Digital Physics

Note 1. John Archibald Wheeler, “Information, Physics, Quantum: The Search for Links,” in Complexity, Entropy, and the Physics of Information (1990). Wheeler’s “It from bit” became a rallying cry for information-theoretic approaches to physics.

Note 2. Jacob D. Bekenstein, “Black Holes and Entropy,” Physical Review D 7 (1973): 2333-2346. The Bekenstein bound limits information density in any region of space.

Note 3. Rolf Landauer, “Irreversibility and Heat Generation in the Computing Process,” IBM Journal of Research and Development 5 (1961): 183-191.

Note 3a. Vanchurin, Vitaly, “Towards a theory of quantum gravity from neural networks,” arXiv:2111.00903v3 (2022).

Note 3b. Alexander, Stephon, William J. Cunningham, Jaron Lanier, Lee Smolin, Stefan Stanojevic, Michael W. Toomey, and Dave Wecker, “The Autodidactic Universe,” arXiv:2104.03902 (2021).

Note 4. Gerard ’t Hooft, “Dimensional Reduction in Quantum Gravity” (1993); Leonard Susskind, “The World as a Hologram,” Journal of Mathematical Physics 36 (1995): 6377-6396.

Note 6. Stephen Wolfram, A New Kind of Science (2002). Wolfram’s computational physics programme has been developed further in his Physics Project (wolfram.com/physics).

Note 8. Giulio Tononi, “Integrated Information Theory of Consciousness: An Updated Account,” Archives Italiennes de Biologie 150 (2012): 293-329. For a more accessible introduction, see Tononi and Christof Koch, “Consciousness: Here, There and Everywhere?” Philosophical Transactions of the Royal Society B 370 (2015): 20140167.

Note 10. Maldacena, Juan, “The Large N Limit of Superconformal Field Theories and Supergravity,” Advances in Theoretical and Mathematical Physics 2 (1998): 231-252. The foundational paper of the Anti-de Sitter/Conformal Field Theory correspondence, which conjectures an exact equivalence between a theory of gravity in a curved higher-dimensional space and a quantum field theory on its boundary. The correspondence remains unproven in general, but it is the most rigorously studied realization of the holographic principle.

Note 11. Laplace, Pierre-Simon, A Philosophical Essay on Probabilities (1814). The source of the thought experiment now known as “Laplace’s Demon”: an intelligence that, knowing the position and momentum of every particle, could compute the entire past and future of the universe. Laplace offered it not as a claim about reality but as a statement of deterministic philosophy.

Note 12. Weberszpil, J. and Sotolongo-Costa, O., “Entropy as a Clock: Foundations and Parametrizations of Emergent Time,” International Journal of Theoretical Physics 64 (2025): 48. Unifies entanglement entropy growth, thermal modular flow, and the Page-Wootters relational time mechanism into a single entropic-time framework.

Note 16. Conway, John and Kochen, Simon, “The Free Will Theorem,” Foundations of Physics 36(10) (2006): 1441–1473. Strengthened in Conway, J. and Kochen, S., “The Strong Free Will Theorem,” Notices of the American Mathematical Society 56(2) (2009): 226–232. Proves that if experimenters’ choices of measurement settings are not determined by prior information (the “free will” assumption), then particles’ responses to those measurements are also not determined by any prior information. The theorem rests on the Kochen-Specker theorem and on Bell-type correlations, and the quantum predictions behind both have been experimentally confirmed.

Note 17. Jacobson, T. (1995). “Thermodynamics of spacetime: The Einstein equation of state.” Physical Review Letters, 75, 1260–1263. arXiv:gr-qc/9504004. Derives Einstein’s field equations from the proportionality of entropy to the area of local causal horizons, combined with the Clausius relation and the Unruh temperature.

Note 18. Jacobson, T. (2016). “Entanglement equilibrium and the Einstein equation.” Physical Review Letters, 116, 201101. arXiv:1505.04753. Updates the 1995 derivation using entanglement entropy rather than Clausius entropy, obtaining the same result (Einstein’s equations as an equation of state) from quantum-information-theoretic premises.

