This paper proposes a quantum renormalization group (RG)-based theoretical framework aimed at cooperatively emerging spatial geometry and temporal causal structurefrom a background-independent quantum gravity ground state. Traditional renormalization methods primarily focus on coarse-graining spatial degrees of freedom, while time istreated as an external parameter. This study introduces spatio-temporal joint renormalization transformations, unifying temporal evolution with spatial scaling behavior, andconstructs RG transformations based on the spatio-temporal tensor network category,where the transformations are implemented by a series of local quantum operations thatpreserve causal structure. The research proves that at specific fixed points of the RGflow, the system spontaneously emerges with a clear arrow of time, manifested as thebreaking of scale invariance of causal correlation functions. Through numerical simulations of a (1+1)-dimensional quantum lattice model, the cooperative scaling behavior ofspacetime structure is verified. This paper provides a concrete dynamical mechanism for“time emerging from quantum processes” and offers a new path for quantum gravity andholographic duality research.
This paper proposes a quantum renormalization group (RG)-based theoretical framework aimed at cooperatively emerging spatial geometry and temporal causal structurefrom a background-independent quantum gravity ground state. Traditional renormalization methods primarily focus on coarse-graining spatial degrees of freedom, while time istreated as an external parameter. This study introduces spatio-temporal joint renormalization transformations, unifying temporal evolution with spatial scaling behavior, andconstructs RG transformations based on the spatio-temporal tensor network category,where the transformations are implemented by a series of local quantum operations thatpreserve causal structure. The research proves that at specific fixed points of the RGflow, the system spontaneously emerges with a clear arrow of time, manifested as thebreaking of scale invariance of causal correlation functions. Through numerical simulations of a (1+1)-dimensional quantum lattice model, the cooperative scaling behavior ofspacetime structure is verified. This paper provides a concrete dynamical mechanism for“time emerging from quantum processes” and offers a new path for quantum gravity andholographic duality research.
This paper proposes a theoretical and experimental scheme for experimentallysimulating the phenomenon of ”temporal structure emergence” on near-term quantum processors (such as superconducting qubit arrays). By translating the theoretical framework of ”time emerging from quantum information” into executablequantum circuits and algorithms, this research designs a set of hybrid quantumclassical algorithms based on variational quantum simulation (VQS) to simulatethe dynamics of simplified spacetime tensor networks and extract order parameterscharacterizing the arrow of time and causal structure. The experimental objectivesinclude preparing initial entangled states with no preset time direction, realizing evolution akin to renormalization group flow through parameterized quantumcircuits, and measuring causal correlation functions to observe the emergence ofthe arrow of time as a spontaneous symmetry-breaking phenomenon. The articleelaborates on the physical implementation, algorithmic steps, error analysis, andexpected results, demonstrating the feasibility of a proof-of-principle verification onnoisy intermediate-scale quantum (NISQ) devices, thereby providing an actionableexperimental entry point for testing quantum gravity and foundational theories ofspacetime
We develop a derivation of dynamics of Quantum Mechanics and measurement within a fundamentally timeless and scale-invariant framework. Building on a prior reconstruction of quantum kinematics from renormalization consistency, we extend the analysis to dynamical evolution and the measurement process using three minimal principles: a global constraint (timelessness), a variational formulation, and resolution independence. We show that relational time emerges through conditional decomposition of the global state, leading to an effective Schrödinger evolution under suitable conditions on clock subsystems. Renormalization group (RG) transformations, treated as maps on the state space induced by coarse-graining, generate an effective non-unitary dynamics for reduced states. Under standard assumptions of weak coupling and scale separation, this dynamics takes Lindblad form, providing a structural origin for decoherence. A central result of this work is the formulation of a stability selection principle: configurations contributing to observable physics correspond to those that are simultaneously stationary under the action and stable under RG flow. This principle unifies dynamical selection (via the variational structure) and structural selection (via renormalization), and provides a mechanism for the emergence of classical behavior. Measurement is described as an effective sector dynamics arising from decoherence and coarse-grained evolution. The Lindblad equation admits a stochastic representation in which the system evolves between dynamically independent sectors, leading to an effective description of definite outcomes without introducing a fundamental collapse postulate. The Born rule follows from the unique quadratic measure consistent with refinement structure and is realized dynamically within this framework. Together, these results provide a structural account under minimal assumptions of quantum dynamics, decoherence, and measurement as emergent features of a timeless, renormalization-consistent theory governed by stability across scales.
