This paper explores the non–coarse-grained regime of ordered dynamics, revealing a naturally discrete structure underlying quantum gravitational behavior. Discreteness arises from finite record resolution and bounded influence rather than imposed lattice assumptions. Keywordsquantum gravity; discreteness; operational structure; non-coarse-grained dynamics
This paper explores the non–coarse-grained regime of ordered dynamics, revealing a naturally discrete structure underlying quantum gravitational behavior. Discreteness arises from finite record resolution and bounded influence rather than imposed lattice assumptions. Keywordsquantum gravity; discreteness; operational structure; non-coarse-grained dynamics
We propose a novel operator-based formulation of quantum gravity grounded in two foundational principles: a discrete causal lattice and algebraic microcausality. Departing from traditional continuum approaches and wavefunction-based quantum mechanics, this framework models spacetime and matter as emergent phenomena arising from the algebraic structure of displacement operators. In this first part of a two-part series, we construct the foundational framework and demonstrate how key features of quantum mechanics—such as the uncertainty principle, de Broglie relations, and entanglement—emerge naturally without invoking wavefunctions, path integrals, or metric-based geometry. Operator non-commutativity on the causal lattice gives rise to a self-consistent quantum structure with natural ultraviolet finiteness, intrinsic time directionality, and a microcausal interpretation of measurement. This foundational part lays the groundwork for gravitational dynamics, cosmology, and the grand unification principles of gravity and the Standard Model to be explored in the sequel to quantum gravity part II.
Understanding how classical physics emerges from quantum mechanics remains a central problem in the foundations of physics. Here we derive a classical limit from finite-resolution measurements, modeled by continuous coarse-grained POVMs. When the resolved phase-space area is large compared with Planck's constant, the accessible statistics of any quantum state admit an effective classical description: coarse-grained observables become approximately jointly measurable and the induced probability density is positive. We derive the exact evolution equation for this density and show that, in the strong coarse-graining regime, its non-Liouville corrections are suppressed up to an Ehrenfest time, resulting in classical Hamiltonian flow generated by a Hamiltonian smoothed over the measurement cell. When the Hamiltonian varies negligibly across such a cell, the smoothed Hamiltonian reduces to the classical Hamiltonian whose quantization produced the quantum dynamics, thereby closing the quantization--classical-limit loop. Repeated finite-resolution measurements then generate stochastic records confined, with high probability, to tubes around classical trajectories. Our results provide a unified operational framework for the quantum-to-classical transition in microscopic and macroscopic systems, and establish the consistency of the quantization--classical-limit cycle.
The Generalized No-Chaos TheoremBounded Complexity Growth at Any Fixed Coarse-GrainingWithin the Quantum Gravity Consistency Framework (QGCF) - any finite-dimensional, closed, unitarily evolving system satisfying (QGCF1)-(QGCF4) - three logically distinct results are established. First, ontic chaos is impossible in three senses: spectral quasiperiodicity, Poincaré recurrence, and exact vanishing of the asymptotic quantum Lyapunov exponent λL (§5). Second, dynamical (Lyapunov/0-1-test) chaos at any FIXED coarse-graining scale ℓ is also impossible, unconditionally - a direct corollary of the same almost-periodicity that gives λL=0 (§7); the one established quantum-chaos diagnostic that survives this obstruction is level (spectral) statistics, checked preliminarily against a non-integrable/integrable benchmark pair (§7.3). Third, and logically independent of the first two: no observer realized as a proper subsystem can simultaneously reconstruct the complete global state from its own accessible data and evolve under an autonomous unitary law, once genuinely (entangling-ly) coupled to the rest of the system (§8) - a consequence of purification non-uniqueness (Hughston-Jozsa-Wootters) and the non-unitary character of open marginal dynamics, requiring no assumption beyond a generic (non-fine-tuned) boundary coupling. The two failures named in that theorem arise from the same physical mechanism - entanglement across the observer's own boundary - but are shown not to be strictly the same fact, only generically co-occurring. Thermalization of coarse blocks (Calabrese-Cardy growth to a Page-value plateau) is real, established physics, distinct from all of the above, and is presented (§9) together with the timescale τ*(ℓ) it requires, and its role as the only available justification for treating the system's effective Hamiltonian Ĥeff as possessing a meaningful temperature Tc, given that the axioms admit no literal external bath. A note on generality. QGCF does not require a literal discrete graph. (QGCF3) is stated as a disjunction: a discrete, graph-based realization (QGCF3a), giving the sharper results of §8.2 and §8.6.2 exactly; or a continuum realization with only a locality/metric structure (QGCF3b), giving the standard, approximately-causal Lieb-Robinson bound instead. Everything else in this document §5-§7, §8.3-§8.6.1, §9 - depends only on finite-dimensionality and unitarity (QGCF1-QGCF2), or on (QGCF4)'s boundedness/genericity, and holds identically under either branch of (QGCF3).