Note 19. Verlinde, E. (2011). “On the origin of gravity and the laws of Newton.” Journal of High Energy Physics, 2011:29. arXiv:1001.0785. Derives Newton’s laws of gravity from holographic screens and entropic forces, arguing that gravity is emergent, arising from changes in information associated with material bodies.

Note 20. Verlinde, E. (2017). “Emergent gravity and the dark universe.” SciPost Physics, 2(3), 016. arXiv:1611.02269. Extends entropic gravity to cosmological scales, interpreting the dark matter phenomenon as the elastic response of the entropy associated with dark energy.

Note 21. Zurek, W.H. (2009). “Quantum Darwinism.” Nature Physics, 5, 181–188. arXiv:0903.5082. Proposes that the classical world emerges through environmental selection: quantum states that can proliferate redundant copies of themselves in the environment survive decoherence, while fragile states are transformed. Redundancy is the mechanism of objectivity — multiple observers agree because they access independent fragments of the environment that carry the same information.

Note 22. Pikovski, I., Zych, M., Costa, F. & Brukner, Č. (2015). “Universal decoherence due to gravitational time dilation.” Nature Physics, 11, 668–672. arXiv:1311.1095. Demonstrates that gravitational time dilation universally decoheres composite quantum systems by entangling their center-of-mass position with internal degrees of freedom, producing classicality without any external environmental interaction.

Note 22b. Fields, Chris and Michael Levin, “Metabolic limits on classical information processing by biological cells,” Biosystems 209: 104513 (2021). doi:10.1016/j.biosystems.2021.104513.

Note 23. Brouwer, M.M. et al. (2017). “First test of Verlinde’s theory of emergent gravity using weak gravitational lensing measurements.” Monthly Notices of the Royal Astronomical Society, 466(3), 2547–2559. arXiv:1612.03034. Measured lensing profiles around 33,613 isolated central galaxies from the KiDS/GAMA survey overlap. Verlinde’s parameter-free prediction showed good agreement across four stellar mass bins. The authors cautioned this was “only a first step.”

Note 24. Yoon, Y. et al. (2023). “Understanding galaxy rotation curves with Verlinde’s emergent gravity.” Classical and Quantum Gravity, 40(2), 02LT01. arXiv:2206.11685. Analysis of 175 disk galaxies from the SPARC database found good agreement between emergent gravity predictions and observed radial accelerations: a mean logarithmic offset μ[log(g_obs) − log(g_Ver)] = −0.060 ± 0.004 dex (scatter 0.137) using Verlinde’s de Sitter acceleration scale a₀ = cH₀, improving to −0.027 dex (scatter 0.129) with the quasi de Sitter value a₀ = 5.41 × 10^-10 m s^-2. The authors note that a₀ remains an approximation in either case, and that a₀/6 computed with the quasi de Sitter value departs from Milgrom’s MOND constant by 30% (against 10% for the de Sitter value).

Note 25. Tamosiunas, A. et al. (2019). “Testing emergent gravity on galaxy cluster scales.” arXiv:1901.05505. Found that emergent gravity fits to X-ray and weak lensing data are significantly worse than general relativity plus cold dark matter (GR+CDM) for galaxy clusters, with mass predictions exceeding observations by roughly a factor of two at ~1 Mpc scales.

Note 27. Carney, D., Karydas, M., Scharnhorst, T., Singh, R., and Taylor, J.M., “On the quantum mechanics of entropic forces,” Physical Review X 15 (2025): 031038. arXiv:2502.17575. Two explicit microscopic models (a qubit lattice and a nonlocal qubit bath) in which gravitational-strength 1/r² attraction between massive objects arises from entropy maximization alone. Carney describes the models as ad hoc proofs of principle rather than realistic candidates. For criticism: Van Raamsdonk, M., quoted in Musser, G., “Is Gravity Just Entropy Rising? Long-Shot Idea Gets Another Look,” Quanta Magazine (June 2025), arguing the models lack distinctive gravitational features (equivalence principle, free-fall phenomenology). For Verlinde’s continued development of the programme, see notes 19 and 20.

