't Hooft's 2006 paper "The Mathematical Basis for Deterministic Quantum Mechanics" argued that the boundedness of the quantum Hamiltonian can arise from information loss: ontological states of an underlying deterministic theory merge into equivalence classes, and a limit cycle of period PP appears quantum-mechanically as an energy eigenstate with energy E=h/PE=h/P. The paper closed by identifying its urgent, unanswered questions: no explicit deterministic model was known in which the energy is extensive (an integral of a local energy density over space), no scheme existed for weak interactions between nearly independent systems (the given interaction construction produces level shifts but no transitions between energy states), and the phenomenon of destructive interference had not been explained within the deterministic framework. This problem asks for exactly that construction (or a proof of impossibility within the framework): an explicit deterministic model whose emergent quantum Hamiltonian is bounded below and extensive, which supports weakly interacting subsystems with transitions between energy eigenstates, and whose predictions reproduce destructive interference — the step the source itself identifies as the gateway to locally deterministic quantum field theories. This core remains open: existing explicit models are free, non-extensive, or probabilistic.

Background

Since 1999, Gerard 't Hooft has pursued a program of deterministic (also called ontological or superdeterministic) quantum mechanics, in which ordinary quantum mechanics is not fundamental but emerges from an underlying classical, deterministic theory. The 2006 source develops the mathematical core of this program, and its conventions are needed to state the problem. (i) Ontological states are the states the classical system is really in; variables taking definite values in them are called beables (John Bell's term). (ii) Information loss: the deterministic evolution law is allowed to be non-invertible, so that many distinct ontological states become indistinguishable and merge into equivalence classes; it is these classes, not the individual ontological states, that serve as the basis of the emergent Hilbert space. The conjectured physical origin is information loss at the Planck scale, for example during the formation and annihilation of virtual black holes. (iii) Limit cycles: an isolated deterministic subsystem eventually enters a periodic evolution with period PP, and the corresponding quantum state is an energy eigenstate with E=h/PE=h/P; since periods are positive, the emergent Hamiltonian is automatically bounded from below. The simplest illustration is the cogwheel model: a finite cycle of NN deterministic states maps onto a quantum rotor (a quantum particle on a circle, whose discrete spectrum follows from quantized angular momentum); the harmonic oscillator and pairs of oscillators with relatively prime periods illustrate how energies of independent subsystems must add. A cellular automaton — a lattice of classical cells updated by a fixed local rule — is the natural candidate for a spatially extended model of this kind. What the 2006 paper achieved is thus a mechanism and a set of existence results, including an 'existence proof' for interactions in which one subsystem's cycle frequency is shifted by the state of the other. What it explicitly lacked, in its own words, is: (a) extensivity — in quantum field theory the Hamiltonian is H=∫d3x T00(x⃗)H=\int d^3x\,T^{00}(\vec{x}), an integral over space of a local energy density, which is what guarantees that energies of widely separated subsystems add and that the total grows with volume; in the 2006 framework every finite model's Hamiltonian is instead confined to a fixed interval such as [0,2π/δt)[0,2\pi/\delta t) for a time step δt\delta t, so no extensive model had been constructed; (b) genuine interactions — the existence proof produces shifted levels but 'no transitions between the different energy states', i.e. no scattering, no emission or absorption; (c) interference — quantum probabilities contain phase-dependent cross terms (in a two-alternative setting, a term 2Re α∗β⟨ψ1∣ψ⟩⟨ψ∣ψ2⟩2\mathrm{Re}\,\alpha^*\beta\langle\psi_1|\psi\rangle\langle\psi|\psi_2\rangle) with no analogue in ordinary classical statistics, and the paper could not yet exhibit them inside the deterministic framework. Two further constraints frame the goal. First, the paper's analysis of a completely random deterministic map (its Appendix A) shows that random models have a logarithmic, effectively featureless energy spectrum, nothing like a quantum field theory's; hence 'realistic models will have to be far from random', with a rich set of low-energy states far below the fundamental cutoff. Second, Lorentz transformations and general coordinate transformations were deferred: the paper states that once extensive models are found, 'we will be only one step away from generating locally deterministic quantum field theories'. Any such construction must also remain consistent with Bell-inequality experiments, which in this framework is handled by strong (superdeterministic) correlations in the initial state rather than by intrinsic randomness. Note that this is a minority foundations program: the mainstream view treats quantum mechanics as fundamental, so 'solving' here means constructing the model (or proving impossibility within the framework), not overthrowing quantum predictions.

