Completing the deterministic lattice reformulation of superstring theory: world-sheet continuum limit, emergent Lorentz invariance, state counting, and deterministic interactions

In lectures published as 'Beyond Relativistic Quantum String Theory or Discreteness and Determinism in Superstrings' (Erice 2012, publ. 2014), Gerard 't Hooft exhibited a mapping between a classical, deterministic string — integer-valued left- and right-moving modes on a world-sheet lattice, a target space restricted to a discrete lattice of length $\sqrt{2\pi\alpha'}$, and Boolean world-sheet variables mapped to fermions — and the standard quantized (super)string. For the free string the mapping is exact within the world-sheet Brillouin zone, but the equivalence with standard superstring theory is incomplete in several precise ways: the continuum limit and gauge fixing of the world-sheet lattice; the emergence of continuous Lorentz invariance (currently only a conjecture from the discrete Lorentz group plus transverse rotations) and of supersymmetry; the state-counting problem for the longitudinal modes; edge states at $\eta=\pm1/2$ where the canonical commutators fail; deterministic interactions and second quantization (which force an oriented string with coupling fixed to $g_s=1$); and unincorporated boundary conditions and GSO projection. The open problem is to complete — or rigorously refute — this deterministic reformulation of superstring theory. No resolution of these items is known in the literature to date.

Background

String theory describes one-dimensional objects whose history sweeps a two-dimensional surface, the world sheet, inside a higher-dimensional space-time called the target space ($D=10$ for superstrings). In a fixed light-cone gauge (a convenient coordinate choice in which one time-like and one space-like target-space direction are singled out), the physical degrees of freedom are the $D-2$ transverse embedding coordinates, which decompose into left- and right-moving modes obeying linear wave equations; the remaining longitudinal coordinates are not independent but are fixed by the residual world-sheet gauge constraints (the Virasoro constraints), and getting these constraints right is what forces the critical dimensions $D=26$ or $10$ and ensures Lorentz invariance.

Since the 1990s 't Hooft has pursued a deterministic underpinning of quantum mechanics: a set of commuting, discrete variables with definite values ('ontological' states, or beables) evolve by a classical reversible law, for example a cellular automaton — a lattice system whose variables are updated site by site by a fixed local rule — and quantum mechanics arises through a change of basis in which this evolution acts unitarily. Two 2012 papers (arXiv:1204.4926 and arXiv:1205.4107) made this precise for one-dimensional lattice systems: an automaton of integer-valued left/right movers is mapped exactly onto a bosonic quantum field theory, with the fractional parts of the integers, viewed as positions on a torus (the dual lattice), playing the role of conjugate quantum variables.

In the source paper this machinery is applied to string theory. The transverse string modes are taken to be integers $A_{L,R}^a$ on a world-sheet lattice; within the Brillouin zone $|\kappa|<1/2$ of world-sheet momenta $\kappa$ (the momentum range available on a lattice of unit spacing) the Hamiltonian is identical to the continuum one, so the free theory is exactly reproduced. The integer parts of the target-space coordinates define a space-time lattice of length $a_\ell=\sqrt{2\pi\alpha'}$, where $\alpha'$ is the string slope parameter (the inverse string tension) — a length of order the Planck length if $\alpha'$ is Planckian. Boolean (two-valued) variables attached to world-sheet sites become world-sheet fermions via the Jordan–Wigner transformation (a standard lattice construction producing anticommuting fermion operators from two-valued variables), and with exactly $D-2$ of them the bosons and fermions combine into the supermultiplets (multiplets pairing bosonic and fermionic states) of the superstring, which also removes the longitudinal modes. The construction is invariant under the discrete Lorentz group $O(D-1,1,\mathbb{Z})$ — Lorentz transformations with integer matrix entries — and admits one deterministic interaction: two oriented strings exchange arms when they meet at a lattice point, which fixes the string coupling to $g_s=1$.

