The Cellular Automaton Interpretation of Quantum Mechanics

Gerard 't Hooft

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The Cellular Automaton Interpretation of Quantum Mechanics

Gerard 't Hooft

arXiv:1405.1548v3May 7, 2014quant-ph

TL;DR

The paper addresses whether quantum phenomena can arise from an underlying classical, deterministic description rather than being fundamentally quantum. It develops cellular-automaton models represented with quantum tools and concludes that such models can reproduce quantum mechanics while offering an alternative interpretation of its conceptual puzzles.

Problem
The paper questions whether quantum mechanics must be the fundamental description of nature and seeks a classical account compatible with its observed phenomena and interpretive challenges.
Method
The paper formulates deterministic cellular automata and maps their ontological states into a Hilbert-space framework with a special ontological basis.
Results
The paper concludes that several cellular-automaton models reproduce quantum mechanics exactly, while also clarifying issues including collapse and measurement.
Takeaways & Limitations
Quantum mechanics may function as a mathematical tool for analyzing systems that are classical and deterministic at their core.

Abstract

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When investigating theories at the tiniest conceivable scales in nature, almost all researchers today revert to the quantum language, accepting the verdict from the Copenhagen doctrine that the only way to describe what is going on will always involve states in Hilbert space, controlled by operator equations. Returning to classical, that is, non quantum mechanical, descriptions will be forever impossible, unless one accepts some extremely contrived theoretical constructions that may or may not reproduce the quantum mechanical phenomena observed in experiments. Dissatisfied, this author investigated how one can look at things differently. This book is an overview of older material, but also contains many new observations and calculations. Quantum mechanics is looked upon as a tool, not as a theory. Examples are displayed of models that are classical in essence, but can be analysed by the use of quantum techniques, and we argue that even the Standard Model, together with gravitational interactions, might be viewed as a quantum mechanical approach to analyse a system that could be classical at its core. We explain how such thoughts can conceivably be reconciled with Bell's theorem, and how the usual objections voiced against the notion of `superdeterminism' can be overcome, at least in principle. Our proposal would eradicate the collapse problem and the measurement problem. Even the existence of an "arrow of time" can perhaps be explained in a more elegant way than usual. Discussions added in v3: the role of the gravitational force, a mathematical physics definition of free will, and an unconventional view on the arrow of time, amongst others.

21 The cellular automaton

The book proposes that quantum mechanics can describe systems classical and deterministic at their core, using cellular-automaton models and a special ontological basis. These models reproduce quantum mechanics while reframing measurement, collapse, and related interpretive problems, though important theoretical difficulties remain.

Motivation: The Cellular Automaton Interpretation treats quantum mechanics as a description of underlying reality that may be simpler and deterministic.The author presents this as a proposed interpretation rather than a replacement for quantum mechanics.
Interpretive consequences: The interpretation aims to clarify the collapse question, measurement problem, Schrödinger’s cat, entanglement, and the arrow of time without changing practical quantum calculations.The author also argues that Bell’s theorem and objections to determinism may be addressed through the structure of the underlying theory.
Ontological basis: A special ontological basis assigns reality to mutually orthogonal Hilbert-space states, restricting the resulting theories to a subset of quantum theories.The universe occupies one ontological state with probability 1, while the other basis states have probability 0.
Results and scope: Several models reproduce quantum mechanics without modifying its equations, although the explicit constructions remain too simple to explain all quantum consequences.The author acknowledges that these models do not yet provide sufficiently refined accounts of interesting interacting systems.
Modeling strategy: The approach constructs classical models whose evolution follows unambiguous laws and can nevertheless be analyzed using quantum-mechanical techniques.The models use discrete variables and may be quantum mechanical and classical at the same time.
Open problems: The main unresolved difficulties concern energy signs and the locality and lower-bound properties of effective Hamiltonians.The book postpones detailed treatment and suggests that gravitational effects may be relevant to acceptable models.

3. Interpreting quantum mechanics

The Cellular Automaton Interpretation proposes classical ontological states beneath quantum descriptions, treating quantum mechanics as a computational tool rather than the fundamental theory. It addresses superposition, decoherence, and Bell-type correlations through deterministic models and an ontology-conservation proposal.

