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In the first part of this Dissertation, we study non-perturbative aspects ofquantum electrodynamics on Riemannian manifolds by using heat kernel asymptoticexpansion techniques. Here, we established the existence of a newnon-perturbative heat kernel asymptotic expansion for a Laplace type operatoron homogeneous Abelian bundles with parallel curvature, and we evaluatedexplicitly the first three coefficients. As an application, we computed theimaginary part of the non-perturbative effective action in quantumelectrodynamics and derived a generalization of the classical Schwinger'sresult for the creation of scalar and spinor particles in an electromagneticfield induced by the gravitational field. We also discovered new infrareddivergences due to the gravitational corrections, which represents a completelynew physical effect. In the second part of the Dissertation, we studied someaspects of a newly developed non-commutative theory of the gravitational fieldcalled Matrix Gravity. There are two versions of Matrix Gravity, in the firstone, called Matrix General Relativity, the action functional is obtained bygeneralizing the Hilbert-Einstein functional to matrix-valued quantities. Inthe second one, called Spectral Matrix Gravity, the action is constructed formthe first two spectral invariants of a non-Laplace type second order partialdifferential operator. For the first version, we found the dynamical equationsof the theory, while, for the second version, we computed the firstnon-commutative corrections to Einstein equations in the weak deformationlimit. Furthermore, we developed the kinematics of test particles in MatrixGravity and found the first and second order corrections to the usualRiemannian geodesic flow.

Non-Perturbative Aspects of Quantum Electrodynamics on Curved Space and Investigations in Matrix Gravity