Note 26. Oppenheim, J., “A postquantum theory of classical gravity?” Physical Review X 13 (2023): 041040. arXiv:2203.17155. Proposes a consistent framework in which gravity remains classical while coupling stochastically to quantum matter. The theory predicts a precise, testable trade-off: the diffusion (noise) in the gravitational field is bounded below by the decoherence it induces in quantum systems. See note 26b for the paper that derives the experimental signatures.

Note 26a. Feynman, Richard P., “The Role of Gravitation in Physics,” Chapel Hill Conference (1957); reprinted in Feynman Lectures on Gravitation, ed. Morinigo, F.B., Wagner, W.G., and Hatfield, B. (Addison-Wesley, 1995).

Note 26b. Oppenheim, J., Sparaciari, C., Šoda, B., and Weller-Davies, Z., “Gravitationally induced decoherence vs space-time diffusion: testing the quantum nature of gravity,” Nature Communications 14 (2023): 7910.

Note 28. Page, D.N. and Wootters, W.K., “Evolution without evolution: Dynamics described by stationary observables,” Physical Review D 27 (1983): 2885–2892. The foundational paper proposing that time emerges from quantum entanglement between subsystems of a static universe.

Note 29. Foti, C., Coppo, A., Barni, G., Cuccoli, A. and Verrucchi, P., “Time and classical equations of motion from quantum entanglement via the Page and Wootters mechanism with generalized coherent states,” Nature Communications 12 (2021): 1787. Derives both the Schrödinger equation and Hamilton’s classical equations of motion purely from entanglement between a system and a quantum clock.

Note 30. Coppo, A., Cuccoli, A. and Verrucchi, P., “A magnetic clock for a harmonic oscillator,” Physical Review A 109 (2024): 052212. Extends the Page-Wootters framework to show that classical phase-space trajectories emerge naturally when the clock is macroscopic.

Note 31. Pearson, A.N. et al., “Measuring the Thermodynamic Cost of Timekeeping,” Physical Review X 11 (2021): 021029. Using a silicon nitride membrane as a mesoscopic clock, demonstrated that the entropy cost of timekeeping scales linearly with clock accuracy.

Note 32. Wadhia, V. et al., “Entropic Costs of Extracting Classical Ticks from a Quantum Clock,” Physical Review Letters 135 (2025): 200407. Double quantum dot experiment showing that reading the clock costs up to a billion times more energy than the clock’s internal ticking mechanism.

Note 34. Julian Barbour, Tim Koslowski, and Flavio Mercati, “Identification of a gravitational arrow of time,” Physical Review Letters 113 (2014): 181101. See also Barbour’s The Janus Point: A New Theory of Time (2020) for the full development of the argument.

Note 35. Wolpert, David, Rovelli, Carlo, and Scharnhorst, Stefan, “Disentangling the Boltzmann Brain Hypothesis and the Second Law,” Entropy 27(12) (December 2025): 1227. Published via MDPI; Wolpert affiliated with the Santa Fe Institute. Demonstrates that standard arguments connecting the past hypothesis, the second law, and Boltzmann brains involve subtle circular reasoning: which moment is treated as “fixed” determines whether entropy increases or decreases. The result does not overturn the second law (which remains observationally ironclad) but exposes the logical gap in standard derivations of time’s arrow from initial conditions.

Note 36. Weinberg, S. (1964). “Photons and Gravitons in S-Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass.” Physical Review 135, B1049–B1056. The foundational bootstrap derivation showing that a massless spin-2 particle must couple universally. Universal coupling yields the equivalence principle, and with it general relativity, from self-consistency alone. See also Weinberg, S. (1965). “Photons and Gravitons in Perturbation Theory: Derivation of Maxwell’s and Einstein’s Equations.” Physical Review 138, B988–B1002, for the extension to graviton self-interactions.

Note 37. Rodina, L. (2016). “Uniqueness from gauge invariance and the Adler zero.” arXiv:1612.06342. Modernized and generalized Weinberg’s bootstrap proof, showing that locality, unitarity, and Lorentz invariance uniquely fix the graviton’s interactions. Rodina quote from Natalie Wolchover, “Why the Laws of Physics Are Inevitable,” Quanta Magazine, December 9, 2019.