(This work presents an initial conceptual formulation of the renormalization-based approach to quantum theory within the TQF framework. A subsequent version 2 develops formulation of the Quantum reconstruction based on a linear state space, providing derivation of quantum-mechanical kinematics—including the emergence of the Hilbert space structure and the Born rule—from resolution independence. Readers primarily interested in the formal derivation are encouraged to consult the later versions, while the present work may be viewed as outlining the guiding ideas and initial physical intuition behind the quantum reconstruction.) The mathematical structure of quantum mechanics is traditionally introduced axiomatically, with the linearity of the state space, the Hilbert space structure, and the Born rule postulated independently. In this work, we investigate whether these features can instead emerge from a fundamental symmetry principle: the absence of a preferred scale, implemented as invariance ofphysical observables under coarse-graining transformations. We formulate a framework in which physical states are represented as functionals over configuration space and are subject to a family of coarse-graining transformations forming a renormalization group (RG) semigroup. Physical observables are required to be invariant under these transformations, while dynamical operators are allowed to transform covariantly withinthe induced RG flow. Under general consistency conditions—including associativity, symmetry, continuity, and invariance under refinement—we show that the admissible composition law of states admits an additive representation up to an invertible reparametrization, leading to a linear structure of the state space in reparametrized variables. Imposing consistency of weight assignments under coarse-graining further constrains the norm, selecting a quadratic form and inducing an inner product structure. As a result, the space of states acquires the structure of a complex Hilbert space without being postulated a priori. Within this framework, probability emerges as a measure induced by the state functional. By analyzing the transformation of weight assignments under coarse-graining, we identify an effective renormalization group flow in the space of admissible probability rules and show that the quadratic rule arises as a distinguished and heuristically stable fixed point. This provides adynamical origin for the Born rule. We further analyze the semiclassical regime, in which coarse-graining suppresses interference between configuration sectors, leading to decoherence and the emergence of effective classical behavior. In this limit, classical dynamics and an effective notion of time arise from the phase of the state functional, and an approximate Schr¨odinger equation is recovered for fluctuations. These results suggest that the core structural features of quantum mechanics can be understood as universal fixed-point properties of an underlying renormalization symmetry acting on the space of states. This perspective provides a conceptual bridge between quantum mechanics and scale-invariant quantum field theories, including the Timeless Quantum Field framework.
The problem of the origin of time, i.e., ”how time emerges from the foundationof quantum gravity,” is a central challenge in modern physics. This paper proposesa background-independent and computable emergence mechanism: time is not ana priori coordinate but rather a chain of partially ordered slices spontaneouslyorganized by quantum geometric states under the action of discrete renormalizationgroup flow. We construct a joint categorical framework that unifies the entanglement renormalization of spin networks and the geometric evolution of discrete Ricciflowas afunctorial evolution sequence. It is theoretically proven that when the jointevolution follows macroscopic monotonicity of entanglement entropy, this sequencenaturally generates a causal chain with a natural partial order. The macroscopic”time measure” is given by a topological invariant—the H0 barcode length in persistent homology—and its ”arrow of time” is locked by the irreversible growth ofentanglement entropy, without requiring additional thermodynamic or boundarycondition assumptions. We conduct large-scale numerical experiments on a 323periodic lattice, verifying the core theoretical prediction: the barcode lifetime Land the system’s low-energy spectral gap ∆ exhibit a universal inverse relationship, satisfying ∆ · L = 0.144(5), with a coefficient of determination R2 = 0.999.This mechanism directly leads to two observable cosmological predictions: 1) A”barcode modulation” in the low-ℓ power spectrum of the cosmic microwave background radiation, with characteristic scale ∆ℓ ≈ 3 and amplitude A ≈ 0.05; 2) Aprimordial stochastic gravitational wave background with peak frequency in the10−9–10−8 Hz range. These predictions can be tested by next-generation observational experiments such as CMB-S4 and SKA-PTA, thereby transforming the”origin of time” from a philosophical speculation into a computable, simulatable,and falsifiable physical problem.
