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 $\Gamma_{\mathrm{decoh}}\propto \mathcal{B}(E,T_{\mathcal{U}})L^{5-2d_{\mathcal{U}}}$ 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.