Abstract
A deterministic reformulation of quantum mechanics is thought to be able to bypass the usual philosophical interpretations of probability and stochasticity of the standard quantum mechanical scenarios. Recently ’t Hooft proposed a different perspective based on the ontological formulation of quantum mechanics, obtained by writing the Hamiltonian of a quantum system in a way to render it mathematically equivalent to a deterministic system. The ontological deterministic models consist of elementary cells, also called cellular automata, inside which the quantities describing the dynamics oscillate in periodic orbits, extending and replacing the quantum mechanical classical language based on harmonic oscillators. We show that the structure of the cellular automaton sets finds a clear physical interpretation with the Majorana infinite-component equation: the cellular automata are elementary building blocks generated by the Poincaré group of spacetime transformations with positive-definite energy down to the Planck scales, with a close relation to the Riemann Hypothesis.
This is a preview of subscription content, log in via an institution to check access.
References
P. Marage and G. Wallenborn, “The Debate between Einstein and Bohr, or how to interpret quantum mechanics,” in: The Solvay Councils and the Birth of Modern Physics (Science Networks. Historical Studies, Vol. 22, P. Marage and G. Wallenborn, eds.), Birkhäuser, Basel (1999), pp. 161–174.
J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Phys. Phys. Fiz., 1, 195–200 (1964).
J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Collected Papers on Quantum Philosophy), Cambridge Univ. Press, Cambridge (2004).
M. Bell and S. Gao (eds.), Quantum Nonlocality and Reality: 50 Years of Bell’s Theorem, Cambridge Univ. Press, Cambridge (2016).
V. Scarani, Bell Nonlocality, Oxford Univ. Press, Oxford (2019).
D. Bohm, “A suggested interpretation of the quantum theory in terms of “hidden” variables. I,” Phys. Rev., 85, 166–179 (1952).
G. ’t Hooft, “Deterministic quantum mechanics: The mathematical equations,” (Submitted to the Article Collection: “Towards a Local Realist View of the Quantum Phenomenon”), Front. Phys. Physics, 8, 253, 12 pp. (2020).
I. Licata, “Quantum mechanics interpretation on Planck scale,” Ukr. J. Phys., 65, 17–30 (2020).
P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford Univ. Press, Oxford (1947).
S. Hossenfelder, “Testing super-deterministic hidden variables theories,” Found. Phys., 41, 1521–1531 (2011), ar**v: 1105.4326; “Testing superdeterministic conspiracy,” J. Phys.: Conf. Ser., 504, 012018, 5 pp. (2014).
S. Hossenfelder and T. Palmer, “Rethinking Superdeterminism,” Front. Phys., 8, 139, 13 pp. (2020).
R. A. Bertlmann and A. Zeilinger (eds.), Quantum (Un)speakables: From Bell to Quantum Information, Springer, Berlin (2002); Quantum (Un)speakables II. Half a Century of Bell’s Theorem, Springer, Cham (2017).
A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?,” Phys. Rev., 47, 777–780 (1935).
G. ’t Hooft, “Explicit construction of Local Hidden Variables for any quantum theory up to any desired accuracy,” ar**v: 2103.04335.
G. ’t Hooft, The Cellular Automaton Interpretation of Quantum Mechanics (Fundamental Theories of Physics, Vol. 185), Springer, Cham (2016).
D. Schumayer and D. A. W. Hutchinson, “Colloquium: Physics of the Riemann hypothesis,” Rev. Modern Phys., 83, 307–330 (2011); Erratum, 769–769.
E. Majorana, “Teoria relativistica di particelle con momentum internisico arbitrario,” Nuovo Cimento, 9, 335–344 (1932).
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 2: The Classical Theory of Fields, Pergamon Press, Oxford–London (1962).
F. Tamburini, I. Licata, and B. Thidé, “Relativistic Heisenberg principle for vortices of light from Planck to Hubble scales,” Phys. Rev. Res., 2, 033343, 5 pp. (2020).
F. Tamburini and I. Licata, “Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis,” Phys. Scr., 96, 125276, 26 pp. (2021).
M. V. Berry and J. P. Keating, “The Riemann zeros and eigenvalues asymptotics,” SIAM Rev., 41, 236–266 (1999).
M. V. Berry and J. P. Keating, “\(H=xp\) and the Riemann zeros,” in: Supersymmetry and Trace Formulae: Chaos and Disorder (Cambridge, UK, September 8–20, 1997, NATO Science Series B, Vol. 370, I. V. Lerner, J. P. Keating, and D. E. Khmelnitskii, eds.), Kluwer Academic Press, New York (1999), pp. 355–367.
A. Connes, “Trace formula in noncommutative geometry and the zeros of the Riemann zeta function,” Selecta Math. (N.S.), 5, 29–106 (1999).
A. Strumia, “Interpretation of quantum mechanics with indefinite norm,” Physics, 1, 17–32 (2019).
C. M. Bender, D. C. Brody, and M. P. Müller, “Hamiltonian for the zeros of the Riemann zeta function,” Phys. Rev. Lett., 118, 130201, 5 pp. (2017); ar**v: 1608.03679.
R. de la Madrid, “The role of the rigged Hilbert space in quantum mechanics,” Eur. J. Phys., 28, 287–312 (2005); ar**v: quant-ph/0502053.
F. Tamburini and I. Licata, “General relativistic wormhole connections from Planck-scales and the ER=EPR conjecture,” Entropy, 22, 3, 14 pp. (2020).
I. Licata, “The big computer. Complexity and computability in physical universe,” in: Determinism, Holism, and Complexity (V. Benci, P. Cerrai, P. Freguglia, G. Israel, and C. Pellegrini, eds.), Springer, New York (2003), pp. 117–123.
G. Preparata and S.-S. Xue, “The standard model on Planck lattice: Mixing angles vs quark masses,” Nuovo Cimento A, 109, 1455–1460 (1996).
G. Preparata and S.-S. Xue, “Do we live on a lattice? Fermion masses from the Planck mass,” Phys. Lett. B, 264, 35–38 (1991).
G. Preparata, “Quantum gravity, the Planck lattice and the Standard Model,” (Proceedings of the VII Marcel Grossman Meeting on General Relativity, Stanford, July 24–30, 1994, R. Ruffini, G. Mac Keiser, and R. T. Jantzen, eds.), World Sci., Singapore (1997), pp. 102–115; ar**v: hep-th/9503102.
J. Magueijo, A Brilliant Darkness: The Extraordinary Life and Mysterious Disappearance of Ettore Majorana, the Troubled Genius of the Nuclear Age, Basic Books, New York (2009).
R. Casalbuoni, “Majorana and the infinite component wave equations,” PoS (EMC2006), 37, 004, 15 pp. (2007); ar**v: hep-th/0610252.
S. W. Hawking, M. J. Perry, and A. Strominger, “Soft hair on black holes,” Phys. Rev. Lett., 116, 231301, 9 pp. (2016).
S. W. Hawking, “Black hole explosions?,” Nature, 248, 30–31 (1974).
F. Tamburini, M. D. Laurentis, I. Licata, B. Thidé, “Twisted soft photon hair implants on black holes,” Entropy, 19, 458, 9 pp. (2017).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The authors declare no conflicts of interest.
Additional information
Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 214, pp. 308–317 https://doi.org/10.4213/tmf10375.
Rights and permissions
About this article
Cite this article
Tamburini, F., Licata, I. Majorana tower and cellular automaton interpretation of quantum mechanics down to Planck scales. Theor Math Phys 214, 265–272 (2023). https://doi.org/10.1134/S0040577923020101
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577923020101