A Field-Theoretic Approach To Quantum Mechanics And Geometric Fields

This work presents a unified, geometry-first formulation of modern theoretical physics, treating quantum mechanics, field theory, and differential geometry not as separate subjects, but as successive layers of a single mathematical structure.The first section develops quantum mechanics from the perspective of physical boundary conditions and operator emergence, emphasizing how discrete quantum behavior arises naturally from continuous systems in limiting cases. Rather than treating quantum theory as a standalone postulate system, it is framed as a transitional regime between classical intuition and geometric field structure.The second and substantially larger section constructs the full tensorial framework underlying modern field theories. Beginning with index calculus and progressing through manifolds, connections, curvature, and the Riemann–Ricci–Weyl decomposition, the text develops general relativity as a natural consequence of coordinate-free geometric consistency. Curvature is not introduced as an assumption, but as an inevitable outcome of nontrivial parallel transport.The final chapters extend this geometric framework into modern theoretical physics, covering gravitational collapse, singularity theorems, black hole mechanics, thermodynamic interpretations of horizons, and the transition from classical geometry to quantum field behavior. Concepts such as Hawking radiation, horizon entropy, and holographic duality are presented as structural consequences of field-theoretic consistency rather than isolated physical phenomena.Across all sections, the central theme is that physical laws are constraints on geometric structure, and that what we call “forces” or “particles” are manifestations of deeper invariants in a unified mathematical field.The result is a compact but dense monograph that bridges quantum theory and gravitation through a common geometric language, aimed at readers comfortable with advanced mathematics and theoretical physics. Read more