Quantum Statistics of Indistinguishable Particles (Series III)

The conventional framework for quantum statistics is built upon gauge theory, where particle exchanges generate path-dependent phases. However, the apparent consistency of this approach masks a deeper question: is gauge invariance truly sufficient to satisfy the physical requirement of indistinguishability? We demonstrate that gauge transformations, while preserving probabilities in a formal sense, are inadequate to capture the full constraints of identical particles, thereby allowing for unphysical statistical outcomes. This critical limitation necessitates a reconstruction of the theory by strictly enforcing indistinguishability as the foundational principle, thus moving beyond the conventional topological paradigm. This shift yields a radically simplified framework in which the statistical phase emerges as a path-independent quantity, \( \alpha = e^{\pm i\theta} \), unifying bosons, fermions, and anyons within a single consistent description. Building upon the operator-based formalism of Series I and the dual-phase theory of Series II, we further present an exact and computationally tractable approach for solving N-anyon systems.