At the Frontier of Spacetime pp 247-308 | Cite as

# Smooth Quantum Gravity: Exotic Smoothness and Quantum Gravity

## Abstract

Over the last two decades, many unexpected relations between exotic smoothness, e.g. exotic \(\mathbb {R}^{4}\), and quantum field theory were found. Some of these relations are rooted in a relation to superstring theory and quantum gravity. Therefore one would expect that exotic smoothness is directly related to the quantization of general relativity. In this article we will support this conjecture and develop a new approach to quantum gravity called *smooth quantum gravity* by using smooth 4-manifolds with an exotic smoothness structure. In particular we discuss the appearance of a wildly embedded 3-manifold which we identify with a quantum state. Furthermore, we analyze this quantum state by using foliation theory and relate it to an element in an operator algebra. Then we describe a set of geometric, non-commutative operators, the skein algebra, which can be used to determine the geometry of a 3-manifold. This operator algebra can be understood as a deformation quantization of the classical Poisson algebra of observables given by holonomies. The structure of this operator algebra induces an action by using the quantized calculus of Connes. The scaling behavior of this action is analyzed to obtain the classical theory of General Relativity (GRT) for large scales. This approach has some obvious properties: there are non-linear gravitons, a connection to lattice gauge field theory and a dimensional reduction from 4D to 2D. Some cosmological consequences like the appearance of an inflationary phase are also discussed. At the end we will get the simple picture that the change from the standard \(\mathbb {R}^{4}\) to the exotic \(R^{4}\) is a quantization of geometry.

## Notes

### Acknowledgments

I have to thank Carl for 20 years of friendship and collaboration as well numerous discussions. Special thanks to Jerzy Król for our work and many discussions about fundamental problems in math and physics. Now I understand the importance of Model theory. Many thanks to Paul Schultz for reading and corrections.

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