QUANTIZATION INDUCED BY GEOMETRY

Gerald Fischer

Key words: Geometric objects, quantization, geometric observables.

1991 Math. Subject Classification: 53B50, 81Q99.

Introduction

This  contribution is guided by the idea of storing information
about a geometric system in a suitable ring of objects of
this system. This idea is mainly used in algebraic geometry
\cite{EH} where the information about algebro geometric objects,
the varieties, is carried by schemes which are
special ringed spaces $({\cal G},spec\, {\cal G})$ which consist of a
structure sheaf ${\cal G}$ and a topological space given by the
spectrum of the ring ${\cal G}$. Quite similar in quantum theory
the information about a physical system is given by a ring
of observables ${\cal R}$ represented as operators
${\cal O}$ on a Hilbert space ${\cal H}$ \cite{BLT}.
The eigenvalue equations correspond to the physical measurements
which yield elements in the ground field ${\C}$ as measurement values.
In  slight contrast to usual ring theory here we define the
spectrum $spec\, R$ of a ring $R$ as the set of its maximal left
ideals and  a  scheme  as a ringed  space which looks like
$(R\, ,\, spec\, R)$. In this context the pairs
$({\cal O}\, ,\,{\cal H})$ in quantum theory and $({\cal G}\,,\,
 spec\,{\cal G})$ in algebraic geometry represent the same idea  only
in different domains of research. We apply the idea of a scheme to
differential geometry what forces us to introduce the term of a
geometric observable and structure sheaves formed by these observables.