Several gaps, catalogued by the author in the paper's concluding section, separate this from a complete equivalence with standard superstring theory. The lattice constraints do not coincide with the continuum Virasoro constraints, and the standard light-cone gauge choice cannot be imposed naively because the longitudinal integers fluctuate from site to site; on a finite world-sheet lattice the commutators of the excited modes are modified and the continuous Lorentz algebra does not close — the emergence of continuous Lorentz invariance from the discrete group plus the exactly invariant transverse rotations is only conjectured. Transverse data do not uniquely fix the longitudinal integers (a state-counting problem: one risks over- or under-counting physical states). At the edge configurations $\eta_A(x)=\pm1/2$ of the fractional variables, the canonical commutation relations fail and wave functions must vanish. The interacting, multi-string (second-quantized) theory with a proper vacuum is not constructed — the author's later book proposes second quantization precisely to handle the vacuum and edge states — and boundary conditions for open and closed strings and the GSO projection (the projection that removes tachyonic states from the superstring spectrum) are, as the author admits, hardly touched upon.

Problem Statement

Can 't Hooft's deterministic, discrete superstring model be completed into a theory exactly equivalent to standard quantum superstring theory — or is there a precise internal obstruction? The model is as defined in the source: integer-valued transverse left- and right-movers on a world-sheet lattice with Brillouin zone $|\kappa|<1/2$; fractional variables $\eta$ generating the dual lattice; a target-space lattice of length $a_\ell=\sqrt{2\pi\alpha'}$; $D-2$ Boolean variables per world-sheet site mapped to fermions; invariance under the discrete Lorentz group $O(D-1,1,\mathbb{Z})$; and a leg-exchange interaction. The single overarching objective — establishing (or refuting) the exact equivalence with the standard superstring — decomposes into the following jointly necessary questions raised by the source:

1. World-sheet continuum limit and gauge fixing. Formulate the passage to the continuum on the world sheet and the gauge fixing of the longitudinal modes so that the lattice constraints reproduce the standard continuum superstring constraints; the standard light-cone gauge condition cannot be imposed naively because the longitudinal integers fluctuate from site to site.

2. Emergent symmetries. Prove or disprove that continuous Lorentz invariance — and with it supersymmetry — emerges in the quantum theory from discrete Lorentz invariance together with the exactly invariant continuous transverse rotations; at present this is only a conjecture, and on a finite lattice the Lorentz algebra does not close.

3. State counting. Specify the world-sheet gauge or lattice-occupation rule so that each set of transverse data corresponds to exactly one physical state, neither over- nor under-counted.

4. Edge states. Control the configurations $\eta_A(x)=\pm1/2$, where the canonical commutation relations fail and wave functions must vanish, and clarify their role in the full theory, including the vacuum.

5. Interactions and second quantization. Construct a deterministic interacting, multi-string (second-quantized) theory — strings exchanging arms at intersection points — incorporating boundary conditions for open and closed strings and the GSO projection, or show that it cannot be done deterministically.

6. Coupling, orientation, and the lattice. Clarify whether the fixed coupling $g_s=1$ and the necessity of oriented strings are predictions of the deterministic formulation or artifacts to be removed, and whether the target-space lattice is a physical structure or a mathematical device.

A complete positive answer (with the evolution deterministic and reversible in the ontological basis) establishes the equivalence in all of the respects raised above: the world-sheet continuum limit and gauge fixing of the longitudinal modes (question 1); the emergence of continuous Lorentz invariance and supersymmetry, with closure of the symmetry and constraint algebras (question 2); state counting matching the standard superstring spectrum (question 3); controlled edge states at $\eta=\pm1/2$ and their role in the vacuum (question 4); a deterministic interacting, second-quantized theory including boundary conditions and the GSO projection (question 5); and a determination of whether the fixed coupling $g_s=1$, the oriented-string requirement, and the target-space lattice are dictated by the deterministic formulation or removable (question 6). A complete negative answer identifies an obstruction internal to this class of deterministic lattice models that survives the world-sheet continuum limit. Partial progress on individual items is valuable but does not by itself settle the problem.