The Einsteinian view: The interpretation replaces unobservable quantum states with classical ontological states whose collective evolution produces observable statistical outcomes.The observer no longer defines physical degrees of freedom or their values within the proposed formalism.
The collapsing wave function and Schrödinger’s cat: Quantum mechanics is treated as a mathematical tool: superposed states solve equations, but only definite ontological states represent physically realized configurations.This distinction is intended to remove the logical difficulty illustrated by Schrödinger’s cat.
Decoherence and Born’s probability axiom: The interpretation makes decoherence unnecessary by declaring superpositions of classical states computational templates rather than states of the real universe.Coefficients may instead represent probabilities over initial ontological configurations.
Bell’s theorem, Bell’s inequalities and the CHSH inequality: Bell-type objections arise because local hidden-variable theories appear unable to represent entangled states and reproduce their correlations.The cited discussion frames entanglement as a central reason such theories are commonly rejected.
Bell’s theorem, Bell’s inequalities and the CHSH inequality: The proposed response combines deterministic settings with an ontology-conservation law, allowing Bell-theorem violations without abandoning the underlying classical description.The author connects this proposal to superdeterminism and defines measurement choices as constrained by physical history.
Bell’s theorem, Bell’s inequalities and the CHSH inequality: The mouse-dropping correlation function is presented as a deterministic model that reproduces the quantum expression while encoding correlations among Alice’s setting, Bob’s setting, and a shared angle.The author acknowledges that the model requires ontological information about future settings and treats its conservation as the relevant constraint.

4. Deterministic quantum mechanics

Deterministic quantum mechanics presents quantum mechanics as a cross-section of classical and quantum descriptions, grounded in an ontological basis whose states evolve classically. The approach uses basis transformations and superpositions as calculational tools while treating macroscopic outcomes and probabilities as consequences of underlying ontological states.

Deterministic quantum mechanics is neither a modification of standard quantum mechanics nor classical theory, but a cross-section of both.
The approach can begin with a completely classical theory or with conventional quantum mechanics containing a special ontological basis.In the quantum formulation, the ontological basis is a special Hilbert-space basis whose elements map into other basis elements under sufficiently dense time evolution.
The proposed ontological basis makes time evolution appear classical and identifies large-scale observables, including detector data and planetary orbits, as diagonal beables.The construction is linked to classical observables while retaining the original quantum theory and its freedom to use other bases.
Basis transformations make ontological states appear as quantum superpositions in descriptions suited to long-distance and long-time calculations, allowing interference phenomena.
Measurement and wave-function collapse are claimed to occur automatically because ontological states never form superpositions, so cats emerge definitively dead or alive.The author presents this resolution of collapse, measurement, and Schrödinger’s cat problems as a central argument for the Cellular Automaton Interpretation.
Using superpositions to calculate transition amplitudes still yields classical final outcomes, while unitary transformations make Born’s rule the sole probability expression.The framework treats quantum templates as superpositions of ontological states and classical outcomes as distributions over those states.

6. Quantum gravity

The section connects deterministic models with quantum mechanics through information loss, equivalence classes, and Hilbert-space representations. It proposes that irreversible classical dynamics can yield time-reversible quantum laws while retaining an underlying thermodynamic arrow of time.

Quantum gravity: At the Planck scale, special relativity, quantum mechanics, and gravity are simultaneously relevant, yet no complete synthesis has been achieved.The section motivates quantum-gravity work by the unresolved coexistence of these three theories.
Information loss and quantum structure: Information-equivalence classes map the states of an information-losing classical model into the ontological basis of a quantum theory.The construction groups states with common future behavior into classes whose evolution can be represented by a unitary matrix.
The arrow of time: Information loss can make the underlying classical dynamics time-irreversible while the emergent quantum description remains exactly time-reversible.The quantum observer can preserve symmetries such as CPT invariance even when classical states merge irreversibly.
Bell correlations: Info-equivalence classes may create apparent non-locality in the effective quantum theory, potentially helping reconcile the model with Bell-theorem violations.The proposed non-locality is described as apparent and induced by the class structure.
The arrow of time: Large-scale information loss can reduce the ontological basis while leaving quantum laws time-reversible and classical measurement-linked states thermodynamically asymmetric.The section uses this distinction to explain how a classical arrow of time may coexist with reversible quantum equations.
Open issues: The interpretation of complex time and energy bounds remains highly speculative.The proposed relation between an energy-like quantity and an imaginary time component is explicitly identified as speculative.

