When a quantum system couples to a scale-invariant environment, what form must its decoherence take? We prove that the answer is unique: under locality, Lorentz invariance, unitarity, and continuous scale invariance, the effect of any such environment is mathematically equivalent to that of an \emph{unparticle bath} -- a scale-invariant continuum of states -- characterized entirely by the scaling dimension $d_{\mathcal{U}}$ of the coupled operator. This is not a modelling choice but a consequence of conformal symmetry. All decoherence and dissipation exponents are fixed by $d_{\mathcal{U}}$ through exact consistency relations, providing falsifiable predictions independent of microscopic details. We validate the framework using multi-channel transport data from the unitary Fermi gas, where two genuinely independent observables yield a consistent $d_{\mathcal{U}} = 7/4$. We further show that quantum Ising criticality, inflationary cosmology, and high-energy astrophysical neutrinos -- spanning more than 25 orders of magnitude in energy -- are unified as specific realizations of the same structure. A decoherence phase transition at $d_{\mathcal{U}} = 5/2$, where quantum coherence is \emph{protected} rather thandestroyed at long times, is a qualitative prediction inaccessible to any memoryless dynamical description.
When a quantum system couples to a scale-invariant environment, what form must its decoherence take? We prove that the answer is unique: under locality, Lorentz invariance, unitarity, and continuous scale invariance, the effect of any such environment is mathematically equivalent to that of an \emph{unparticle bath} -- a scale-invariant continuum of states -- characterized entirely by the scaling dimension dU of the coupled operator. This is not a modelling choice but a consequence of conformal symmetry. All decoherence and dissipation exponents are fixed by dU through exact consistency relations, providing falsifiable predictions independent of microscopic details. We validate the framework using multi-channel transport data from the unitary Fermi gas, where two genuinely independent observables yield a consistent dU=7/4. We further show that quantum Ising criticality, inflationary cosmology, and high-energy astrophysical neutrinos -- spanning more than 25 orders of magnitude in energy -- are unified as specific realizations of the same structure. A decoherence phase transition at dU=5/2, where quantum coherence is \emph{protected} rather thandestroyed at long times, is a qualitative prediction inaccessible to any memoryless dynamical description.
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 dU. 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 dU. The scaling dimension governs a rich phase structure, including a thermalization transition at dU=3/2, the Ohmic boundary at dU=2, and a decoherence transition at dU=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 dU=3/2, producing 1/f noise, while the (2+1)D case yields dU≈1.413 from the conformal bootstrap. In inflationary cosmology, massless scalar and graviton baths in de Sitter spacetime give dU=2, predicting linear decoherence growth consistent with the quantum-to-classical transition. For high-energy astrophysical neutrinos, the decoherence rate Γdecoh∝B(E,TU)L5−2dU 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.
This paper proposes a self-consistent and rigorous framework for quantum cosmology—Scale-Invariant Hierarchical Quantum Cosmology (SIHQC). Its core dynamical mechanism isenvironment-induced decoherence driven by energy flow. When the energy flow is zero, theUniverse as a closed quantum system resides in a pure coherent superposition state, where allmoments, spatial positions, and quantum branches coexist completely. When the energy flowexceeds the critical threshold, the decoherence process projects the Universe onto anobservable classical effective spacetime, naturally emerging a single "present", a single spatialposition, and a definite classical history.This framework postulates that the Universe originated from a scale-invariant hierarchicalquantum initial state with a strictly self-similar hierarchical structure. This state is rigorouslydefined via the inductive limit of Hilbert spaces, which completely circumvents the divergenceproblem caused by infinite tensor products in traditional quantum cosmology. Based on thetheory of open quantum systems and the mechanism of environment-induced decoherence,this paper systematically addresses fundamental issues in quantum cosmology, including thestructure of the initial singularity, the origin of the arrow of time, and the quantum-to-classicaltransition, using energy flow as the sole driving variable. It also discusses its implications for thedynamics of cosmic expansion.A universal and constant global decoherence threshold of the Universe ε0 = 1/τP exists in theframework, uniquely determined by the Planck time. The current temperature of the CosmicMicrowave Background (CMB) is a natural manifestation of this threshold in the low-energyUniverse, rather than an artificially input parameter. This model is fully compatible withstandard quantum mechanics, open system dynamics, thermodynamics, the classical limit ofgeneral relativity, and the ΛCDM cosmological model. It only introduces two free parametersconstrained by observations (the branch number and the hierarchy activation exponent ),and provides four physical inferences with clear falsifiability
Predicting universal transport coefficients in far-from-equilibrium quantum systems remains a fundamental challenge. A paradigmatic example is the non-thermal fixed point (NTFP) of isolated Bose gases, where coherence spreads as ℓ2(t)=Cℏt/m with a universal constant C. While the scaling exponent z=2 is well established, the amplitude C has remained elusive because the underlying particle cascade n(k)∼k−4 leads to a divergent kinetic energy, threatening the very existence of a constant speed limit. Here we resolve this paradox and present the first analytical, parameter-free prediction of a universal amplitude C. A deep interplay between symmetry and dissipation is uncovered. The emergent weak U(1) symmetry at the NTFP enforces a conserved total current, forcing the low-energy phase dynamics to obey a diffusive Langevin equation with noise entering as the divergence of a stochastic current. This structure, combined with dynamical decoherence of high-momentum modes, yields a universal power-law momentum distribution f~(v)∼(1+v2)−3 (with v=kℓ) that naturally regularizes the ultraviolet divergence. From this, a parameter-free geometric baseline C=3 is obtained, independent of microscopic details. The experimental value C=3.4(3) [Martirosyan et al., Nature 647, 608 (2025)] is then shown to be quantitatively consistent with universal logarithmic corrections arising from a marginally irrelevant coupling at the fixed point. A new paradigm is thus established for predicting transport coefficients in strongly correlated non-equilibrium systems: symmetry constraints determine the low-energy effective theory, dynamical decoherence provides a natural ultraviolet completion, and scaling analysis delivers testable predictions moving beyond scaling exponents to quantitative amplitude prediction.