References

1. Gerard 't Hooft, "Beyond Relativistic Quantum String Theory or Discreteness and Determinism in Superstrings", in What We Would Like LHC to Give Us (Subnuclear Series Vol. 50, World Scientific, 2014), pp. 243-264. DOI: 10.1142/9789814603904_0016, https://doi.org/10.1142/9789814603904_0016 (originating chapter)
2. Gerard 't Hooft, "Discreteness and Determinism in Superstrings", arXiv:1207.3612 [hep-th] (2012), https://arxiv.org/abs/1207.3612 (author's preprint version of the originating chapter)
3. Gerard 't Hooft, "Relating the quantum mechanics of discrete systems to standard canonical quantum mechanics", arXiv:1204.4926 [quant-ph] (2012), https://arxiv.org/abs/1204.4926 (predecessor paper, Ref. [8] of the source)
4. Gerard 't Hooft, "Duality between a deterministic cellular automaton and a bosonic quantum field theory in 1+1 dimensions", arXiv:1205.4107 [quant-ph] (2012), https://arxiv.org/abs/1205.4107; published as Foundations of Physics 43 (2013) 597-614, DOI: 10.1007/s10701-013-9702-z, https://doi.org/10.1007/s10701-013-9702-z (predecessor paper, Ref. [9] of the source)
5. Gerard 't Hooft, "On the Foundations of Superstring Theory", Foundations of Physics 43 (2013) 46-53. DOI: 10.1007/s10701-012-9682-4, https://doi.org/10.1007/s10701-012-9682-4
6. Gerard 't Hooft, "Superstrings and the Foundations of Quantum Mechanics", Foundations of Physics 44 (2014) 463-471. DOI: 10.1007/s10701-014-9790-4, https://doi.org/10.1007/s10701-014-9790-4
7. Gerard 't Hooft, "The Fate of the Quantum", arXiv:1308.1007 [quant-ph] (2013), https://arxiv.org/abs/1308.1007
8. Gerard 't Hooft, "The Cellular Automaton Interpretation of Quantum Mechanics", Fundamental Theories of Physics Vol. 185, Springer (2016); preprint arXiv:1405.1548 [quant-ph]. DOI: 10.1007/978-3-319-41285-6, https://doi.org/10.1007/978-3-319-41285-6
9. Gerard 't Hooft, "Deterministic Quantum Mechanics: the Mathematical Equations", arXiv:2005.06374 [quant-ph] (2020), https://arxiv.org/abs/2005.06374 (primary category quant-ph, cross-listed hep-th)
10. Gerard 't Hooft, "Projecting local and global symmetries to the Planck scale", arXiv:2202.05367 [hep-th] (2022), https://arxiv.org/abs/2202.05367 (also published in Dialogues Between Physics and Mathematics, Springer (2022), pp. 215-226, DOI: 10.1007/978-3-031-17523-7_9, https://doi.org/10.1007/978-3-031-17523-7_9)
11. Gerard 't Hooft, "An ontological description for relativistic, massive bosons", arXiv:2306.09885 [quant-ph] (2023), https://arxiv.org/abs/2306.09885
12. Federico Centrone, Marco Barbieri, Alessio Serafini, "Quantum Coherence in Noisy Cellular Automata", arXiv:1801.02057 [quant-ph] (2018), https://arxiv.org/abs/1801.02057 (adjacent citing work)
13. K. V. Bayandin, "Causal discrete field theory for quantum gravity", arXiv:2001.10819 [gr-qc] (2020), https://arxiv.org/abs/2001.10819 (adjacent citing work)
14. Fabrizio Tamburini, Ignazio Licata, "Majorana Tower and Cellular Automaton Interpretation of Quantum Mechanics down to Planck Scales", arXiv:2111.01532 (2021); Theoretical and Mathematical Physics 214 (2023) 265-272. DOI: 10.1134/S0040577923020101, https://arxiv.org/abs/2111.01532, https://doi.org/10.1134/S0040577923020101 (adjacent citing work)
15. Gerard 't Hooft, "Beyond relativistic quantum string theory", Modern Physics Letters A 29 (2014) 1430030. DOI: 10.1142/S0217732314300304, https://doi.org/10.1142/S0217732314300304 (author's journal summary of the same Erice lecture)
16. Michael B. Green, John H. Schwarz, Edward Witten, "Superstring Theory", 2 volumes, Cambridge University Press (1987; 25th-anniversary reissue on Cambridge Core, 2012). Vol. 1 DOI: 10.1017/CBO9781139248563, https://doi.org/10.1017/CBO9781139248563; Vol. 2 DOI: 10.1017/CBO9781139248570, https://doi.org/10.1017/CBO9781139248570 (standard light-cone superstring conventions)
17. Joseph Polchinski, "String Theory", 2 volumes, Cambridge University Press (1998). Vol. 1 ("An Introduction to the Bosonic String") DOI: 10.1017/CBO9780511816079, https://doi.org/10.1017/CBO9780511816079; Vol. 2 ("Superstring Theory and Beyond") DOI: 10.1017/CBO9780511618123, https://doi.org/10.1017/CBO9780511618123 (standard superstring conventions)