8. More problems

This section develops consequences of information loss, holography, and deterministic foundations for quantum theory. It presents mechanisms for quantum probabilities, collapse, locality, and the arrow of time while identifying gravity and scale hierarchies as unresolved challenges.

Consequences for quantum theory: The proposal treats the universe as a single classical, discrete ontological history rather than an infinity of distinct universes.This framing supports the claim that measurement outcomes remain associated with a single ontological state.
Consequences for quantum theory: Born probabilities are claimed to hold exactly, without corrections, within the proposed framework.The supplied passage states the result but does not provide the derivation here.
Measurement: Wave-function collapse is claimed to arise automatically because experiments end in a single ontological state rather than a superposition.The proposal requires no nonlinear correction to Schrödinger’s equation.
Locality: The underlying theory may be local even though transformations among classical states can generate effective non-local behavior.The section connects this possibility to info-equivalence classes and quantum correlations.
The hierarchy problem: The Planck scale near 10^19 GeV, nuclear scale near 1 GeV, and neutrino scale near 10^-11 GeV form a hierarchy whose origin remains obscure.The extreme separation of scales makes a detailed theory involving gravity, quantum mechanics, and relativity difficult to formulate.
Gravity: Gravity is proposed as a crucial ingredient because local time translations require a local Hamiltonian density.The text presents this as a likely route toward resolving remaining difficulties, not as an established result.
Holography: Holography is formulated as bulk information disappearing while surface information remains accessible for characterizing equivalence classes.The black-hole entropy discussion associates the maximum information in a volume with the largest black hole fitting inside it.

Calculation Techniques

The calculation techniques construct quantum descriptions from deterministic models including cogwheels, rotators, fermions, and neutrino configurations. These examples test internal consistency and identify both useful correspondences and limits of the models.

Cogwheel models: The Cogwheel Model supplies explicit procedures for constructing quantum Hamiltonians and studying continuum limits from deterministic systems.The calculations connect cogwheels with harmonic rotators and other classically described structures.
Cogwheel models: A periodic cogwheel approaches the quantum harmonic oscillator with the same period in the continuum limit, subject to stated subtleties.The limit is performed in two steps through a harmonic rotator.
Neutrino model: The simplified free two-component neutrino model yields classically moving flat membranes or sheets rather than particles.This result holds when mass terms and interactions are excluded.
Scope and limitations: The models are local and realistic but limited in scope, omitting features such as all particle species, symmetry groups, and special and general relativity.They are presented as transparent indications of directions rather than complete descriptions of the real world.
Cogwheel models: A cogwheel with N states represents SU(2) with total angular momentum ℓ satisfying N = 2ℓ+1, while its harmonic rotator has a bounded Hamiltonian.The representation is unitary and includes a natural ground state.
Spectral correspondences: Discrete Hamiltonians have finitely many eigenstates with a lower-bounded spectrum, allowing the lowest-energy state to serve as a ground state or vacuum.The eigenvalues lie in [δE, 2π/T + δE] for the stated discrete periodic model.
Spectral correspondences: In the periodic continuum limit, deterministic eigenvalues become integer multiples of 2π/T, matching the spectrum of a harmonic oscillator with period T.The text describes the harmonic oscillator as a deterministic system in disguise.

13. The continuum limit of cogwheels, harmonic rotators and oscillators

The section shows how continuum periodic deterministic systems can be mapped to harmonic rotators and oscillators, while examining spectral, locality, and positivity constraints in the resulting Hamiltonians.

Continuum limits: As N→∞, cogwheels yield continuum limits with either fixed time step or fixed period and vanishing time quantum.The two limits produce different descriptions of the continuum spectrum and ontological variables.
Mapping criteria: Mappings are required to be time-independent so that solving one model’s evolution law also solves the other’s physical evolution.Without constraints on the mapping, any model could be mapped onto any other, making the relation physically meaningless.
Spectral mapping: The mapping is made in two steps: a finite harmonic rotator first maps perfectly onto the harmonic oscillator, followed by a continuous limiting procedure.The rotator’s upper energy ceiling avoids pathologies associated with spectra bounded below but not above.
Locality and positivity: Accurate quantum behavior near the vacuum can force the effective Hamiltonian to be non-local over distances of order T^2MPl.The non-locality arises from slowly convergent Fourier coefficients needed to reproduce a spectral jump at zero energy.
Locality and positivity: Using a rapidly convergent expansion near the spectrum’s center reduces the Hamiltonian error to order (ωδt)^(R+2), so only a few neighbors may suffice.Negative-energy states can subsequently be interpreted as antiparticles after second quantisation.

