This work presents a mathematically consistent, logically falsifiable unified framework for characterizing scale-invariant topological properties of open quantum systems across atomic, biological, and cosmological scales. We define a generalized non-Hermitian topological invariant invariant under scale transformations and renormalization-group flow, construct a compatible scale-adapted de Rham cohomological complex, and establish its rigorous correspondence with the first cohomology class of the parameter manifold. The framework is motivated by structural analogies across vastly separated physical regimes, while remaining purely theoretical and ontologically neutral, providing a rigorous mathematical foundation for interdisciplinary research in mathematical physics and quantum science.
This work presents a mathematically consistent, logically falsifiable unified framework for characterizing scale-invariant topological properties of open quantum systems across atomic, biological, and cosmological scales. We define a generalized non-Hermitian topological invariant invariant under scale transformations and renormalization-group flow, construct a compatible scale-adapted de Rham cohomological complex, and establish its rigorous correspondence with the first cohomology class of the parameter manifold. The framework is motivated by structural analogies across vastly separated physical regimes, while remaining purely theoretical and ontologically neutral, providing a rigorous mathematical foundation for interdisciplinary research in mathematical physics and quantum science.
We present a mathematically rigorous framework for non-Hermitian topological phases in open quantum-biological systems, enabling room-temperature, topologically protected information transfer between quantum states and biological molecular dynamics. The theory is built on generalized non-Hermitian Chern numbers, Berry phase matching, and unitary conformal mapping, with a strict fidelity constraint of F≥99.95% and a non-Hermitian backscattering-free condition for edge-state transport. This work unifies non-Hermitian quantum mechanics, topological condensed matter physics, and biological quantum dynamics, offering testable predictions for coherence enhancement in photosynthetic complexes, topological protection in ion channels, and DNA topological information encoding. All results are derived from open axiomatic quantum theory with no confidential content, providing a new theoretical foundation for next-generation quantum-biological interfaces.
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
This paper explores the rigorous formulation of entanglement spectra within the framework of C*-algebras, proposing their utility as robust topological invariants for classifying quantum phases of matter. Entanglement spectra, derived from the eigenvalues of reduced density matrices, have emerged as powerful diagnostic tools for characterizing topological order, often revealing hidden patterns beyond conventional symmetry-breaking paradigms. By embedding quantum systems and their subsystem decompositions into the abstract mathematical structure of C*-algebras, we provide a generalized and mathematically rigorous approach to define and analyze these spectra. This C*-algebraic perspective allows for a unified treatment of finite and infinite quantum systems, including those arising in quantum field theory, and naturally incorporates the algebraic notion of locality and quantum states. We demonstrate how specific features of the entanglement spectrum, such as degeneracy patterns or gaps, can be viewed as invariants under C*-algebraic deformations that preserve the underlying topological order. This framework suggests a deep connection between the algebraic properties of observables, the structure of entanglement, and the classification of topological phases, potentially linking to established mathematical tools like K-theory. The work establishes a theoretical foundation for leveraging entanglement spectra as fundamental topological indicators within a highly abstract yet physically relevant algebraic setting.
Topological phases are generally characterized by topological invariants denoted by integer numbers. However, different topological systems often require different topological invariants to measure, and theses definition usually fail at critical points. Therefore, it's challenging to predict what would occur during the transformation between two different topological phases. To address these issues, we propose a general definition based on fidelity and trace distance from quantum information theory: manifold distance (MD). This definition does not rely on the berry connection but rather on the information of the two manifolds - their ground state wave functions. Thus, it can measure different topological systems (including traditional band topology models, non-Hermitian systems, and gapless systems, etc.) and exhibit some universal laws during the transformation between two topological phases. Our research demonstrates for different topological manifolds, the change rate (first-order derivative) or susceptibility (second-order derivative) of MD exhibit various divergent behaviors near the critical points. Compared to the strange correlator, which could be used as a diagnosis for short-range entangled states in 1D and 2D, MD is more universal and could be applied to non-Hermitian systems and long-range entangled states. For subsequent studies, we expect the method to be generalized to real-space or non-lattice models, in order to facilitate the study of a wider range of physical platforms such as open systems and many-body localization.
Topological phases are generally characterized by topological invariants denoted by integer numbers. However, different topological systems often require different topological invariants to measure, such as geometric phases, topological orders, winding numbers, etc. Moreover, geometric phases and its associated definitions usually fail at critical points. Therefore, it's challenging to predict what would occur during the transformation between two different topological phases. To address these issues, in this work, we propose a general definition based on fidelity and trace distance from quantum information theory: manifold distance. This definition does not rely on the berry connection of the manifolds but rather on the information of the two manifolds - their ground state wave functions. Thus, it can measure different topological systems (including traditional band topology models, non-Hermitian systems, and topological order models, etc.) and exhibit some universal laws during the transformation between two topological phases. Our research demonstrates that when the properties of two manifolds are identical, the distance and associated higher-order derivatives between them can smoothly transition to each other. However, for two different topological manifolds, the higher-order derivatives exhibit various divergent behaviors near the critical points. For subsequent studies, we expect the method to be generalized to real-space or non-lattice models, in order to facilitate the study of a wider range of physical platforms such as open systems and many-body localization.
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.