Note 38. Baumann, D., quoted in Wolchover (2019). See also Baumann, D. et al. (2020). “The Cosmological Bootstrap: Weight-Shifting Operators and Scalar Seeds.” Journal of High Energy Physics 2020:204. arXiv:1910.14051. Applies bootstrap methods to constrain the physics of the very early universe, demonstrating that symmetry and consistency conditions strongly limit what could have happened in the first moments after the Big Bang.

Note 39. Gisin, N. (2020). “Mathematical languages shape our understanding of time in physics.” Nature Physics 16, 114–116. Argues that intuitionist mathematics, in which real numbers are finite unfolding processes rather than completed infinite objects, naturally expresses the passage of time and the creation of new information. See also Gisin, N. (2019). “Indeterminism in Physics, Classical Chaos and Bohmian Mechanics: Are Real Numbers Really Real?” Erkenntnis 86, 1469–1481. arXiv:1803.06824. Gisin quotes from Natalie Wolchover, “Does Time Really Flow? New Clues Come From a Century-Old Approach to Math,” Quanta Magazine, April 7, 2020.

Note 39a. Vonnegut, K., Slaughterhouse-Five, or The Children’s Crusade (New York: Delacorte, 1969). “Millepedes” is Vonnegut’s spelling. The Tralfamadorian account of death (“So it goes”) and the comparison of seeing all time to seeing “a stretch of the Rocky Mountains” come from the same book. Wording checked against published transcriptions; page numbers vary by edition.

Note 39b. Feynman, R.P. (1949). “The Theory of Positrons.” Physical Review 76, 749–759. DOI: 10.1103/PhysRev.76.749. The same worldline reinterpretation was proposed independently by Ernst Stueckelberg (1941), which is why it is often called the Feynman–Stueckelberg interpretation.

Note 39c. Feynman, R.P. (1965). “The Development of the Space-Time View of Quantum Electrodynamics.” Nobel Lecture, December 11, 1965. Feynman recalls Wheeler’s call (“Feynman, I know why all electrons have the same charge and the same mass.” “Why?” “Because, they are all the same electron!”) and adds that he took the observation that positrons could be represented as electrons going from the future to the past more seriously than the single-electron idea itself.

Note 39d. Ellis, G.F.R. (2006). “Physics in the real universe: time and spacetime.” General Relativity and Gravitation 38, 1797–1824. arXiv:gr-qc/0605049. Quoted from the abstract. Ellis ties the growth of the block to the emergence of complex systems, including life, and to proper time along families of world lines rather than to any preferred surface of simultaneity. Developed further in Ellis, G.F.R. and Rothman, T. (2010). “Time and spacetime: the crystallizing block universe.” International Journal of Theoretical Physics 49, 988–1003.

Note 39e. Boethius, The Consolation of Philosophy (c. 524), Book V, prose 6. Boethius defines eternity as the complete, simultaneous, and perfect possession of unending life, and argues that God’s foreknowledge is better understood as knowledge of a present that never passes, which sees events without compelling them, as our seeing a man walk does not make him walk.

Note 39f. Hartshorne, C., Omnipotence and Other Theological Mistakes (Albany: State University of New York Press, 1984). Hartshorne lists the belief that omniscience includes complete knowledge of the future among the “mistakes”: a perfect knower knows the actual as actual and the possible as possible, and a future that is partly indeterminate is known as partly indeterminate. The position develops A.N. Whitehead’s Process and Reality (1929). Later “open theist” writers reached a similar view of foreknowledge from within Christian theology.

Note 40. Del Santo, F. and Gisin, N. (2019). “Physics without determinism: Alternative interpretations of classical physics.” Physical Review A 100, 062107. arXiv:1909.03697. Reformulates classical mechanics using intuitionist mathematics, obtaining the same predictions as standard equations while casting events as genuinely indeterminate. The reformulation bridges the conceptual gap between classical determinism and quantum randomness.