Timeless Quantum Field Theory from Renormalization Symmetry: Quantum Mechanics, Gravity, and Cosmology

We propose a Timeless Quantum Field (TQF) framework in which the fundamental description of nature is globally stationary and invariant under renormalization transformations. The framework is based on three principles: global stationarity, a variational formulation of dynamics, and the absence of a preferred scale, implemented as renormalization symmetry.We show that invariance under coarse-graining in state space leads to the emergence of Hilbert space structure, linearity, and the Born rule, providing a derivation of quantum mechanics from renormalization principles. On the dynamical side, we construct a scale-invariant field theory including a unified gauge sector and quadratic curvature terms. All physical scales, including the Planck scale, arise dynamically via dimensional transmutation. From this action, we derive a Hamiltonian constraint of the form HˆTQF Ψ = 0, providing a candidate quantum gravitational dynamics consistent with the renormalization-based quantum structure. The semiclassical limit recovers Einstein equations with an emergent gravitational coupling. Applying the framework to cosmology, we obtain an inflationary scenario driven by the scalar degree of freedom associated with Rˆ2 gravity. The resulting predictions are consistent with current observations and exhibit small deviations due to renormalization group flow. In particular, we obtain ns ≈ 0.965, r ≈ 0.004, a running αs ∼ −10ˆ−4, and a running of running βs ∼ −10ˆ−5, with corrections determined by the beta functions of the underlying theory. This work provides a unified perspective in which quantum mechanics, gravitational dynamics, and cosmological structure all emerge from a single renormalization symmetry principle.