15. Fermions

The fermion constructions derive quantum field behavior from deterministic permutations and represent a massless neutrino through commuting beables describing moving planar sheets.

Deterministic fermion construction: A deterministic model with M states evolves by a permutation, and its fermionic extension preserves the same occupied-state permutations while enlarging the state space to 2^M.The resulting Hamiltonian is bounded from below.
Deterministic fermion construction: The Jordan-Wigner transformation generates the minus signs in fermionic anticommutators from an underlying deterministic system.Without this transformation, the required anticommutator expressions would not arise.
Neutrino model: For m = 0, the model admits complete beables and represents a massless two-component Majorana or chiral “neutrino.”The model ignores interactions, hence the quotation marks around “neutrino.”
Neutrino model: The beable basis describes planar sheets at distance r, oriented by q̂, moving transversely at light speed with direction set by the Boolean s.The remaining beables stay constant during this deterministic motion.
Operator representation: The constructed operators x⃗, p⃗, and σ⃗ obey the required commutation relations when expressed in the ontological basis.The Hamiltonian generates rotations of the beables q_i while leaving other specified beables unchanged.
Locality and correlations: Although the sheets extend infinitely, the equations of motion are local; non-locality appears instead in the model’s space-like correlations.These correlations connect points sharing the same sheet orientation and sign, and may be relevant to Bell-inequality behavior.

16. PQ theory

PQ theory decomposes real coordinates and momenta into integer and angular parts, then uses torus variables and phase transformations to construct equivalent quantum descriptions.

Integer-angle decomposition: PQ theory decomposes p and q into integer and angular components: p = 2πK + κ and q = X + ξ/2π.K and X are integers, while κ and ξ are angles.
Torus transformation: The torus Hilbert space generated by |κ, ξ⟩ is equivalent to the Hilbert space generated by real-line position or momentum eigenstates, apart from one exceptional state.This establishes the central equivalence used by the PQ construction.
Torus transformation: A phase function φ(κ, ξ) converts twisted boundary conditions on the torus into strictly periodic boundaries for the transformed wave function.The required phase function has an unavoidable movable topological singularity.
PQ basis: The transformed commutator is [qop, pop] = i(I − |ψe⟩⟨ψe|), reflecting the exceptional edge state.The edge-state correction modifies the canonical commutation relation on the full space.
PQ basis: The states |Q, P⟩ form an orthonormal basis because their waveforms are self-Fourier and orthogonal under integer shifts.Their wave functions resemble wavelets but have these additional orthogonality and transform properties.
Integer-angle decomposition: The formalism aims to transform integer-based systems with permutation evolution into quantum systems through a Hamiltonian satisfying Pop = e^−iHopδt.It is intended to map systems based on integers to real-number descriptions and back.

17. Models in two space-time dimensions without interactions

The section constructs deterministic counterparts for free bosonic and string models in restricted settings, obtaining exact mappings in two dimensions and lattice-constrained string equivalence, while identifying higher-dimensional stability limits.

Higher-dimensional limitations: PQ theory has not successfully extended the bosonic-field construction beyond two dimensions, because equations such as the wave equation are difficult to apply to integers.The text reports partial success only for massless fermions in three spatial dimensions.
Two-dimensional bosons: The quantised massless bosonic field theory has an exact mapping to cellular-automaton states, aside from possible edge states.The mapping is established by treating left- and right-movers separately.
Two-dimensional bosons: In one spatial dimension, left- and right-moving integer fields obey deterministic propagation laws AL(x,t) = AL(x + t) and AR(x,t) = AR(x − t).These propagation rules match the corresponding quantum field modes.
Two-dimensional bosons: Both the quantum boson model and integer left/right-mover model are mathematically equivalent to a classical scalar field defined modulo 2π.The latter model is ontologically distinct from the preceding two models.
Higher-dimensional limitations: In higher dimensions, lattice models can develop exponentially growing modes when the dispersion relation exceeds its allowed cosine range, making them unstable.The instability is tied to the absence of a non-negative conserved energy function.
Higher-dimensional limitations: Stable multi-oscillator models with non-negative energy are strongly restricted to integer matrices with specific diagonal and off-diagonal patterns.Under Tij = δij, the described form is the most general stable model identified in the passage.
String models: The comparison of string states focuses on transverse variables, while X− remains a constraint because it determines the string’s spacetime location.X+ can serve as the independent target-time variable, whereas X− has no independent dynamical role but may cause difficulties.
String models: The classical string is equivalent to the fully quantised bosonic string when its coordinates lie on a square lattice satisfying the essential lattice-parameter condition.Under that condition, Hilbert-space basis elements propagate according to discrete classical string equations, and the mapping extends to superstrings.

