  In the first part of this Dissertation, we study non-perturbative aspects of
quantum electrodynamics on Riemannian manifolds by using heat kernel asymptotic
expansion techniques. Here, we established the existence of a new
non-perturbative heat kernel asymptotic expansion for a Laplace type operator
on homogeneous Abelian bundles with parallel curvature, and we evaluated
explicitly the first three coefficients. As an application, we computed the
imaginary part of the non-perturbative effective action in quantum
electrodynamics and derived a generalization of the classical Schwinger's
result for the creation of scalar and spinor particles in an electromagnetic
field induced by the gravitational field. We also discovered new infrared
divergences due to the gravitational corrections, which represents a completely
new physical effect. In the second part of the Dissertation, we studied some
aspects of a newly developed non-commutative theory of the gravitational field
called Matrix Gravity. There are two versions of Matrix Gravity, in the first
one, called Matrix General Relativity, the action functional is obtained by
generalizing the Hilbert-Einstein functional to matrix-valued quantities. In
the second one, called Spectral Matrix Gravity, the action is constructed form
the first two spectral invariants of a non-Laplace type second order partial
differential operator. For the first version, we found the dynamical equations
of the theory, while, for the second version, we computed the first
non-commutative corrections to Einstein equations in the weak deformation
limit. Furthermore, we developed the kinematics of test particles in Matrix
Gravity and found the first and second order corrections to the usual
Riemannian geodesic flow.
