smooth structures on R^4 influence quantum gravity, path integrals, and cosmological constants, revealing their potential role in singularities and observable quantities like volume and Wilson loops.

Contribution

It provides a detailed calculation of the smoothness structure contribution to quantum gravity path integrals for exotic R^4s, linking exotic smoothness to cosmological and quantum phenomena.

Findings

Exotic R^4 structures affect quantum gravity path integrals.

Large and small exotic R^4s are constructed via different topological methods.

Exotic smoothness induces a cosmological constant in all cases.

Abstract

Since the first work on exotic smoothness in physics, it was folklore to assume a direct influence of exotic smoothness to quantum gravity. In the second paper, we calculate the "smoothness structure" part of the path integral in quantum gravity for the exotic R^4 as non-compact manifold. We discuss the influence of the "sum over geometries" to the "sum over smoothness structure". There are two types of exotic R^4: large (no smooth embedded 3-sphere) and small (smooth embedded 3-sphere). A large exotic R^4 can be produced by using topologically slice but smoothly non-slice knots whereas a small exotic R^4 is constructed by a 5-dimensional h-cobordism between compact 4-manifolds. The results are applied to the calculation of expectation values, i.e. we discuss the two observables, volume and Wilson loop. Then the appearance of naked singularities is analyzed. By using Mostow rigidity, we obtain a justification of area and volume quantization again. Finally exotic smoothness of the R^4 produces in all cases (small or large) a cosmological constant.