Timeless Quantum Field: Quantum Mechanics from Renormalization Symmetry

The mathematical structure of quantum mechanics is traditionally introduced axiomatically, with the linearity of the state space, the Hilbert space structure, and the Born rule postulated independently. In this work, we investigate whether these features can instead emerge from a fundamental symmetry principle: the absence of a preferred scale, implemented as invariance ofphysical observables under coarse-graining transformations. We formulate a framework in which physical states are represented as functionals over configuration space and are subject to a family of coarse-graining transformations forming a renormalization group (RG) semigroup. Physical observables are required to be invariant under these transformations, while dynamical operators are allowed to transform covariantly withinthe induced RG flow. Under general consistency conditions—including associativity, symmetry, continuity, and invariance under refinement—we show that the admissible composition law of states admits an additive representation up to an invertible reparametrization, leading to a linear structure of the state space in reparametrized variables. Imposing consistency of weight assignments under coarse-graining further constrains the norm, selecting a quadratic form and inducing an inner product structure. As a result, the space of states acquires the structure of a complex Hilbert space without being postulated a priori. Within this framework, probability emerges as a measure induced by the state functional. By analyzing the transformation of weight assignments under coarse-graining, we identify an effective renormalization group flow in the space of admissible probability rules and show that the quadratic rule arises as a distinguished and heuristically stable fixed point. This provides adynamical origin for the Born rule. We further analyze the semiclassical regime, in which coarse-graining suppresses interference between configuration sectors, leading to decoherence and the emergence of effective classical behavior. In this limit, classical dynamics and an effective notion of time arise from the phase of the state functional, and an approximate Schr¨odinger equation is recovered for fluctuations. These results suggest that the core structural features of quantum mechanics can be understood as universal fixed-point properties of an underlying renormalization symmetry acting on the space of states. This perspective provides a conceptual bridge between quantum mechanics and scale-invariant quantum field theories, including the Timeless Quantum Field framework.