In this work first we show that the three main formulations of physics, namely, Newton’s second law of motion, Maxwell field equations of electromagnetism and Einstein field equations of gravitation can be formulated in similar covariant forms so that the formulations differ only by the nature of the geometrical objects that represent the corresponding physical entities. We show that Newton’s law can be represented by a scalar, the electromagnetic field by a symmetric affine connection or a dual vector, and the gravitational field by a symmetric metric tensor. Then with the covariant formulation for the gravitational field we can derive differential equations that can be used to construct the spacetime structures for short-lived and stable quantum particles. We show that geometric objects, such as the Ricci scalare curvature and Gaussian curvature, exhibit probabilistic characteristics. In particular, we also show that Schrödinger wavefunctions can be used to construct spacetime structures for the quantum states of a quantum system, such as the hydrogen atom. Even though our discussions in this work are focused on the microscopic objects, the results obtained can be applied equally to the macroscopic phenomena.

spacetime structures, quantum particles, general relativity, schrödinger wavefunctions, covariant formulations