The Generalized No-Chaos TheoremBounded Complexity Growth at Any Fixed Coarse-GrainingWithin the Quantum Gravity Consistency Framework (QGCF) - any finite-dimensional, closed, unitarily evolving system satisfying (QGCF1)-(QGCF4) - three logically distinct results are established. First, ontic chaos is impossible in three senses: spectral quasiperiodicity, Poincaré recurrence, and exact vanishing of the asymptotic quantum Lyapunov exponent λL (§5). Second, dynamical (Lyapunov/0-1-test) chaos at any FIXED coarse-graining scale ℓ is also impossible, unconditionally - a direct corollary of the same almost-periodicity that gives λL=0 (§7); the one established quantum-chaos diagnostic that survives this obstruction is level (spectral) statistics, checked preliminarily against a non-integrable/integrable benchmark pair (§7.3). Third, and logically independent of the first two: no observer realized as a proper subsystem can simultaneously reconstruct the complete global state from its own accessible data and evolve under an autonomous unitary law, once genuinely (entangling-ly) coupled to the rest of the system (§8) - a consequence of purification non-uniqueness (Hughston-Jozsa-Wootters) and the non-unitary character of open marginal dynamics, requiring no assumption beyond a generic (non-fine-tuned) boundary coupling. The two failures named in that theorem arise from the same physical mechanism - entanglement across the observer's own boundary - but are shown not to be strictly the same fact, only generically co-occurring. Thermalization of coarse blocks (Calabrese-Cardy growth to a Page-value plateau) is real, established physics, distinct from all of the above, and is presented (§9) together with the timescale τ*(ℓ) it requires, and its role as the only available justification for treating the system's effective Hamiltonian Ĥeff as possessing a meaningful temperature Tc, given that the axioms admit no literal external bath. A note on generality. QGCF does not require a literal discrete graph. (QGCF3) is stated as a disjunction: a discrete, graph-based realization (QGCF3a), giving the sharper results of §8.2 and §8.6.2 exactly; or a continuum realization with only a locality/metric structure (QGCF3b), giving the standard, approximately-causal Lieb-Robinson bound instead. Everything else in this document §5-§7, §8.3-§8.6.1, §9 - depends only on finite-dimensionality and unitarity (QGCF1-QGCF2), or on (QGCF4)'s boundedness/genericity, and holds identically under either branch of (QGCF3).
In this thesis, we lay out the goal, and a broad outline, for a program that takes quantum mechanics in its minimal form to be the fundamental ontology of the universe. Everything else, including features like space-time, matter and gravity associated with classical reality, are emergent from these minimal quantum elements. We argue that the Hilbert space of quantum gravity is locally finite-dimensional, in sharp contrast to that of conventional field theory, which could have observable consequences for gravity. We also treat time and space on an equal footing in Hilbert space in a reparametrization invariant setting and show how symmetry transformations, both global and local, can be treated as unitary basis changes. Motivated by the finite-dimensional context, we use Generalized Pauli Operators as finite-dimensional conjugate variables and define a purely Hilbert space notion of locality based on the spread induced by conjugate operators which we call "Operator Collimation." We study deviations in the spectrum of physical theories, particularly the quantum harmonic oscillator, induced by finite-dimensional effects, and show that by including a black hole-based bound in a lattice field theory, the quantum contribution to the vacuum energy can be suppressed by multiple orders of magnitude. We then show how one can recover subsystem structure in Hilbert space which exhibits emergent quasi-classical dynamics. We explicitly connect classical features (such as pointer states of the system being relatively robust to entanglement production under environmental monitoring and the existence of approximately classical trajectories) with features of the Hamiltonian. We develop an in-principle algorithm based on extremization of an entropic quantity that can sift through different factorizations of Hilbert space to pick out the one with manifest classical dynamics. This discussion is then extended to include direct sum decompositions and their compatibility with Hamiltonian evolution. Following this, we study quantum coarse-graining and state-reduction maps in a broad context. In addition to developing a first-principle quantum coarse-graining algorithm based on principle component analysis, we construct more general state-reduction maps specified by a restricted set of observables which do not span the full algebra (as could be the case of limited access in a laboratory or in various situations in quantum gravity). We also present a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. Throughout the thesis, we discuss implications of our work in the broader goal of understanding quantum gravity from minimal elements in quantum mechanics.