18. Symmetries

The section argues that Hilbert-space mappings let deterministic lattice systems exhibit broader symmetry groups than their underlying ontological lattices, including continuous translations and rotations.

Noether’s theorem links continuous symmetries with conserved quantities such as momentum, energy, and angular momentum.
Attaching Hilbert-space basis elements to classical states extends Noether-style symmetry analysis to deterministic systems.For time-independent evolution, energy is obtained from an eigenvalue of the evolution operator.
Lattice models ordinarily retain only discrete crystallographic rotations, such as the finite cubic group O(3, Z), rather than all rotations O(3, R).
Superpositions in Hilbert space can restore continuous translations and rotations absent from the classical lattice description.
Fractional translations are constructed through momentum-space Fourier methods, although only integer translations correspond to ontological lattice shifts.
The proposed rotation transformations form an acceptable rotation group that converges to ordinary rotations in the continuum limit.The construction acts on low-frequency modes in the primary momentum-space circle and handles additional circles separately.
In the PQ lattice, Hilbert-space rotations act normally on states orthogonal to edge states, despite the ontological lattice lacking continuous rotation invariance.
The section interprets quantum symmetry generators as combinations of observable beables and noncommuting changeable parts.In the PQ formalism, integer translations are beables while fractional translations are changeables.

19. The discretised Hamiltonian formalism in PQ theory

The discretised Hamiltonian formalism seeks stable, local, conserved energy functions for deterministic lattice systems and constructs evolution rules that approach Hamiltonian dynamics under smoothness conditions.

19.1. The vacuum state, and the double role of the Hamiltonian (cont’d): A physically acceptable Hamiltonian must be conserved, bounded from below, and local, making a lower energy bound central to stability and approximation methods.
19.1. The vacuum state, and the double role of the Hamiltonian (cont’d): The stability argument requires care because the absence of a stabiliser does not imply that a dynamical system destabilises.The solar system is cited as a counterexample to that inference.
19.2. The Hamilton problem for discrete deterministic systems: For discrete systems, adding a conserved integer quantum E_class = 2πN/δt can restore an absolutely conserved energy.
19.2. The Hamilton problem for discrete deterministic systems: Splitting total energy into classical and quantum parts creates a tension: either the quantum contribution exceeds its bounds or classical energy loses extensivity.
19.3. Conserved classical energy in PQ theory: Requiring a properly bounded energy restricts the simplest formalism to linear or periodic oscillator chains with strictly harmonic forces.
19.4. More general, integer-valued Hamiltonian models with interactions: The proposed integer-valued Hamiltonians can define genuine quantum systems when the classical energy is bounded from below.
19.4. More general, integer-valued Hamiltonian models with interactions: The 1+1-dimensional updating rule preserves discrete energy exactly and approaches standard Hamilton equations in the continuum limit.The construction requires sufficiently smooth energy contours containing more than one lattice point.
19.4. More general, integer-valued Hamiltonian models with interactions: Exceptional extrema and saddle points require additional prescriptions to make contour-following evolution unique and reversible.The construction also permits a cellular automaton with local update ordering and suggests spurious mapping nonlocality can arise in the quantum description.

20. Quantum Field Theory

The section frames quantum field theory as a local, causal framework and examines how cellular-automaton descriptions might approximate it, while highlighting continuum, symmetry, renormalization, and gravitational difficulties.