The mathematical structure of quantum mechanics is traditionally introduced axiomatically, with the linearity of the state space, the Hilbert space structure, and the Born rule postulated independently. In this work, we investigate whether these features can emerge from a fundamental symmetry principle: the absence of a preferred scale. We formulate a framework in which physical states are represented as functionals over configuration space and are subject to a family of coarse-graining transformations forming a renormalization group (RG) semigroup. Physical observables are required to be invariant under these transformations, expressing scale neutrality at the level of state descriptions. Under general consistency conditions—including associativity, symmetry, continuity, and invariance under refinement—we show that the admissible composition law of states is uniquely additive, leading to a linear structure of the state space. Imposing consistency of weight assignments under coarse-graining further constrains the norm, selecting a quadratic form and inducing an inner product structure. As a result, the space of states acquires the structure of a complex Hilbert space without being postulated a priori. Within this framework, probability emerges as a measure induced by the state functional.By analyzing the transformation of weight assignments under coarse-graining, we define a renormalization group flow in the space of admissible probability rules and show that the quadratic rule arises as a distinguished and heuristically stable fixed point. This provides a dynamicalorigin for the Born rule. We further analyze the semiclassical regime, in which coarse-graining suppresses interference between configuration sectors, leading to decoherence and the emergence of effective classical behavior. In this limit, classical dynamics and an effective notion of time arise from the phase of the state functional, and an approximate Schr¨odinger equation is recovered for fluctuations. These results suggest that the core structural features of quantum mechanics can be understood as universal fixed-point properties of an underlying renormalization symmetry. This perspective provides a conceptual bridge between quantum mechanics and scale-invariant quantum field theories, including the Timeless Quantum Field framework.
We develop a complete geometric framework in which each quantum particle possesses its own private spacetime—a world-block—constructed from Fermi–Walker coordinates. The intrinsic spatial metric on each proper-time slice is a dynamical field governed by an action with a universal stiffness constant A0. A single world-block exhibits kinematic non-locality: the metric perturbation at a point depends on the entire wavefunction through an integral relation, while maintaining dynamic locality via causal wave equations. This duality captures the essential non-locality of quantum mechanics without violating relativistic causality. When two particles interact, their world-blocks are stitched along a common boundary, forming a single compound world-block in ordinary four-dimensional spacetime. The stitching imposes local matching conditions that yield a non-separable strain field, providing a geometric account of quantum entanglement. Measurements correspond to fixing boundary conditions on one part of the compound block; the correlated outcome on the distant part is automatically determined by the shared global geometry, not by any superluminal signal. Thus, the apparent non-locality of EPR correlations is explained as a manifestation of geometric connectivity within a single 4D manifold, consistent with Bell’s theorem because the geometry itself is non-local. In the continuum limit of many overlapping blocks, coarse-graining restores an effective local description, and Newton’s law of universal gravitation emerges exactly, with Newton’s constant given by G = 3c4/(8πA0). The model offers a unified, deterministic, and fully relativistic foundation for quantum mechanics and gravity, without invoking extra dimensions or stochastic elements. Experimental signatures in ultrafast interferometry and possible connections to dark energy are discussed. The framework aligns with recent developments in emergent gravity and provides a concrete geometric realization of spacetime from quantum entanglement.