We demonstrate the existence of a universal, structurally unavoidable non-Markovian correction to effective exponential decay laws in open physical systems. The result is derived without assuming any fundamental dynamics, background time, spacetime structure, or microscopic model. The correction emerges solely from three generic ingredients: (i) the non-closure of correlational descriptions, (ii) projection to observable subsystems with unavoidable information loss, and (iii) spectral truncation of the induced structural operator. Under these minimal and representation-independent conditions, any effective exponential decay is necessarily accompanied by a long-time algebraic correction of the form P(t)∼e−γt(1+α/t+O(t−2)), with a fixed exponent and sign that cannot be removed without violating structural consistency. This result is independent of gravity, cosmology, and specific microscopic dynamics, and applies generically to quantum-coherent, weakly dissipative regimes commonly described by Markovian semigroup models. The prediction provides a direct, experimentally testable signature distinguishing strictly Markovian descriptions from structurally non-closed correlational frameworks.
Quantum decoherence is both the fundamental mechanism underlying the quantum-to-classical transition and a major challenge for the development of scalable nanoscale quantum technologies. This work introduces a Quantum Phase Space (QPS) framework that provides a unified geometric description of decoherence. The framework is based on a dual structure consisting of an overcomplete continuous frame of minimum-uncertainty states that defines the QPS geometry, and an orthonormal basis that satisfies the strict mathematical requirements for decoherence within the Spectrum Broadcast Structure (SBS) objectivity criterion for pointer states. Within this framework, the variance-covariance matrix of the QPS ground states serves as a universal indicator of decoherence regimes:it is time-independent for Markovian (memoryless) dynamics, and time-dependent for non-Markovian dynamics with memory and information backflow. To illustrate the formalism, the Hu-Paz-Zhang (HPZ) model is generalized to include simultaneous position and momentum couplings to the environment, leading to a generalized non-Markovian master equation characterized by a spectral-density matrix. The corresponding evolution equations for the first moments(⟨p(t)⟩,⟨x(t)⟩) and for the ground covariance matrix establish a direct connection between microscopic environmental properties, classical-like trajectories and the QPS geometry. The proposed framework provides a unified theoretical foundation for modeling decoherence in nanoscale systems involving simultaneous position and momentum interactions with the environment. The QPS framework may thus bridge fundamental theory and practical quantum engineering, offering a promising coherent pathway to understand, control, and exploit decoherence at the nanoscience frontier.
We investigate a candidate, scale-free coherence invariantΠ ≡ℏ τmd2 ,built from an inertial (or effective modal) mass m, a characteristic coherence or separationlength d, and a coherence or correlation time τ . The original motivation was the possibilitythat a single dimensionless parameter might organize coherence behaviour across a widerange of physical regimes. In this work we take a more modest and technically explicitposition.First, P we present the theoretical framework that motivates Π: a phase–ledger C(t) =i Φi(t) whose conservation, ˙C = 0, leads to local quadratic invariant manifolds in native,dimensionless coordinates. We show, via Buckingham–Π analysis, that for systems whoserelevant variables are (m, d, τ ; ℏ) there is a unique dimensionless combination of this form.Second, we test the coherence invariant hypothesis on a set of heterogeneous butconcrete examples drawn from nine “rungs”: quantum decoherence and matter-waveinterferometry, nonlocal/Bell-type experiments, near- and far-field electromagnetism,astrophysical lines, condensates, cosmic expansion, and dynamic “poke-and-return” motionin atom interferometers and Bose–Einstein condensates. The goal is not to claim a fullyestablished universal law, but to examine where the proposed invariant is (and is not)consistent with data under clearly stated choices of (m, d, τ ).Third, we confront a key limitation: in many platforms there are multiple, physicallyplausible definitions of d and τ . Different choices can shift Π by many orders of magnitude.We therefore emphasize the risk of post-hoc scale selection (“curve fitting”) and propose acanonical scale protocol for future tests: (i) state all plausible definitions of (m, d, τ ) for agiven dataset, (ii) compute Π for each, and (iii) treat the choice that minimizes deviationfrom a hypothesized invariant surface as an empirical question, not a built-in assumption.With these caveats, we find that:• Near-field matter-wave interferometers analysed using the Talbot time as the native τadmit a natural mapping in which an onset band in Π near O(10−1) appears plausible.• Cold-atom and BEC transport experiments, when expressed in terms of their trapscales and transport times, yield very large Π (deep coherence), consistent with theintuitive classification of strongly regulated motion.• In other domains (Bell tests, far-field EM, astrophysical lines, cosmology) the presentevidence is mixed or inconclusive: sensible alternative choices of (m, d, τ ) cannot yetbe ruled out, and the existence of a single cross-domain onset constant is thereforeunproven.iThe main contribution of this paper is thus not a final claim of universality but aclearly defined hypothesis, a normalization and falsifiability protocol, and a set of workedexamples illustrating both the promise and the pitfalls of coherence invariants acrossphysical scales. We highlight specific experiments and re-analyses—especially in matterwaveinterferometry, collisional decoherence, condensates, and atom interferometer/BECdynamics—that could rapidly confirm or falsify key aspects of the framework.Contents