Problem Statement

Within 't Hooft's deterministic quantum-mechanics framework — deterministic evolution of ontological states, information loss partitioning them into equivalence classes, and energy identified as E=h/PE=h/P from limit-cycle periods — construct an explicit model, or prove that no such model can exist, in which all of the following hold: (1) the emergent quantum Hamiltonian is bounded from below and extensive, i.e. expressible as the integral over three-space of a local Hamiltonian density (on a lattice, a sum of local terms), so that the energy of a composite of widely separated subsystems is the sum of the parts' energies plus a small interaction energy, and the total grows with volume rather than being confined to a fixed interval such as [0,2π/δt)[0,2\pi/\delta t); (2) the model supports weak interactions between two nearly independent subsystems, going beyond the source's existence proof: the interaction must produce both shifted energy levels and transitions between different energy eigenstates, reproducing the standard structure of weakly coupled quantum systems; (3) the model's probabilistic predictions exhibit destructive interference — for example, two-path transition probabilities containing the characteristic phase-dependent cross terms — derived from the deterministic evolution rather than postulated. The source presents this constructive step as the prerequisite for the goals it defers: incorporating Lorentz and general coordinate transformations ("when that is achieved, we will be only one step away from generating locally deterministic quantum field theories") and obtaining far-from-random spectra with a rich set of low-energy states, as required of a realistic quantum field theory. The deferred goals themselves, and the source's speculative connections to Standard-Model gauge classes, the cosmological constant, and the universe as a whole, are adjacent directions outside the scope of the problem stated here. Both answer forms are admissible: an explicit construction satisfying (1)-(3), or a rigorous impossibility proof showing these requirements are mutually incompatible within the framework.

References

  1. Gerard 't Hooft, "The Mathematical Basis for Deterministic Quantum Mechanics", in Beyond the Quantum (World Scientific, 2007), pp. 3-19. DOI: 10.1142/9789812771186_0001, https://doi.org/10.1142/9789812771186_0001. Preprint arXiv:quant-ph/0604008, https://arxiv.org/abs/quant-ph/0604008. Journal version: "A mathematical theory for deterministic quantum mechanics", J. Phys.: Conf. Ser. 67 (2007) 012015, DOI: 10.1088/1742-6596/67/1/012015.
  2. Gerard 't Hooft, "The Cellular Automaton Interpretation of Quantum Mechanics", Fundamental Theories of Physics vol. 185 (Springer, 2016), open access. DOI: 10.1007/978-3-319-41285-6, https://link.springer.com/book/10.1007/978-3-319-41285-6. Preprint arXiv:1405.1548, https://arxiv.org/abs/1405.1548.
  3. Gerard 't Hooft, "Deterministic Quantum Mechanics: The Mathematical Equations", Frontiers in Physics 8 (2020) 253. DOI: 10.3389/fphy.2020.00253, https://doi.org/10.3389/fphy.2020.00253. Preprint arXiv:2005.06374, https://arxiv.org/abs/2005.06374.
  4. Gerard 't Hooft, "Duality between a deterministic cellular automaton and a bosonic quantum field theory in 1+1 dimensions", Foundations of Physics 43 (2013) 597-614. DOI: 10.1007/s10701-013-9702-z, https://doi.org/10.1007/s10701-013-9702-z. Preprint arXiv:1205.4107, https://arxiv.org/abs/1205.4107.
  5. Gerard 't Hooft, "Discreteness and Determinism in Superstrings", arXiv:1207.3612 (2012), https://arxiv.org/abs/1207.3612. Published version: "Beyond Relativistic Quantum String Theory or Discreteness and Determinism in Superstrings", in What We Would Like LHC to Give Us (Proceedings of the International School of Subnuclear Physics, 50th Course, Erice; World Scientific, 2014), pp. 243-264. DOI: 10.1142/9789814603904_0016, https://doi.org/10.1142/9789814603904_0016.
  6. Gerard 't Hooft, "Explicit construction of Local Hidden Variables for any quantum theory up to any desired accuracy", arXiv:2103.04335 (2021), https://arxiv.org/abs/2103.04335. Published version: in Quantum Mechanics and Fundamentality (Synthese Library; Springer, 2022), pp. 175-191. DOI: 10.1007/978-3-030-99642-0_13, https://doi.org/10.1007/978-3-030-99642-0_13.
  7. Gerard 't Hooft, "An ontological description for relativistic, massive bosons", arXiv:2306.09885 (2023), https://arxiv.org/abs/2306.09885.
  8. C. Wetterich, "Fermionic quantum field theories as probabilistic cellular automata", Physical Review D 105 (2022) 074502. DOI: 10.1103/PhysRevD.105.074502, https://doi.org/10.1103/PhysRevD.105.074502. Preprint arXiv:2111.06728 (2021), https://arxiv.org/abs/2111.06728.
  9. Gerard 't Hooft, "The hidden ontological variable in quantum harmonic oscillators", Frontiers in Quantum Science and Technology 3 (2024) 1505593. DOI: 10.3389/frqst.2024.1505593, https://doi.org/10.3389/frqst.2024.1505593.
  10. C. Wetterich, The Probabilistic World: A Fundamental Approach to Quantum Mechanics and Probabilistic Computing (Springer, Fundamental Theories of Physics, 2025). DOI: 10.1007/978-3-031-83213-0, https://doi.org/10.1007/978-3-031-83213-0.