The chapter reviews quantum-field-theory features needed for the book’s proposed comparison with cellular automata.
The central challenge is reproducing quantum-field-theory locality with classical deterministic systems that use only local interactions.
Relativistic quantum field theories in the Standard Model are local in the quantum sense, but their continuum and field-content constraints complicate extensions involving gravity.
A lattice continuum limit can be easy to formulate yet lose rotation symmetry and Lorentz invariance.
Second quantisation can restore a lower Hamiltonian bound by allowing indefinite particle numbers, a mechanism the author proposes extending to cellular automata.
The proposed route combines quantum-field-theoretic perturbation with interaction-Hamiltonian expansions, followed by renormalisation-group evolution toward Standard Model scales.
Landau poles can make perturbative renormalisation fail when couplings become large.
Spacelike-separated local operators commute, so entanglement does not permit faster-than-light signalling in relativistic quantum field theories.

21. The cellular automaton

The cellular-automaton construction maps reversible local update rules onto Hilbert-space operators and derives Hamiltonians whose commutator expansions remain locally structured.

The chapter constructs quantum Hamiltonians for cellular automata to compare deterministic dynamics with quantum-field-theory Hamiltonians.
Time reversibility permits individual automaton states to serve as Hilbert-space basis elements; without manifest reversibility, information-equivalence classes are required.
21.1.1. The time reversible cellular automaton: The evolution operator is split into commuting updates of even and odd lattice sites, U = U_A · U_B.
21.1.1. The time reversible cellular automaton: Local update operators fail to commute only across adjacent even–odd sites, preserving a nearest-neighbour structure.
21.1.2. The discrete classical Hamiltonian model: Local additions modulo N are represented with translation operators whose exponentials define the corresponding operators A and B.
The Hamiltonian is derived through a Baker–Campbell–Hausdorff expansion as an infinite series of nested commutators.
Because commutators involve only nearby sites, each term can be interpreted as a local Hamiltonian-density contribution.
The commutator expansion is not guaranteed to converge and may fail at energies above the inverse discrete time unit.

22. The problem of quantum locality

The section examines how cellular automata can satisfy locality while producing quantum-field-theoretic descriptions, but identifies major technical obstacles in obtaining relativistic, bounded Hamiltonians and Standard Model-like physics.

Locality: Cellular automata impose locality by updating each cell only from its direct neighbours, limiting signal propagation to a finite velocity.This resembles the relativistic requirement that signals do not exceed a limiting speed c.
Locality: Quantum locality requires operators at space-like separated points to commute, ensuring operations and measurements are order-independent.The Standard Model’s quantum field theories obey this constraint.
Relativistic compatibility: Most cellular automaton models fail to obey special relativity, preventing their Hamiltonians from even approximately resembling the Standard Model.The section treats this as an important technical problem requiring a systematic remedy.
Constructive approach: A proposed route begins with deterministic automata for freely moving particles and then applies second quantisation to approach quantum-field-theoretic models.The target particle content includes fermions, scalar bosons, gauge bosons, and possibly gravitons.
Constructive approach: The proposal aims to obtain local, bounded Planck-scale Hamiltonians whose renormalization-group evolution could produce familiar quantised fields, but this remains an argument rather than a proof.The required transformation spans about 20 orders of magnitude, and the construction’s success is still conjectural.
Symmetry and Bell’s theorem: Discrete deterministic variables break continuous quantum symmetries into discrete subgroups, creating alternative commuting ontological descriptions relevant to Bell’s theorem.The authors argue that inability to predict Alice’s and Bob’s settings removes the theory’s apparent conspiracy aspect.
Broader scope: The framework seeks models that are locally discrete and classical while remaining expressible through quantum field theory, with gravity motivating Hamiltonians built from local energy densities.The gravitational discussion links extrinsic Hamiltonians to quantum gravity.

23. Conclusions of part II

The conclusions present quantum mechanics as a mathematical toolkit for deterministic models while acknowledging unresolved difficulties in constructing systems as complex as the Standard Model and clarifying foundational issues.

