Timeless Quantum Field: Renormalization Group Structure of State Space and the Emergence of Quantum Mechanics

We investigate the origin of the mathematical structure of quantum theory from the principle of resolution independence within the framework of Timeless Quantum Field Theory (TQF). Starting from the absence of a preferred scale, we require that physical predictions remain invariant under changes of resolution, leading to a renormalization group (RG) action defined directly on the space of states. We show that any coarse-graining map consistent with compositional and observational requirements must be linear, idempotent, and contractive, and therefore defines a projection onto a subspace of coarse degrees of freedom. In measure-theoretic realizations, this projection is uniquely represented by conditional expectation. This structure induces a canonical decomposition of the state space and defines a notion of independence based on RG invariance. We introduce an RG-invariant scalar functional on the state space and prove that, under homogeneity and stability under refinement, it is uniquely quadratic. This quadratic functional induces a norm satisfying the parallelogram law, thereby forcing the emergence of an inner product and a Hilbert space structure. The Born rule arises as the natural probabilistic interpretation of this functional. We further exclude nonlinear coarse-graining maps, non-projective transformations, and non-Hilbert state spaces. Under explicit regularity assumptions—including measurability of representations, -finiteness when required, and mild regularity conditions on functional equations—we establish the uniqueness of the resulting kinematical structure. In contrast to existing reconstruction programs, which assume probability, measurement axioms, or Hilbert space structure, the present approach derives these features from renormalization symmetry. We conclude that quantum theory can be understood as the unique geometry of state spaces consistent under changes of resolution.