Scale-Invariant Open Quantum Systems

We develop a complete theoretical framework for open quantum systems coupled to scale-invariant environments. We show that such environments are universally described by unparticle baths characterized by a single scaling dimension d_{mathcal{U}}. This work provides the proof of the uniqueness theorem, the formalism of the resulting non-Markovian dynamics, and applications to several physical systems. From the uniqueness theorem, we derive the non-Markovian memory kernels, the exact noise kernel including vacuum and thermal contributions, and a fractional generalization of the Caldeira-Leggett master equation for arbitrary d_{mathcal{U}}. The scaling dimension governs a rich phase structure, including a thermalization transition at d_{mathcal{U}}=3/2, the Ohmic boundary at d_{mathcal{U}}=2, and a decoherence transition at d_{mathcal{U}}=5/2 in the thermal regime, beyond which long-time quantum coherence is protected. Three realizations are studied. For the quantum Ising model at criticality, coupling to the energy operator in (1+1) dimensions gives d_{mathcal{U}}=3/2, producing 1/f noise, while the (2+1)D case yields d_{mathcal{U}}approx1.413 from the conformal bootstrap. In inflationary cosmology, massless scalar and graviton baths in de Sitter spacetime give d_{mathcal{U}}=2, predicting linear decoherence growth consistent with the quantum-to-classical transition. For high-energy astrophysical neutrinos, the decoherence rate Γ_decohpropto mathcal{B}\(E,T_{mathcal{U}}\)L^{5-2d_{mathcal{U}}} provides an observable signature of the scaling dimension. We also compare the framework with Caldeira-Leggett and Lindblad approaches, analyze the validity regimes, and discuss experimental implications for trapped-ion simulators, neutrino telescopes, and superconducting qubits.