Operator-Theoretic Structures in Gauge Quantum Field Theory

This project develops an operator-theoretic and representation-theoretic framework for addressing foundational spectral problems in non-Abelian gauge quantum field theory. The primary focus is the four-dimensional Yang–Mills mass gap problem. Rather than claiming a complete proof, the project provides a logically closed reduction of the problem to a sharply formulated lattice-level operator inequality, isolating the precise analytic obstruction that remains unresolved. The first component establishes a rigorous operator-theoretic reduction of the Yang–Mills mass gap problem using lattice Hamiltonians, strong resolvent convergence, vacuum isolation, and spectral stability arguments, entirely independent of Euclidean correlation decay or functional integral techniques. The second component develops a complementary analytic framework for energy coercivity under gauge constraints, employing representation-theoretic decompositions and multiscale operator methods. This work isolates and analyzes the coercivity mechanisms required to control low-energy spectral leakage after gauge projection, directly addressing the remaining obstruction identified in the reduction. Together, these works reformulate the Yang–Mills mass gap problem as a precise operator inequality for a vacuum-renormalized transfer matrix, providing a clear and well-defined analytic target for future resolution.