Conclusions: The technical examples demonstrate multiple links between quantum-mechanical models and completely deterministic systems, although many examples are too simple for direct physical applications.The exercises are presented as support for earlier assertions rather than as complete physical models.
Conclusions: Quantum-mechanical techniques such as Fourier expansions, unitary transformations, perturbation theory, and Noether’s theorem can support deterministic models.The authors propose viewing quantum mechanics primarily as a powerful mathematical tool for handling statistical features.
Unresolved difficulties: A systematic strategy for constructing models as complex as the Standard Model has not yet been found; the proposed procedures remain incomplete ingredients.The authors also contend that Bell-type no-go results contain repeatedly discussed loopholes relevant to superdeterminism.
Unresolved difficulties: The interpretation treats free will as a statement about correlations in the initial state, while acknowledging that complete clarification may require further progress in quantum field theory.The authors identify the concern about conspiracy as unresolved rather than settled.
Symmetries: Non-compact symmetry groups complicate cellular-automaton constructions because their quantum Noether charges do not commute and combine beables with changeables.This specifically affects reproducing Poincaré-type symmetries.
Strings: Quantised strings and superstrings exhibit ontological degrees of freedom on a space-time lattice with lattice length parameter a_spacetime = 2π α′.The authors interpret this as reflecting string theory’s finiteness and giving it a physical interpretation.
Overall assessment: The interpretation claims exact reproduction of quantum mechanics while offering clarification of collapse, measurement, classical links, many-worlds, and pilot-wave issues.The authors state that explicit model constructions are not yet refined enough to explain every quantum-mechanical consequence.
Overall assessment: Nonconvergent perturbation expansions motivate a proposed unavoidable margin of error in quantum-field-theory calculations and possibly in quantum mechanics itself.The conclusion is presented as a novel implication of applying second quantisation in constructing the Hamiltonian.

A. Some remarks on gravity in 2+1 dimensions

The gravity discussion uses 2+1-dimensional spacetime to connect classical geometric defects, particle dynamics, and a possible quantum description based on canonically conjugate tessellation variables.

Geometry: Matter-free regions in 2+1-dimensional gravity are flat spacetime pieces, while particles appear as local topological defects.A stationary particle removes a wedge, producing a cone whose deficit angle is proportional to its mass.
Moving particles: For a moving particle, the wedge is Lorentz-contracted and oriented along the motion bisector so the identification introduces no time shift.The particle velocity is defined as tanh ξ, with contraction factor cosh ξ.
Particle energy: The deficit angle H is interpreted as the moving particle’s energy in the presence of gravity because it obeys the appropriate mass-energy relation in the weak-gravity limit.Deficit angles are additive and conserved.
Hamiltonian structure: Tessellating Cauchy surfaces into polygons yields edge lengths L_i and transverse rapidities η_i whose dynamics obey Hamilton equations.The variables η_i and L_i form canonically associated pairs.
Hamiltonian structure: Replacing the Poisson brackets of L_i and η_i by commutators is proposed as the route to quantising the tessellated gravitational system.The edge lengths act like positions and the η_i like their associated momenta.

A.1. Discreteness of time

The section argues that gravity changes how discreteness of time should be understood: local evolution may be discrete, but global time is not a physical variable in a diffeomorphism-invariant universe.

Hamiltonian constraints: A many-particle gravitational Hamiltonian can force the universe to close when total matter energy exceeds 2π, with total Hamiltonian restricted to 4π.This raises questions about varying the Hamiltonian with respect to the particle rapidities η.
Relative time: After splitting the universe into X and Ω\X, the edge lengths in X obey a Schrödinger equation with respect to time relative to Ω\X.The equation is derived from the gravitational Hamiltonian with H and η promoted to operators.
Local discreteness of time: The Hamiltonian is an angle, so its conjugate time is discrete and the well-defined evolution object is U = e^−iH over one time unit.The discussion interprets this as evolution in discretised steps.
Global time: Global time cannot be declared discrete because gravity is diffeomorphism invariant and time is only a coordinate.Consequently, no Schrödinger equation governs the evolution of the entire universe.
Deterministic gravity: The authors suggest deterministic models may be necessary for 2+1-dimensional gravity because sharply defined dynamical variables are needed to define topological winding properties.This proposal follows difficulties formulating a meaningful Schrödinger equation for the universe.
Higher dimensions: The 2+1-dimensional analysis is extended speculatively toward 3+1 dimensions by considering compactification of one spatial dimension.The authors believe the uncompactified limit could describe our universe within the cellular-automaton interpretation.
Open problems: Discretising spacetime under gravity faces obstacles from curvature, non-compact Lorentz transformations, and black-hole states.The discussion therefore considers topologically regular gravitational theories and conformal reformulations as possible ingredients.
Conformal gravity: Exact local Weyl invariance can prevent coupling constants from running toward the small-distance limit χ → 0.The authors also suggest conformal gauge fixing may impose upper limits on information stored in volumes or areas.