Quantum Group Spectral Quantum Field Theory

Abstract: This paper establishes a rigorous group-theoretic foundation for quantum field theory by replacing the formal path integral with a spectral embedding integral over the unitary representation spectrum of the structural quantum group. The construction uses Haar measure, Plancherel measure, Peter–Weyl summation, and Pontryagin duality to define ordinary product-measure integrals on Gq^×QGq×Q. The framework yields a strict path integral, intrinsic UV/IR spectral regularization, a canonical spectral gauge groupoid, and a controlled derivation of the Yang–Mills action and Euler–Lagrange equation. Keywords: quantum groups; spectral measure; unitary representations; Plancherel measure; path integral; quantum field theory; lattice gauge theory; Peter–Weyl theorem; Pontryagin duality; spectral embedding.