Constructing a quantum field theory from spacetime

TL;DR

This paper explores deep connections between exotic smooth structures on spacetime, foliations, noncommutative geometry, and quantization, leading to a novel construction of quantum field theory frameworks from geometric and topological data.

Contribution

It introduces a new approach linking exotic R^4 structures, foliations, and noncommutative algebras to construct quantum field theory models via geometric quantization.

Findings

Relation between exotic R^4 and codimension-1 foliations characterized by Godbillon-Vey invariant

Construction of a hyperfinite III_1 factor algebra from foliation leaf spaces

Quantization of Poisson algebra leading to skein algebra isomorphic to hyperfinite factor II_1

Abstract

The paper shows deep connections between exotic smoothings of a small R^4 (the spacetime), the leaf space of codimension-1 foliations (related to noncommutative algebras) and quantization. At first we relate a small exotic R^4 to codimension-1 foliations of the 3-sphere unique up to foliated cobordisms and characterized by the real-valued Godbillon-Vey invariant. Special care is taken for the integer case which is related to flat PSL(2,R)-$bundles. Then we discuss the leaf space of the foliation using noncommutative geometry. This leaf space contains the hyperfinite III_1 factor of Araki and Woods important for quantum field theory (QFT) and the I_{\infty} factor. Using Tomitas modular theory, one obtains a relation to a factor II_{\infty} algebra given by the horocycle foliation of the unit tangent bundle of a surface S of genus g>1. The relation to the exotic R^4 is used to construct the (classical) observable algebra as Poisson algebra of functions over the character variety of representations of the fundamental group \pi_{1}(S) into the SL(2,C). The Turaev-Drinfeld quantization (as deformation quantization) of this Poisson algebra is a (complex) skein algebra which is isomorphic to the hyperfinite factor II_{1} algebra determining the factor II_{\infty}=II_{1}\otimes I_{\infty} algebra of the horocycle foliation. Therefore our geometrically motivated hyperfinite III_1 factor algebra comes from the quantization of a Poisson algebra. Finally we discuss the states and operators to be knots and knot concordances, respectively.