Geometric algebras and the foundations of quantum theory.
The difficulties associated with the quantization of the gravitational field suggests a
modification of space-time is needed. For example at suffici~ly small length scales the
geometry of space-time might better discussed in terms of a noncommutative algebra.
In this thesis we discuss a particular example of a noncommutative algebra, namely
the symplectic Schonberg algebra, which we treat as a geometric algebra. Thus our
investigation has some features in common with recent work that explores how geometry
can be formulated in terms of noncommutative structures.
The symplectic Schonberg algebra is a geometric algebra associated with the covariant
and the contravariant vectors of a general affine space. The "embedding" of this space
in a noncommutative algebra leads us to a structure which we regard as a noncommutative
affine geometry. The theory in question takes us naturally to stochastic elements
without the usual ad-hoc assumptions concerning measurements in physical ensembles
that are made in the usual interpretation of quantum mechanics. The probabilistic nature
of space is obtained purely from the structure of this algebra. As a consequence,
geometric objects like points, lines and etc acquire a kind of fuzzy character. This
allowed us to construct the space of physical states within the algebra in terms of its
minimum left-ideals as was proposed by Hiley and Frescura [1J. The elements of these
ideals replace the ordinary point in the Cartesian geometry.
The study of the main inner-automorphisms of the algebra gives rise to the representation
of the symplectic group of linear classical canonical transformations. We show that
this group acts on the minimum left-ideal of the algebra and in this case manifests
itself as the metaplectic group, i.e the double covering of the symplectic group. Thus
we are lead to the theory of symplectic spinors as minimum left-ideals in exactly the
same way as the orthogonal spinors can be formulated in terms of minimum left-ideals
in the Clifford algebra .. The theory of the automorphisms of the symplectic Schonberg
algebra allows us to give a geometrical meaning to integral transforms such as: the
Fourier transform, the real and complex Gauss Weierstrass transform, the Bargmann (3)
transform and the Bilateral Laplace transform. We construct a technique for obtaining
a realization of these algebraic transformations in terms of integral kernels. This gives immediately the Feynmann propagators of conventional non-relativistic quantum
mechanics for Hamiltonians quadratic in momentum and position. This then links
our approach to those used in quantum mechanics and optics. The link between the
theory of this noncommutative geometric algebra and the theory of vector bundles is