  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.
