Advanced Perspectives in Mathematical Physics and Quantum Theory
Advanced Perspectives in Mathematical Physics and Quantum Theory Abstract Quantum Field Theory (QFT) stands as one of the most profound achievements of modern physics, unifying quantum mechanics and special relativity into a single framework for describing fundamental interactions. This chapter provides a comprehensive examination of QFT’s mathematical underpinnings, from its origins in classical field theory to operator-valued distributions, Hilbert space formalism, and renormalization techniques. It explores both the successes and limitations of canonical quantization, path integrals, and algebraic approaches, emphasizing the role of functional analysis and distribution theory in providing rigor. Conceptual challenges such as ultraviolet divergences, gauge redundancy, observer dependence, and the absence of a complete non-perturbative formulation in four dimensions are critically assessed. By tracing historical developments and contemporary advances—including applications in particle...
